Sk Goyal maths book

MATHEMATICS

J . H . VJ W

I ' AL.

Director Omega Classes, Meerut

ARIHANT PRAKASHAN ARIHANT

KALINDI, T.P. NAQAR, MEERUT-250 002

\ MATHEMATICS GALAXY Q

S.K. Goyal

A lextg W

S.K. Goyal

Co-ordinate Geometry

Q

IxtvuitUue. Algebra (Vol. I & II)

Q &

A lea SW

& &

a

/ } lea gW ^ Algebra

Calculus (Differential)

Amit M. Agarwal Amit M. Agarwal

Play with Graphs

Amit M. Agarwal

/f lextg W

Amit M. Agarwal

Trigonometry

Problems in Mathematics

S.K. Goyal

Objective Mathematics

S.K. Goyal

AR IHA NT P R AK AS HA N An ISO 9001:2000 Organisation Kalindi, Transport Nagar Baghpat Road, MEERUT-250 002 (U.P.)

© Author All the rights reserved . No part of this publication may be reproduced, sto red in a retrieval system transmitted in any form or by any means- electronic, mechanical, photocopying recording or otherwise, without prior permission of the author and publisher.

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Amit M. Agarwal

A leag W Calculus (Integral) A lea Book 4 Vector & 3D Geometry

Tel.: (0121) 2401479, 2512970, 2402029 Fax:(0121)2401648 email: [email protected] on web: www.arihantbooks.com

if.

S.K. Goyal

ISBN 81 83 480 14 4

W

Price : Rs. 200 .0 0

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Laser Typesetting at: Vibgyor Computer

Printed at: Sanjay Printers

PREFACE This new venture is intended for recently introduced Screening Test in new system of Entrance Examination of IIT-JEE. This is the first book of its kind for this new set up. It is in continuation of my earlier book "Problems in Mathematics" catering to the needs of students for the main examination of IIT-JEE. Major changes have b een e ffe ct ed in the set up the book in the edition. The book has been aivided into 33 chapters. In each chapter, first of all the theory in brief but having all the basic concepts/formulae is given to make the student refresh his memory and also for clear understanding, Each chapt er has both set of multiple cho ic e questions- having one correct alternative, and one or more than one correct alternatives. At the en d of e ach chapter a practice test is provided for the student to assess his relative ability on the chapter. Hints & Solutions of selected questions have been provided in the end of book, J •
\

CONTENTS Complex Numbers Theory of Equations

1 14

5.

Sequences & Series Permutations & Combinations Binomial Theorem

25 36 47

6. 7.

Determinants Probability

8.

Logarithms and Their Properties Matrices Functions

1. 2. 3. 4.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20 .

Limits Continuity & Differentiability Differential Coefficient Tangent and No rm al Monotonicity Maxima & Mini ma Indefinite Integration

57 65 76 79 86 97 104 112 119 124 128 134 141

21.

Definite Integrat ion Area Different ial Equations Straight Lines

22 . 23.

Pair of Straight Lines Circle

177 182

24.

Conic Section-Pa rabola Ellipse Hyperbola

25. 26. 27. 28. 29 . 30 .

Trigonometrical Ratios & Identities Trigonometric Equations Inverse Circ ular Functio ns Properties of Triangles

31 . Heights & Distanc es 32. Vectors 33.

Co-ordinate Geometry-3D HINTS & SOLUTIONS

153 158 164

197 208 220 231 241 247 254 265 271 285 297

COMPLEX NUMBERS i (a + ib) < (or>) c+ id is not defined. For example, the statement 9 + 6/< 2 - /'makes no sense. : (i) Complex numbers with imaginary parts zero are said to be similarly those with real pa rt s ze ro are sa id to be v i = V^T is called the imaginary unit.

(ii) io ta :

/2 = -1, /

Also

4n

3

= - /, /

4

= 1, etc.

/ = 1, / = /, / 4n + 2 = - 1 I n+3 _ _ j f or ar| y j n t e g e r n

In gene ral For exam ple,

4n+1

; 1997 = / 4x 499 + 1 = /

Also, 2. 3.

/ = ——, i A complex numb er z is said to be purely real If lm (z) = 0 an d is sai d t o b e purely imag inary if Re (z) = 0. The complex number 0 = 0 + /'. 0 is both purely real and purely imaginary. Th e sum of four con sec uti ve po wer s of / is zero .

4/JJ-7 4.

in+ 7

n=1 n=4 To find digit in th e unit's pl ace, this me tho d is clea r from following ex amp le : What is the digit in the unit's place of (143)

86

?

The digits in unit's place of different powers of 3 are as follows : 3, 9, 7, 1, 3, 9, 7, 1 (period bei ng 4) remainder in 86 + 4 = 2

5.

So the digit in the unit's place of (143) 86 = 9 [Second term in the sequence of 3, 9, 7,1,...] V^ a = /'Va, whe n 'si is any real numbe r. Keep ing this result in mind th e following compu tation is correct. V- a V- b - i^fa i^fb = F ~{ab =-^fab But th the computation V- a V- b = >/{(- a) ( - b)} = 'lab is wrong. Because the property Va -ib = Jab hold good only if , at leas t on e of VJ o r Vb is real. I t do es not hold good if a, b are negative numbers, i.e., Va, VFare imaginary numbers.

The complex number z = (a, b) = a + ib and z = (a, - b) = a - ib, where a and are real numbers, and b * 0 are sa id to be complex co nju gat e of each other. (Here the complex con jug ate is obtained changing the sign of /). : z is the mirror image of z along real axis. 0) ( ¥ ) = *

/' = V- 1 just

2


(ii) z = z <=> z is purely real (iii) z = - z<=> z i s pur ely imag ina ry (iv) R e (z ) =R e (z ) = (v) Im (z) = (vi) Z1

z- z 2/

+ Z2 = z

1

+Z2

(vii)zi - Z 2 = Z1 - z

2

(viii) (viii) Z1 Z2 = z 1 . z 2

(ix)

Z1

Z1 Z2

Z2

(x) Z1Z2 + Z1Z2 = 2 Re (z1 z 2 ) = 2Re (ziz 2 ) (xi) = (z)

n

(xii) Ifz— /( zi ) then z = / (z i)

z=a+ib,a,be If Ft, then arg z = tan ~ quadrant in which the point (a, b) lies :

1

( b / a ) alwa ys giv es the principal valu e. It de pe nd s on t he

(i) (a, b) e first qua dran t a > 0, b> 0, the principal value = arg z = 6 = ta n" (ii) (a, b) e sec ond quadr ant a < 0,

1

^

b > 0, the principal value

= arg z = 9 = 7t - tan

-1

^ a (iii) (a, b) e third qua dra nt a< 0, b< 0, the principal value ar g z = 9 = - n + tan -1 (iv) (a, b) e fourth qua dra nt a > 0, b < 0, the principal value = ar g z = 9 =

(i) (ii) (iii) (iv)

tan

-7i < 9 < n amp litu de of th e com ple x numb er 0 is not defin ed If zi = Z2 <=> I zi I = I Z2 I and amp zi = amp Z2 . If ar g z=n/2 o r - r c / 2 , is purely imaginar y; if arg z = 0 or±rc , zi s purely real.

If zi , Z2, Z3 be th e affi xes of the vert ices of a triangle described in counter-clockwise sense (Fig. then : (-Z1~ Z2) j a = (Zi - Z 3 ) I Z1 - Z3 I I Z1 - Z2 I Z1 -Z3 = a = Z BAC amp Z1 -Z2

ABC

0

or

Not e th at if a = |

o r t h e n

A( z t)

B(Z2)

Complex Numbers

3 Z\- Z3 .

—

. .

is p ure ly imagin ary .

: Here only principal values of arguments are considered.

(i) I z I > 0 => I zl = 0 iff z = 0 an d I z I > 0 iff z * 0. (ii) - I z I < Re (z) < I z I and - I z I < Im (z) < I z I (iii) IzI

= Iz

l =l-zl = l - z I 2

(V) (iv) IzzZ1Z2= I I=z Il Z1 I I Z2 I (vi)

In gen er al I zi Z2 Z3 Z4 .... z n I = I zi 11 Z2 11 Z3 I ... I zn I I Z1 I Z2 I

(vii) I Z1 + Z2 I II zi I - I z 2 II

+z

n

I < I zi I +1 Z2 I +1 23 I +

+1 zn I

(ix) I z" I = Iz l" (X) I I Z1 I - I Z 2 I | < I Z1 + Z2 I < I Z1 I + I Z2 I

Thu s I zi I + I Z2 I is the gr ea tes t pos sibl e v alue of I zi I + I Z2 I and I I zi I - I Z2 I I is t he leas t po ss ib le va lu e of I zi + 22 I. (xi) I zi + z 2 I2 = (zi ± z 2 ) (ii + z 2 ) = I zi I2 +1 z 2 I2 ± (Z1Z2 + Z1Z2) (xii) Z1Z2 + Z1Z2 = 2 I zi I I Z2I c os (81 -02) where 0i = arg (zi) and 9= arg

(Z2).

(xiii) I zi + z 2 I2 + I zi - z 2 I2 = 2 j I zi I 2 + I z 2 I 2 } (xiv) i.e., unit modulus If zis unimodular then I zl = 1. In ca se of unimodula r let z = co s 0 + /'sin 9, 9 <= R. :

is always a unimodular complex numbe r if z * 0.

§ 1.6. (i)

Arg (zi zz) = Arg (zi) + Arg (z 2 ) In general Arg (Z1Z2Z3 zn) = Arg (zi) + Arg (zz) + Arg{Z3)+ ... + Arg (z„) /

(ii) Arg Z1

Z2

(iii) Arg I-

z

\

= Arg (zi ) - Arg (z 2 )

|= 2A rg z

(iv) Arg (z") = n Arg (z) ( v ) l f A r g ^ j=9, th en Arg ^

2kn - 9 where ke I.

=

(vi) Arg z = - Arg z § 1.7. 2

Unity has n roots viz. 1, co , co , to these n roots is zero.

3

-1

ro"

which are in G.P., and about which we find. The sum of

(a) He re 1 + to + co2 + ... + co"~ 1 = 0 is the basic concept to be understood. (b) The product of these

n roots i s ( - 1)" ~ 1 .

§ 1.8. (a) If n is a positive or negative integer, then

9 Objective Mathematics

(c os 6 + /' sin 9) " = co s n8 + / sin n9 (b) If n is a positive integer, then 1/

(cos 9 + /'sin 9)

"= cos

2/ot + 9 n

+ / sin

2/ot + 9

n

whe.e k =0, 1, 2, 3 (n-1) The value corresponding to k = 0 is called the principal value. Demoivre's theorem is valid if n is any rational number.

a + ib are :

The square roots of z =

h i I- a 2

Izl + a 2 z I + a

and

I*

4

1. The square root of

i are : ±

2. The square root of 3. The square root of

^

/' are : ±

+ a

for b > 0

for b < 0.

J (Here 6= 1 ). 1 -/"

(Here £> = - 1).

co are : ± isf

4. The square root of co

2

are : ± co

§ 1.10. Cube roots of unity are

(1)

1 + o) + co2 = 0

(2)

co 2 =1

(3)

co 3 n =1,co

1 co, co2

3

co

= co

= co.

2

(4) co = o> and (co)2 = co (5) A comple x numbe r a + ib, for which I a : £> I = 1 : V3" or V3~: 1, can alw ays be ex pr es se d in te rm s of /', co or co2. (6) Th e cu be r oots of unity when r epr ese nte d on complex plane lie on verti ces of an equilater al inscribed in a unit circle, having centre srcin. One vertex being on positive real axis. (7) a + £xo + cco2 = 0 => a = b = b = c if a, b, c are real. (8) com = co3, co = e 2n' /3, co = e~ 2 , c ' / 3 § 1.11. {i) If zi an d Z2 ar e two comp lex nu mbe rs, t hen the di sta nce b etw ee n zi a nd Z2 is I zi - Z2 I. (ii) Se gm en t joining points A (zi) and B (Z2) is divided by point P (z) in th e ratio mi : m2 m-\Z2+ IV2Z1 , , z ' , mi an d mz are real. then (mi + m2) (iii) Th e equ at io n of th e line joining zi an d Z2 is given by z

z

1

Z1

Z1

1

zz

z2

1

= 0 (non parametr ic form)

Complex Numbers

5

(iv) Thr ee points z\ , Z2 and Z3 are collinear if

(v) a z + a z

Z1

z\

1

Z2

Z2

1

Z3

Z3

1

=0

= real des cri bes equation of a straight line.

: The comple x and real slope s of the line a z + az+ b= 0 are (b s Ft) • — and a

(a)

. ^ ar e respec tively. Im (a) ' If ai an d 02 ar e complex sl op es of two lines on th e Argand pl ane th en * If lines ar e perpe ndic ular th en a i +
(vi) I z - zo I = r is equa tion of a circle, who se ce ntr e is Zo and radi us is r and I z - zo I < r represents interior of a circle I z - zo I = rand I z - zo I > r represents th e exter ior of th e circl e I z - zo I = r. (vii)

z z + az + a z + k = 0 ; (k is real) repr es ent circle with ce ntr e - a an d ra dius ^l a I

(viii) ( z - z i ) (z - Z2) + (z - Z2) (z - z 1) =0 A (zf") and B (z 2 ).

is

equat ion

of

circle

with

2

- k.

dia met er

AB

where

(ix) If I z - zi I + I z - Z2 I = 2 a w he re 2 a > I zi - Z2 I the n point z de sc ri be s a n el lips e havi ng foci at zi and Z2 and ae (x)

zi and Z2 an dae (xi)

Ft.

If I z - zi I - I z - Z2 I = 2a whe re 2a < I Zi - Z2 I the n point z de sc ri be s a hype rbol a having foci at

Ft.

Equat ion of all circle which are ort hogon al to I z - z\ I = n and I z - Z2 I = Let th e circle b e I z - a I = rcut given circles orthogonally a - zi I r 2 + n2

...(1)

and

•••(2)

z2

+ /f

I a - Z2 I

On solving r£ - r? = a (z 1 - z 2) + a (zi - £2) + I Z2 I2 - I zi I 2 and let a = a + ib. z - Zi Z-Z2

(xii)

= k is a circle if k * 1, an d is a line if k = 1 .

(xiii) Th e equa tio n I z - z\ I2 +1 z - Z2 I 2 = k , will represent a circle if k > ~ I zi - Z2 I2 (Z 2 - Z 3 ) (Z1 - Z4)

(xiv) If Arg

= ± 71, 0, then points zi , Z2 , 23 , Z4 ar e concy clic .

(Z1 - Z3 ) (Z2 - Z4) \

/

(i) lota (/) is neither 0, nore greater than 0, nor less than 0. (ii) Amp z - Amp ( - z) = ± n acc ordin g a s Amp (z ) is positive or ne gati ve. (iii) Th e tri angl e w ho se ver tic es a re zi , Z2 , Z3 is equila tera l iff 1 Z1 - Z2

or ( iv )

If

1 z

+

a+na

2

1 Z2 - Z

1 3

•=0

Z 3 - Z1

Z12 + Z | + Z3 2 = Z1Z2 + Z2Z3 + Z3Z1 =

+ 4)

a + \(,a

+ 4)

and

6


Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. One of the values of (a) e

_7 C/ 2

i' i s (i = V- 1) 2

(b)^

(c) ^

(d ) e *

2. If xr = cos (7t/3 r ) - i sin (n/3r), then valu e of

xx. x2....

(b)-l

= 1 and arg

Zl - 22

then Z]/Z 2 is always (a) zero

(b) I z r (d) None of these

2

Z\ — 6 + i and z 2 = 4 - 3 i. Let z be

complex number such that

Zi - z

(c) I z — (5 + i) I = 5 (d) I z - (5 + /) I = VT 5. The numbe r of solutions of the equation

z + I z I2= 0,

whe re z e C is (b) two (d) infinite ly many

6. If z = (k + 3) + iV(5-^

zis (a) a stra ight line (c) an ellips e

2

) ; then the locus o f

(b) a circ le (d) a parab ola

7. The locus of z which sotisfies the inequality logQ.31 z - 1 I > log 0 . 3 1 z - /1 is given by (a) * + y < 0

(b )x + y > 0

(c)x-y>0 8. If Zi an d z

t-y <0 (d)j are any two complex numbers,

2

then Iz, + Vz

(where

[.]

=0

denotes

the

greatest

integer

(a) 0

(b) 10

(c) 100

(d) not def ine d

11. The centre of square ABCD is at z = 0. A i s Zi. Then the centroid of triangle ABC is (a) Z) (cos n ± i sin 7t) ZI

= — ; then z satisfies

(a) I z - ( 5 - 0 1 = 5 (b )l z- (5 -/ )l = V5

(a) one (c) three

[arg I z I] dx is : E

(b) — (cos n ± i sin n)

Z-Zi

arg

10- If z * 0, then J

function)

2

(c)|lzl

^ mn (m e 7)

100

- z, tz, z - iz, is (a)itzl

21-22

^ Zl + 22

(d ) i

3. The area of the triangle on the Argand plane formed by the complex numbers

4. Let

2

(b) a rational number (c) a positive real number (d) a purely imaginary number

is

(a)l (c) - i

z,+z

(c) Z\ (cos rc/2 ± i sin 7C/2) Zl

(d) — (cos tc/2 ± i sin

n/2)

12. The point of intersection of the curves arg ( Z - 3i) = 371/4 and a rg (2z + 1 - 2i) = n / 4 is (a) 1 / 4 ( 3 + 9i) (b) 1 / 4 (3 - 9;") (c) 1 / 2 ( 3 + 2/) (d) no point 13. If S (n) = i n + i where i = V- 1 and n is an integer, then the total number of distinct values of 5 (n) is (a) 1 (c)3

(b)2 (d)4

14. The smallest positive integer n for which 1+ i *

1- i (a) 1 (c )3

= - 1, is

/

(b) 2 (d) 4

15. Conside r the follow ing statements : 2-z\

(a) Zi II (c) IZ) + z 2 1

I + I Zi - V z 2 - z \ I is equal to (b )l z

2l

(d) No ne of these

9. If Z\ and z 2 are complex numbers satisfying

S, : - 8 = 2i x4 i = V(- 4) x V( - 16 )

S2 :

4) x V( - 16) =

4) x ( - 16)

5 3 : V (- 4) x ( - 16) = V64" Sa : V64 = 8

Complex Numbers

7

incorrect one is

of these statements, the

a + [ko + ya>2 + 5co2 ^

(b) S2 only

(a) Sj only (c) Sonly 3

(d) No ne of thes e

17. If

(b) 4 +1 a/3~ (d) 4 — / a/3"

regular polygon of n sides whose centre is . Im (z,) srcin and if 1, then n is equal Re (z,) to

19. For x an d

(b )z

x2, )>i,y2 e R.

Zi =X i+ n' i

z2=x2

(c) 1 , 0 (d) - 1, 1 25. If 1, co and co2 are the three cube roots of unity,

then

the

roots

of

the

equation

(x - l ) 3 - 8 = 0 are

2 ,...,

co""

1

ar e n, nth roots of unity.

(a)n

and

(b)0

9" - 1

(c)

(d)

9"+l

2 l = lz 3 l

If 8/z 3 + 12z2 - 18z + 21 i = 0 then (a) I z=I 3 / 2 (b) I z I = 2 /3 (c) I z I = 1 (d) I z I = 3/4

(d) I z, I then

1+z

= 0

l -z -i

(a)e"

the

If

r sin 8

2x - 2x + x + 3 equals (a) 3 - (7/2 ) (b) 3 + (/ /2 ) (d) (3 - 0 / 2

23 . If 1. co. co2 are the three cube roots of unity then fo r a . (3. y. 5 e R. the expression

(d) re~rcos

9

z,, z 2 , Z3

are three distinct complex a, b, c are three positive real numbers such that a b c numbers and

I Z2 - ZL I

a

expression

- r sin 0

(b ) re~

r cos 9

I Z3 - Zl I I Zl -

2

(Z2-Z3)

Z+ 1

2

(c) (3 + 0 / 2

If z = re'B, then I e'z I is equal to

(c)e

If the following regions in the complex pl an e, the onl y on e tha t do es not represen t a circle, is

22. If

(b) 1, 1

20. Of z is any non-zero complex number then arg (z) + arg (z ) is equal to

(c) arg

are

+ iy2

(b) I z , I I z 2 1 > I z 3 1

21

and

« -1 the va lu e of (9 - co) (9 - co ) .. . (9 - co will be

Z3 = ^ (Zi + z2 ), then ZL, Z2 and z 3 satisfy (a)lz,l = lz

(a) 0 , 1

2 6 . If l, o), CO

If 0 ;

A and B

then,

(c) 3, 1 + 2co, 1 + 2co2 (d) None of these

(d) No ne of these b

(1 + co)7 = A + B(i>

(b) 3, 2co, 2co2

18. If lz l= l, th en f- |- 7= le qua ls 1+z (c) ' z

24. If co is a com pl ex c ube root of unity

(a) - 1, - 1 - 2co, - 1 + 2co2

(b) 16 (d) 32

(a)Z

-1 (d) CD"

CO

(c) -

respectively equal to

an dz i represe nt adja cent vertices of a

(a) 8 (c) 24

(b)co

(a) 1

16. If the multiplicative inverse of a complex num ber is (V3~+ 4i' )/1 9, the n the com ple x number itself is (a) V3~ -4i (c) V 3 + 4i

is equal to

P + aco 2 + yto + 8co

, then

Z2

1

2

(Z3-Z1)

(Z]-Z 2)

(a) 0

(b) abc

(c) 3 ab c

(d ) a + b + c

30. For all comple x num bers z i, z

2

satisfying

I Z! I = 12 and I z 2 - 3 - 4/1 = 5. the mi ni mu m value of I Z\ - Z 21 is (a )0

(b) 2

(c )7

(d) 17

8


31. If z\,z

2 ,zi

triangle

are the vertices of an equilateral in

the

(z? + zl + zj)=k (a) A: = 1 (c)J t = 3

argand

plane,

39. If

then

(z,z 2 + z2z3 + Z3Z1) is true for

r

vert ices

of

a

triangle which is: (b) right angled isoscele s (d) obtuse angled isosceles

(b) (1 /2 , - VT/2 ) (d) (0, - 3)

a, and (3 y

equals

(a)-

root s

of of

(b) co2

co

(d) 3C07

(a) a circle and a line (b) a radius of a circle (c) a sector of a circle (d) an infinite part line 42. If

the

point s

r epresented

by

comple x

numbers Zj = a + ib, z 2 = c + id and Z\ - Z2 are

(a)zo

(b) 2 £

collinear then

(c) 3zl

(d) 9zl

(a)ad + bc = 0 (c)ab + cd=0

35. If a complex number z lies on a circle of radius 1/2 then the complex number ( - 1 + 4z) lies on a circle of radius (a) 1/ 2

(b) 1

(c)2

(d)4

36. If n is a positive integer but not a multiple of 3 and z = - 1 + i <3, then (z2" + 2Y + 22 ") is (b) - 1

(b)V2,J (d) 2 V2,

|

„_ , 10 f . 2nk . 2nk 0 38. Th e va lue of I sin —— - 1 cos — k= l 11 11 (b) 0 ( dj i

=0 =0

triangle ABC lies at the srcin, then the nine point centre is represented by the co mp le x number Z1+Z2

(c) 1 (d j 3 x 2" 37. If th e vert ices of a tr iangle are 8 + 5i, - 3 + i, - 2 - 3;, the modu lus and the argument of the complex number representing the centroid of this triangle respectively are

(b)ad-bc (d)ab-cd

A, Band C represent 43. Let the comple x numbers Z],Z 2, Z3 res pect ively on the complex plane. If the circumcentre of the

,

equal to

(a) - 1 (c) - i

the

41. The set of poin ts in an arga nd diagra m whi ch satisfy both I z I ^ 4 and arg z = n/3 is

34. If Z], z2 , Z3are the vert ices of an e quilat eral 2 2 2 triangle with centroid Z q, then Z\ + z 2 + Z3

(a)2,-

are

x 3 - 3x 2 + 3x + 7 = 0 (00 is cu be r oot a- 1 B - l yunity), then ~r + — r + - is au- 1 y - 1

(c) 2co

33. The value of VT+ V( - i) is (a) 0 (b) <2 (c) - i (d) i

(a) 0

3 25 (x - iy) where x, y are

real, then the ordered pair (x, v) is given by

40. If

32. The compl ex nu mber s Zj, z 2 and z 3 satisfying

(a) of zero area (c) equilatera l

+ _

(c) ( - 3, 0)

(d) A: = 4

the

2

(a) (0 ,3)

(b) k = 2

——~ = - — a r e z2-z3 2

3 1 V3~ ,50 f

(c)

z3

Zi +Z 2 - z 3

(b) (d)

Zi + z2 - Z3 Zl - z2 - z 3

44. Let a and (3 be two distinct comp lex numb ers such that I a I = I(3 I. If real par t of a is posi ti ve and imaginar y par t of (3 is negati ve, then the com ple x num ber ( a + (3) /( a - (3) may be (a) zero (b) real and nega tive (c) real and positive (d) purely imaginary 45. The complex nu mber z satisfies the condition 25 = 24. The maxim um distance from - — z the srcin of co-ordinates to the point z is (a) 25

fb) 30

(c) 32

(d) Non e of these

Complex Numbers

9

46. The points A, B and C represent the complex numbers z\, Z2, (1 - i) Z\ + iz 2 respectively on the complex plane. The triangle ABC is (a) isosceles but not right angled (b) right angled but not isosceles (c) isosceles and right angled (d) None of these 47.

The system of equation s has (a) no solut ion (c) two solu tion s

55. If

I z + 1 + i I = <2 IzI = 3

value

of 2

+ 4(3co+ 1) (3(0

the

)+

2

expression 2

3(2(0+l)(2co

+1)

+ 1) + ...+ (n + 1) (nco + 1)

(no) 2 + 1) is

(a)

(c) f " ( W 2 + M

50.

T,

If

Re (a + a

2

f - n (d) None of these ( 87C ^

—

. .

+ ;sin

( 871

—

(b )-l /2

(c)0

(d) None of these

200

then

z

satisfying

the

equation

1 + «]Z + a2z + ... + anz = 0 (a) lie outside the circle (b) lie inside the circle

Iz I= | I z I =^

Iz I = |

(a) Refle xive (c) Transitive

(b) Symme tric (d) Anti-s ymmetr ic

(b) (1 ,- 1) (d) (4, 8)

I z, - 1 I < 1, I zz - 21 < 2, I z 3 - 3 I < 3

then I Z] + z2 + z 3 1 (a) is less than 6 (c) is les s than 12

59. The

roots

(z + a(i)

3

of

the

cubi c

(b) is mor e than 3 (d) lies bet wee n 6 arid 12

(a) ^l ap l

(b) V3"l a

(c)V3"ipi

(d) ^-l

I al

60. The number of points in the complex plane that satisfying the conditions I Z - 2 I = 2, z (1 - i ) + I ( l + 0 = 4 is (a )0 (b) 1

53. If I z I = max {I z - 1 I, I z + 1 1} the n (a) I z + z I= |

equa tion

= a 3 (a ^ 0), represent the vertices

of a triangle of sides of length

50

2 i + II f = x + iy then (x, y) is p=1 k=0

(a) (0,1 ) (c) (2, 3) 52. If

numbers

+ a 3 + a 4 + a 5 ) is

(a) 1/2

51 . If

57. If I ak I < 3, 1 < k < «, the n all the com ple x

58. If X be the set of all complex numbers z such that I z I= 1 and d efi ne rela tion if on X by 2n Zi R z2 is I arg z, - arg z 2 1 = — the n R is

+ «

a = cos

4

(d) z

(d) lie in | < I z I < i

l)2

n (n+

. to equ. al

(b) z 2

(c) z 3

(c) lie on the circle

((0 is the cube root of unity)

the

(b) 7 (d) V31~+ 2

(a) z

(d) None of these

2(1 +co)(l +co

(a) 2 + VTT (c )V 31 ~- 2

then

I iz + Z\ I is

tf z = — A/3~+ 56. If - — j then (z ] 0 1 .103.105 + ( )

48. The centre of the circle represented by I z + 1 I = 2 I z — 1 I on the complex plane is (a) 0 (b) 5 / 3 49. The

I z - i I < 2 and z, = + 5 3/

maximum value of

(b) one solution (d) none of these

(c) 1/ 3

(c) lz + z l = l (d) z - z = 5 54. The equation I z + i I - 1 z - /1 = k represents a hyperbola if k >2 (a) - 2 < k2< (b) (c) 0 < k < 2 (d) No ne of thes e

(b) z + z = 1

(c) 2

(d) mo re than 2

10


Each question in this part, has one or more than one correct answer(s). a, b, c, d corresponding to the correct answer(s).

67. If I z - 1 I + I z + 3 I< 8, the n the ra nge of values of I z - 4 I is (a) (0 ,7 ) (b) (1, 8)

61. If arg (z) < 0, then arg ( - z) - arg (z) = (a) n /

\ (Cl-J

(b) - n 71

m f

(c) [1 ,9 ]

62. If Oq, a,, a2, ..., a „ _, be the

n, nth roots of

the unity, then the valu e of

n -1 a, I — is i = 0 (3 — a ; )

equal to , N (a) (c)

sin whe re z is non real, can be the angle of a tr ia ngle if (a) Re (z) = 1, I m (z) = 2

(d)

3" - 1

3" - 1 n+2

69. If then

(a) I co - 5 - i I 1 (d) I arg (co- 1) I < Jt/2 64. co is a cub e root of unity and n is a positive inte ger sat isf yin g 1 + co" + co 2" = 0; then n is of the type (b) 3m + 1

(c) 3m + 2

(d) No ne of these .

z+ 1 If r is a purel y imag ina ry numb er; then z z+ I lies on a (a) straight line (b) circle (c) circle with radius = (d) circle passing through the srcin.

66. The equation whose roots are mh power of the roots of the equation,

x — 2x cos 0 + 1 = 0, is given by 2

2

(a) (jc + cos nQ) + sin n0 = 0 (b) (x - cos «0) 2 + sin 2 ;i8 = 0 +

(z )< l

(d) Wone of these

3" - 1

(a) 3m

m

(c) Re (z) + / m (z) = 0

(b)

3" - 1 n+ 1

(c) .v2 + 2xcos

(d) [2, 5]

-1

68.

(b) Re ( z ) = l, - l < /

n

z 63. If satisfies I z - 1 I < I z +3 I co = 2z + 3 - isa tis fie s

65.

For each question, write the letters

1 = 0

(d) x - 2x cos «0 + 1 = 0

tan a - i (sin a / 2 + cos a / 2 )

1 + 2i sin a / 2 imaginary, then a is given by (a) mi + 71/4 (b) tin -

(C ) 2/771

is

purely

n/A

( d ) 2/!7t +

71/4

70. If z 1 and z 2 are non zero complex numbers such that I Zi - Z 2 I = I Z] ! + I z 2 I then (a) I arg z, - arg z 2 I = n (b) arg z, = arg z 2 (c) Zi + kz2~ 0 for some positive numbe r k (d)z, z

2

+ Zi z 2 < 0

71. If z is a com ple x num ber a, , a2, a3, bh b2, b3 all are real then

and

ajZ + b\ z

a2z + b 2z

a^z + b^z

îZ +

b2z + a2z

b^z + a^ z

b2z + a2

+

fliZ

b\z + a\ 2

2

(a) (a i a 2 a 3 + b, b2b t ) I z I (b) I z I2 (c)3 (d) None of these 72. Let ^

e

e

r'Jt/2 n

.

B

2

- ,'jt/6

~

2

- / 5K/6

JTe ^ 3 poi nts ' fo V3~ be- three rm in g a tr iangle ABC in the argand plane. Then A ABC is : (a) equilateral (b) isosceles (c) scal ene (d) Non e of these

n

_=

Complex Numbers

73.

11

1+z 1 +z

If | z l = 1, then

+ 1| ^

| is equa l

(b) Ib I> 3 (d) No ne of thes e

3

(b)cosec

3

(c) cosec

3

8.e'

(d) cosec

2

8. e ~ 24 i + n/

8. e ^

8 . e - ' ( 2 4 - W2

z-1

n. ; = — is equal to 4

(a) M

VT 2 n (d) None of these <2 84. The digit in the unit's place in the value of

2

(b) - to

2

(c) - co3

(d) - co

4

(c)

(739) 49 is

n and

I a,-1< 1, A,-> 0 fo r i = l , 2 , 3,.. ,

+ ^ + A3 + ... + A„ = 1, the n the val ue of I A ^ j + A2a2 + ... + A na n I is (b) < 1

(c) > 1

(d) No ne of thes e I2 + I

2I

2

(a )3 (c) 9

(b) 4 (d) 2

85 . If zj , Z2,

(a) = 1 I

that

(b)l (d)-

z+i) )

(36_7r/2)

I

su ch

83. Perimeter of the locus represented by arg

* ™

(a) - co

78. Iflz, +

co nsta nt

(amp z) - — is equal to z (a) i

76. If 1 + co + co2 = 0 then co 1994 + co1995 is

77. If

compl ex

(c)-l

75. The trigono metric f orm of z = (1 - i cot 8)' is 3

a

82. If I z - 3/1 = 3 and am p z e ^ 0, ~ ] then cot

74. If all the roots of z 3 + az + bz + c = 0 are of unit modulus, then

(a)cosec

is

(a) a + a = 1 (b) a + a = 0 (c) a + a = - 1 (d) The absolute value of the real root is 1

n (arg (z)) (arg (z)) n (arg (z/2)) n (arg (z/2))

(a) I a< I3 (c) I c
a

a z + z + a = 0 has a rea l root the n

to : (a) 2 cos (b) 2 sin (c) 2 cos (d) 2 sin

81. If

4

Z4 are roots of the equation 3

2

a0z + axz + a2z + a^z + a4 = 0, where a0, ah a2, a3 and aA are real, then

then

(a) z\,z2, z 3 , z 4 are also roots of the equation (a) — is pure ly real (c) z, z 2 + z 2 Zi = 0

(b) — is purely imagi nary (d) amp

z,.

: z2' 2 are the four compl ex number s

79. If

taken in ord er su ch that Z] - Z4 = Z2 z 2 - z, ^

and

= — then the quadrilateral is a (b) square

(c) rectang le

(d) cyclic quadrilateral

zj , Z2

be

two

com ple x

represented by points on the circle and i z I = 2 respectively then

numb ers IzI= 1

(a) max I 2z, + z 2 I = 4 (b) min I z, - z 2 I = 1 (c)

Z2 + ~

S3

z 3, z4

equation

jt - z2 and IZ] + z2 I

(a) at least one of z,, z

2

1 —

1 +

—

then

is unimodular

(b) both Z], Z2 are unimodular (c) Z], z 2 is unimodular

(a) rho mbus 80. Let

2,

(d) None of these 86. if

represented by the vertices of a quadrilateral

amp

(b) zi is equal to at least one of z,, z

(c) - I ,, - z 2 , - z 3 , - Z4 are also roots of the

K

(d) No ne of thes e

(d) None of these 87. If

z = x+ iv

1 and co = -

-iz

Z-l

implies that in the complex plane. (a) z lies on imaginary axis (b) z lies on real axis (c) z lies on unit circle (d) none of these

then

I co I = 1

12

Objectiv e Mathemati cs

88

- If Sr = j sin a- d {ir x) when

(a )0 (c)2

(; = V^ T ) then

4n - 1

Let 3 - i and 2 + i be affixes of two points A and B in the argand plane and P represents z~x + iy then the of the complex number locus of P if I z - 3 + /1 = I z - 2 - i i is

L Sr is (n e N) r= 1 (a) - cos x + c

(b) cos

(c) 0 For

x + c

(d) not def ine d

complex

numbers

Zi = x 2 + iy2

we

< x2 and Vi S y 2

+ iy ]

and

Z\ n z2

of

Z\=X\

write then

for

all

(b) 1 (d)3

(a) Circle on AB as diameter (b) The line AB (c) The perpendicular bisector of (d) None of these

complex

numbers z with 1 n z we ha ve 1 - z n 1+z

AB

is

Practice Test Time: 30 Min.

MM : 20

(A) There are 10 parts in this question. Each part has one or more than one correct

5. Let f(z) = sin z an d g{z) = cos z. If * denotes a composition of functions then the value of

1. The number of points in the complex plane that satisfying the conditions | z - 2 | = 2 , 2 (1 - i) + z (1 + i) = 4 is (a) 0 (c) 2

(f+ig)*(f~ig)

(b) 1 (d) mor e th an 2 3

2

2

| z + | si n 0

3

(d) less than | s in 9i | + | sin 0

2

| + | sin 0

3

| + | sin 0 4 then

| sin 03

+ | sin 0

4

|

2-i 3 +i

i) - z (i - 1) is - 1+i (a) (b) 2 2 - 1 i(i+ 1) (c) (d) 1 + i 2 Let S be th e set of complex nu mb er z whic h satisfy 2

(c) | + Ai

qua dra tic

eia/p\

in the straight line z (1 +

1/2

the

equ atio n

(1 + i) x - (7 + 3 i) x +(6 + 8t) = 0 then the other root must be (a) 4 3i 4(b) 1 i (c) 1 + i (d) i (1 - i)

The reflection of the complex number

l og 1 / 3 |log (a) 1 ± i

(d) i e

(c) ie 6.If one root of

| z+

| sin 0 4 | = 3 , from z = 0, are

2

(b) ie'

e

(a) greater than 2/3 (b) less than 2/3 (c) grea ter th an | sin ©i | + | sin 0

is

(a) ie

2. The dist anc es of th e roots o f th e e quat ion | sin Gj | z + | si n 0

answer(s). [10 x 2 = 20]

(| z | + 4 | z | + 3)} < 0 the n S i s i (b) 3 (d) Empty set

e3ia/p2

is

em/p

...

Lim fn (k) is n —»<*>

(a) 1 (b) - 1 i (c) i (d) g The common roots of 3

th e

equ ati ons

2

z +(1 +i)z + (l + j ) z + j = 0 1993

4 - 3i

z +z (a) 1

1994

.

and

„

+ 1 = 0 ar e (b) co

(c) co2

(d) (o

981

The argument and the principle value of 2+i the complex number Ai +( 1 + i ) 2 are (a) tan

1

(-2)

(b) - tan"

1

2

(c) tan

1

[ ^

(d) - tan"

1

[ |

Complex Numbers

10. If | z

x

13

| = 1, | z

2

| = 2, | z

3

| = 3 an d

| 9zxz 2 + 4zjz3 +Z3Z

2

(b) 36

(a) 6 (c) 216

I z l + z 2 + 2 3 I = 1 then

(d) None of these

| is equal to

Max. Marks 1. First attempt 2. Second attempt 3. Third attempt

must be 100%

Answers 4. (b) 10. (a)

6.(b)

5. (d) 11. (d) 17. (a)

12. (d) 18. (a)

22. (a)

23. (b) .

24. (b)

28. (a)

29. (a)

30. (b)

35. (c) 41. (c)

36. (a) 42. (b)

1. (a) 7. (c) 13. (c)

2. (c)

3. (c)

8. (d) 14. (b)

9. (d) 15. (b)

19. (d) 25. (c)

20. (a) 26. (c)

21. (d) 27. (a)

31. (a)

32. (c)

33. (b)

34. (c)

16. (a)

37. (b)

38. (d)

39. (b)

40. (d)

43. (c)

44. (d)

45. (a)

50. (b)

51. (b)

46. (c) 52. (c)

47. (a)

49. (b)

53. (c)

48. (b) 54. (a)

55. (b)

56. (c)

57. (a)

58. (a)

59. (b)

60. (c)

61. (a)

62. (a)

63. (b), (c), (d)

64. (b), (c)

65. (b), (c), (d)

66. (b), (d)

67. (c)

68. (b)

69. (a), (c), (d)

70. (a), (c), (d)

71. (d)

72. (a)

73. (a)

74. (a)

75. (a)

76. (a), (d)

77. (b)

78. (b), (c),

79. (c), (d)

80. (a), (b), (c)

81. (a), (c), (d)

82. (a), (d)

83. (d)

84. (c)

85. (a), b)

86. (c)

87. (b)

88. (b)

89. (a)

4. (d)

5. (d)

90. (c)

1. (c) 7. (c)

2. (a)

3. (b), (c), (d)

8- (b), (c), (d)

9. (a), (b)

10. (a)

6. (c), (d)

THEORY OF EQUATIONS An equation of the form ax+2 +c =bx0, where a , b, c e c and a * 0, is a quadratic equation. The numbers a, b, c are called the coefficients of this equation.

a such that aa

A root of the quadratic equation (1) is a complex number

D=

b2

- 4ac is the

...(1)

2

+ ba+ c = 0. Recall that

discriminant of the equation (1) and its root are given by the formula. -£> + VD x = — , 2a

1.

If a, b, c e Ra nd a * 0, then (a) If D < 0, the n equation (1) ha s no roots. (b) If D > 0, then equation (1) has real and distinct roots, namely, - b + VD -b-VD and then a x 2+a (bx+c x - x i ) =(X-X2) (c) If D - 0, the n equat ion (1) ha s real and equ al r oots *t=x 2 = - b-

...(2)

ax 2 + bx + c = a(x-x1)

and the n

2

.

...(3)

2

To represent the quadratic ax + bx + c in form (2) or (3) is to expand it into linear factors. If a, b, c e Q an d D is a perfect square of a rational number, then the roots are rational and in case it be not a pe rf ec t sq uar e th en th e roots are irrational. 3. If a, b, c e R and p + iq is one root of equation (1) (q * 0) then the other must be the conjugate p and vice-versa, (p, q e R a nd /' = V- 1). It a, b, c e Q and p + Vg is one root of equation (1) then the other must be the conjugate p -Jq vice-versa, (where p is a rational and Jq is a surd). 5. If a = 1 an d b, c e I and the root of equation (1) are rational numbers, then these roots must be integers. 6. If equa tion (1) ha s more than tw o roots (compl ex numbers), then (1) be co me s an ident ity i.e., a=b=c=0

Consider two quadratic equations : ax 2 + bx + c= 0 and (i)

a'x

2

If two co mm on roo ts th en a_ _

b^ _ c^

a' ~ b' ~ d

+ b'x + d = 0

iq and

Theory of Equations

(ii)

15

If on e co mm on root the n

(ab' - a'b) (be?- b'c) = (eel -

(Interval in which roots lie) Let f (x) = ax 2 + bx+ c= 0, a,b,ceR,a> 0 and k,h ,k2e R and /ci < k 2 . Then the following hold good : (i) If both the roots of f(x) = 0 are greater than k. then D> 0, f(k)>0 (ii)

an d

da) 2.

a,p

be

the

root s.

Sup pos e

> k,

2a

If bot h th e root s of f(x) = 0 are less than k then D> 0, f(k) w > 0 and

< k,

2a Ifk lies between the roots of f(x) = 0, then D> 0 and f(k)<0. (iv) If exactly on e root of f(x) = 0 lies in the interval (ki , k 2) then D> 0, f(ki) f(k 2) < 0, (v) If bot h root s of f(x) = 0 are confined between k-[ an d k 2 then D > 0, f (/c-i)> 0, f(k2) > 0 (iii)

i.e.,

k-\ <

g + P

<

k2.

: Let f(x) be a function defi ned on a close d interval [a, b] such that

(vi)

(i) f (x) is continuous on [a, b] (ii) f(x) is derivable on (a, b) and (iii) f (a) = f(b) = 0. Then c e (a, b) s.t . f (c) = 0. (vii) : Let f(x) be a function defined on

[a, b], such that

(i) 1(x) is continuous on [a, b], and (ii) f(x) is deri vable on (a , b). Then c e (a , b) s.t. f'(c) =

— ^ ^ U

•

3

(Generalised Method of intervals) _ Let where /ci, k2,

F(x) = (x-

ai)*1

(x-a2f2

..., kne Nand ai, a2, as

(x-a3)k3...

(x-an-i)kn~'

(x-an)k"

...(1)

anfixed natural numb er s satis fying the condition

ai < a2 < a3 ... < a n - i < an First we mark the numbers ai, a2, ..., an on the real axis and the plus sign in the interval of the right of the largest of these numbers, i.e., on the right of an. If k n is even we put plus sign on the left of an and if kn is odd then we put minus sign on the left of an. In the next interval we put a sign according to the following rule : When passing through the point an-1 the polynomial F(x) ch an ge s sign if kn-1 is an odd num be r and the polynomial f(x) ha s sa me sign if kn~ 1 is an e ven numbe r. Then we co nsi der the next interval and put a sign in it using the same rule. Thus we consider all the intervals. The solution of F(x)> 0 is th e union of all intervals in which we have put the plus sign and the solution of F(x)< 0 is th e union of all inte rval s in which we have put the minus sign.

1. An equation of the form (x- a) (xwhere

b) (x- c) ( x - d ) = A, can be solved

a< b< c< d, b- a = d- c,


16

by a chang e of variable. (x - a) + (x - b) + (x - c) + (x - d)

i.e.,

(a +

y=x

or

c+ d)

2. An equation of the form

b) (x - c) ( x- d) = /Ax2

(x- a) (x-

where a£> = cd, can be reduced to a collection of two quadratic equations by a change of variable 3. An equation of the form

(x - a) 4 + (x- b) 4 = A i.e., making a substitution (x-a) + (x-fr) y 2

can also be solved by a change of variable,

The equation of the form I f (x) + g (x) I = I f (x) I + I g (x) I is equivalent of the system

f(x)g(x)>0 5. An equation of the form am

+ t /(*) = c

a, b, c e R

where and a, b, c satisfies the condition a 2 + b 2 = c then solution of the equation is f (x) = 2 and no other solution of this equation. 6. An equation of the form 900

(ft*)}

is equivalent to the equation {f(x)}

1. logi* ^ = ^ log 2.

a

9W

= 10 9 (J ° 109 nx)

where f{x)> 0

b

lo

f 9a 9 = glogaf [x + /?] = n + [x], n e I when [.] de not es the gre ate st integer.

5.

x = [x] + {x}, {} denotes the fractional part of x

6.

[x] -

7.

(x) =

1 — + rx n 1

x +

[x],

+

21 + ... +r n

x+

" - i i = [nx] X

if 0 < {x} < £

[x] + 1,if

!<{x}<1

where (x) denotes the nearest integer to x

i.e., thus

(x) > [x] (1 38 29 )= 1; (1 543) = 2; (3) = 3

y = x+

ab

Theory of Equations

17

Each question in this part has four choices out of which just one is correct. Indicate you choice of correct answer for each question by writing one of the letters a, b, c, d which ever is appropriate. 1- Let

= ax2 + bx + c

f(x)

/( 1) >-

l,/(3

)<-4a

and

nda*0t

/ ( - 1) < 1, hen

(b) a < 0 *

(a) a > 0

(a)0 (c )4

(e) sign of 'a' can not be determined (d) none of these 2. If a and (3 are the root s of t he equa tion

x - p (x+l)-q a 2 + 2a + 1 a2 + 2a + q

= 0, |

then

the

value

of

(b)4 (d) infinit e

then the value of

(a + b + c) is

(a) lb2 - ac

(b) b2-

(c) b2 — 4ac

(d) 4b2-

2

roots

of

-4 J : + 5 ) = ; c _ la

(a) 1 and 2 (c) 3 and 4

equation

sign, is (a) 7/4 (c)-l/2

2 1 * I2 — 51*1 + 2 = 0 is

(d) - 5

2* /2 + (V2 + l) x = (5 + 2 < l f solutions

of

(b) 1 (d) infini te

7. Th e num ber of value s of a for which

(a) infinit e

(b) six

(c) four

(d) one

(c) two solution s

is an identity in x is

(a) 0 •

8. T he so lut ion of x - 1 = (x - [x]) (x - {x}) (w he re [x] and {x} a re the inte gral and fractional part of x) is (a)xeR (b)xe/?~[1,2) (c) x e [1 ,2 ) (d) x e R ~ [1, 2]

is

(d) mor e than tw o solutio ns

15. The number of real solutions of the equation

ex = x is

(b) 1 (d)3

2

14. The equa tio n Vx + 1 - Vx — 1 = V4x — 1 has (a) no solut ion (b) one solut ion »

(a2 - 3a + 2) x + (a - 5a + 6) x + a2 - 4 = 0

(a ) 0 (c)2

(b) 1 p

13. The numb er of real solutions of the equa tion

(b)2 (d) infini te

(a) 0 (c) 2

(d) - 5 an d - 6

12. The value of a f or whic h the equa tion (a + 5) x 2 - ( 2 a + 1) x + ( a - 1) = 0 has roots equal in magnitude but opposite in

(b) 2 and 3 (d) 4 and 5

num be r of real 1 „ 1 x is = 2x2 — 4 x2 -4

(b )- 3 a n d - 1 0 '

(c ) - 4 and - 9

re

6. The

pl ac e of 13 and its ro ot s we re fou nd to be (-2) and (-15). The actual roots of the (a )- 2 and 15

lac the

11. Th e coeffi cie nt of x in the quadratic equatio n ax +bx + c =0 was wr ong ly take n as 17 in

equation are:

lac

5. The number of real solutions of the equation (a)0 (c) 4 f

a cos 2x + fc sin x + c = 0

(c) 6

ax + bx + c = 0,

b, c)

10. The number of values of the triplet (a, for which

(a)2

are of the fo rm a / ( a - l ) a nd (a + l ) / a ,

= 4 lcosxl

(b)2 (d) 6

p2 + 20 + q

3. If the roots of the equation,

5 log 5 (,

sin(lll)

is satisfied by all real x is

(b)l (d) None of these

real

2

P2 + 2 P + 1

(a) 2 (c)0

4. The

9- The number of solutions of in [ - 71, Jt] is equal to :

(c) 2 16. If tan a and tan P

(b )l (d) No ne of these are the roo ts of

the

equation ax +bx + c = 0 then the val ue of tan ( a + p) is : (a) b/(a - c)

(b) b/{c - ay*

(c)a/(b - c)

(d )a/(c - a)

18


17. If

a , fJ are the roo ts of t he equa tion x + x Va +13 = 0 th en the val ues of a and (3 are : (a ) a = 1 an d p = - 1

(a) < 0 ' (c) > 0

(b) a = 1 an d p = - 2^ (d) a = 2 an d P = - 2 the

(b) > 0 (d) zero 2

(c) a = 2 and j$ = 1 18. If a , P are

ax + 2bx - 3c = 0 has non (3 c/ 4) < (a + b); then c is

26. If the equation real roots and always

root s

of th e

equation

27. The root of the equati on 2 ( 1 + i)x - 4 (2 - i) x - 5 - 3i = 0 whic h has greater modulus is

2

8x - 3x+ 27 = 0 the n the f ( a 2 / P ) l / 3 + ( p 2 / a ) 1 7 3 1 is : (a) 1/3

(b) 1/4

(c) 1/5

(d) 1/6

19. For a * b, if the equations

valu e

(a) (3 - 5i)/2 (c) (3 + 0 / 2

of

4

28. Let a, b,c

x + ax + b = 0 and

(d) 2

20. Let a , p be the r oots of the equat ion (x - a) (x - b) = c, c * 0. The n the root s of the equ ati on (x - a ) (x - P) + c = 0 are :

21 .If

(b ) b , c ( d ) a + c,b + c

a , P are

the

root s

of th e

(b>

(c ) x2 + x + 2 = 0 22. The

num be r

(d)x of

-

2

+ 19x + 7 = 0

real

+ 1= 0 soluti ons

of

24. The

(b) two * (d) zero

total

num ber

of

the

roots

of

the

equat ion,

,x + 2 ^-jbx/(x- 1) _

= 9 are given by

(a) 1 - log 2 3, 2 ( 2/3 ), 1 »

2

( d) - 2, 1- ( lo g 3) /( lo g 2) The number of real solutions N. of the equation co s (ex) = 2x+2~x (a) Of

is (b)l (d) infin itely many

If the roots of the equation, x + 2 ax + b = 0, are real and distinct and they differ by at most 2m then b lies in the interval (b) [a2 - m2, a1)

(a) (a 2 - m 2 , a 2 )

23. The nu mbe r of solutions of the equa tion 1x1 = cos x is (a) one (c) thre e

of

the n

(b)y=P (d) a < y (d)4

l + lc'-l (a )0 (c)2

ro ot

(c ) 2 , - 2

equa tion

2

The 29 .

(b)log

x + x + 1 = 0, then the e quat ion who se roots are a and p is : (a)x 2+ * + l = 0

a

0 < a < p,

equation, ax + 2 bx + 2c = 0 has a root y that always satisfies (a) y= a (c )y =( a + P)/2

(b )0

(a) a, c (c ) a, bt

is

- bx — c = 0 and

ax ^

value of (a + b) is (c) 1

e R and a * 0. If a is a root of

a2x2 + bx + =c 0, P

x + bx + a = 0 have a common root, then the (a) , - 1

(b) (5 - 3 0 / 2 (d) (1 + 3 0 / 2

2

2

2

(c) (a , a + m )

(d) No ne of these

If x + px + 1 is a fact or of the exp ress ion 3

2

ax +bx + c then solutio ns

of

(a )a

2

+ c2 = -ab 2

(b )a2-c2 =

sin nx = I In I x 11 is (a)2 (c) 6 „

If a, b, c be positive real numbers, the following system of equations in x, y an d z :

(b)4 (d) 8 2

- 2 0 p x+ (25 p 2 + 1 5 p - 66) = 0,

(al (4 /5 , 2) (c)(-1,-4/5)

2

2

2

2

2

a

are less than 2, lies in (b) (2, » ) (d)(-=o,-i)

a

(d) No ne of these

2

p for which both the roots of the

25. The value of equation 4x

= -bc^

-ab

(c) a -c

b

c

2

2

2

2

2

2

b

c

Theory of Equations 2

X • —

a

+

2

T

b

V

I

19

2 -

7 —

+

,2

T

2 =

2

c

(c)3

1, has:

(d) 7, 3

41. If x' + x + 1 is a factor of ox

(a) no solution (b) unique solution, (c) infinitely many solutions (d) finitely many solutions

^

+ bx + cx + d

2

then the real root of

ax' + bx + cx + d = 0 is

(a) - d/a% (c) a/d

(b) d/a (d) None of these

42. x' og, v > 5 implies

34. The number of quadratic equations which remain unchanged by squaring their roots, is

(b) x e ^0 , j j u ( 5

(a) x e (0, 0=)

k

-)

1

(a)n ill (c) four

(b) two y (d) infinitely many.

( C ) X 6

35. Ifâj<_0, the positive root of the equation

(c)a(-i

(d)a(l

ax2 +\bx+ c = 0, equation 2

ax - bx (A: — 1) + c (x — 1) = 0 has roots a 1- a 1-P (a) (b) 1- a' 1- p a ' P a + 1 p+1 (c) (d) a+ l' P+ b a ' p 37. The solution of the equation

(c)2

'"

log

(d) 2~ the

roots are a

2

y

equation

4

- x 2+ 1 = 0

Given that, for all

x - 2x + 4

(d) x

2

2

+ * + l)

2

= 0

x e R, The expression

which

15

(a + Jbf~

the

9 . 3 2 *- 6.3

X

1 5

equation

= 2a,

where

,

(a) ± 2, ± V J

(b) ± 4, ± V l T

(c) ± 3, ± V5

(d) ± 6, ± V20

45. The number of number-pairs (x,y) which will satisfy the equation 2

2

'(b ) 2 (d) No ne of thes e

46. The

solution

log* 2 log

2r

set

of

the

equation

2 = log 4 , 2 is

expression

I

For

(b) ,22} any

(d) None of these real

lies are

the

expression

(b) 2k 1 9

(a )kz (c) 3k 2

(d) None of these

48. The solution of

+ 4

x

[Vx 2 + k 2] can not exceed

2 (k-x)

->2x

9.3 +6 .3+ 4

the

+ (a - -lb) x "

a2 - are b= 1

(c) j

-x + 1= 0

1 lies bet ween — and 3, the va lue 3

x + 2 x+ 4 be tw ee n

(d)0 of

roots

(a) { 2"

2

(a) (x - x + l ) = 0 (b) (x (c)x

(c)l. The

whose

, p 2 , y2 , 8 2 is

2

-3x+ 1) are

(b) 2

(a) - 1 (c) 4

38. If a , P, y, 8 ar e the r oots of x4 + x2+ 1 =0 then

44.

- 9x + 4)

x - x y + y = 4 (x + y - 4) is

3 lcg " A + 3x' og " 3 = 2 is given by (b) 3~ log2 " (a) 3 logj a ,og

2

= V(2x - 2x ) - V ( 2 x (a )3

~X

2

2

2

(b)a(l-V2)<£

36. If a and p are the roots of then the

O)

V(5x 2 - 8x + 3) - V(5x

x - 2a I x — a I - 3a 2 = 0 is (a) a (- 1 - VfT)

(L.O

43. The values of x which satis fy the eqattion

x- 1

+ 1x1 =

• is

1

(a) 3" and 3 (c) - 1 and 1

(b) - 2 and 0 (d)0 and 2

(a) x > 0

(b) x > 0 (d) None of these

49. The number of positive integral solutions of 40. The value of " ^ 7 + ^ 7 ^ ^7 + V7 _ .. , is (a) 5

=

X

2

( 3 X - 4 ) 3 ( X - 2 ) ^ Q is

(x-5) (b)4

(a) Four

5

(2x-7)

6

(b) Thr ee


20

(c) Tw o

(c) Thre e

(d) Only one

50. The number of real solutions of the equation 9 r— | ^ = - 3 + x 2. -x

(a) None, (c) Two 51. The

is

|x

(JC —

3) I

JC

+

J

i)

(3 +

- x2) _

I has

(a) Unique (c) No Solution

(b) Two solutions (d) More than two

52. If xy = 2 (x + y), x
the

Two ja, Three' No solutions Infinitely many solutions

2x z

1+z

l+x2

(a) /* 1

0

»

**

x .2

+2

(b) 15 » (d) No ne of the se

2

(a)(l+a,)

(c) (1 - atf

1 is l+y2

roo ts

of the

equ ati on then

-(ao + a

+ (a 0-

jr — 3 I + 2 2 r,l x-3 1+4 , =x .2

(b) Only one

(c) Tw o

(d) Fou r

a2)2

roots

of

+2

4

(3 - x) + (2 - x) = (5 - 2x) is

the 4

equation

are

(a) all real (b) all imaginary(c) two real & two imaginary (d) None of these

55. If a be a positive integer, the number of

60. The numbe r of ordere d 4-tup le (x,y,z, w) (x,y,z, we[0, 10]) whi ch satisfies the inequality

values of a satisfying : ,Jt/2 2 (cos 3x 3 a — -— + — co sx J 4 4 0 + a sin x - 20 cos x

2)«

(d) None of these 59. The 4

(a) Non e .

2

2

2

2 si n * 3C0S y 4 sin

dx<~-

1S

(a)0 (c) 81

(a) Only one

the

(b) (1 + a{) 2 + (a 0 + a 2 ) 2

(d) 4 +1 ,

(a ) 10 (c) 12

[x + y] is

(1 - a 2 ) (1 - p 2 ) (1 - Y2) is equal to

54. The numb er of nega tive integral solutions of 2

57. If y = 2 [x] + 3 = 3 [ * - 2 ] + 5, then ([x] denotes the integral part of x)

x3 + a0 x + ax x,.+ = 0,a2

(b ) 2

3 (c)

(b) for b = 0 there are three solutions (c) for 0 1 there are no solutions (f) None of these

58. If a , (J, y are

53. The numbe r of real solut ions of the system of equations 2z

which

(a) for b < 0 there are no solutions»

(b) One (d) More than two

equati on

(d) Four

56. For the equatio n I JC - 2x - 3 I = b statement or statements are true

z

2

5 C0S

w > 120 is (b) 144 (d) Infinite

(b) Tw o

Each question in this part, has one or more than one correct answer(s). a, b, c, d corresponding to the correct answerfs).

62. Let/(x) be a quadratic expression which is po si ti ve fo r all real x.

61. The equation ^Kiogj x) - (9/2) log, x + 5] _ » 3 ^

(a) at least one real solution (b) exactly three real solutions (c) exactly one irrational solution (d) complex roots


sha

If

g

(x) =/ (x ) - / ' (x ) +/" (x) ,

(a ) g (x) > 0 (c)g(x)<0

then for any real x,

(b) g (x) > 0 (d)g(x)<0

Theory of Equations

63. For

a> 0,

21

the

root s

of

the

equa tion

logax a + log* a + lo g a J x a = 0, are given by / \

-4 /3

(a)a t \ -1/2 (c) a 64. The real values of

(c) exactly one positive root (d) at least one root which is an integer 72. If 0 < a < b < c, and the ro ots a , (i of the

-3 /4

(b ) a

-1 (d) a X for which the equation,

3x 3 + x - lx + X =0 has two

distinct

real

ax + bx + c= 0

equation

are

non

real

complex numbers, then (a) I a I = I P I

(b) I a I > 1

(c) I p
(d) No ne of these

roots in [0, 1] lie in the interval(s) (a) ( - 2 , 0 )

73. If

(b) [0, 1]

(c) [1,2] 65. If a is

one

(d )( -~ ,o o) ro ot of the

equ ati on

4JC + 2x - 1 = 0. then its oth er ro ot is give n by (a) 4a- 3 3 a

(b) 4 a

(c) a - (1 /2 )

3

+ 3a

of

the

equati on,

2 + 4x + 3), are given by

(a) 2- V3 " (c) 2 + V3~

2a + 3b + 6c = 0 (a. b, c e R)

67. If

the

1 2

largest -.C

9

+

. V

4

-J C+

(a) [0, 00) (c) ( 00)

interval 1>

0

the

roots

of the

equati on

— JC — 1 = 0, then the value of X

f j

+ a

I1 J

] is

(b) - 5 (d) No ne of thes e

has no real root, then

(b)a + 2c>b (d)a + 3c 4b

which

k e I are (a )0 (c) 2 If

a, P

(b) 1 (d )3 be

roo ts

of

x~ - .v- 1 = 0

and

An = a" + p" then A.M. of An _ j and An is (a) 2A„ ] + (c) 2 An _ 2 78. If

a. b. c e R

l/2A„

(b)

+i

(d) Non e of th es e and

the

equality

ax' - bx + c = 0 has compl ex r oots whi ch are

is

(b) ( 0] (d) Non e of these

70. The system of equation .v - I y - 1 I = 2 has (a) No solution (c) Tw o sol utio ns

in

JC3

(a) 2 a + c>b

(d) None of these 68. Let F (x) be a function defined by F (x) = x — [x], 0 * x e R, where [.t] is the greatest integer less than or equal to .v. Then the number of solutions of F(x) + F{\/x) = 1 is/ar e (a) 0 (b) infi nite (c) 1 (d )2

JC

(b) 3/2 (d) 7/2

If c > 0 and the eq uat ion 3 ax 2 + 4 bx + c = 0

then

quadratic equation ax' + bx + c = 0 has (a) At least one in [0, 1] (b) At least one root in (-1, 1] (c) At least one root in [0, 2]

69. The

(a) 1/2 (c) 5/2

(a)-3 (c) - 7

( b ) ( - l + /V3 )/2 (d)(-l-/V3)/2

when

{JC} and [JC] are fractional and integral part of JC then JC is

74. If a , P, y are

(d) - a - (1 /2 )

66. The roots 2 2 (JC + 1) =x(3x

5 {JC} = JC + [JC] and [JC] - {JC} = j

I .v — 1 I + 3y = 4,

(b) A unique solution (d) More than two solutions

71. If A,G an d H are the Arithmetic mean. Geometric mean and Harmonic mean be tw ee n tw o un eq ua l posi tive integers. Th en the equation AY" - I G I .V - H = 0 has (a) both roots are fractions (b) at least one root which is negative fraction

reciprocal of each other then one has (a) I b I b > c and the quadratic equation

(a + b- 2c) ,v : + (b + c - 2a) x + ( r + a - 2b) = 0 has a root in the interval ( - 1, 0) then (a) b + ca> (b)c + a<2b (c) Both roots of the given equation are rational (d) The equation clx2 + 2bx + c = 0 has both negative real roots

22


'a' for which 2 quadratic equation x + (a + 2) x - (a + 3) = 0 then S is given by

80. Let 5 be the set of values of lie

between

the

(a) ( - - , - 5 )

of

the

possess

four

real solutions (b) 0 < a < \

(c) 0 < a < 5

(d) No ne of thes e foll owin g equation

31*112 - I x h = 1 '(b) 2

(c)3

83. Th e eq uat ion l x + l l l x - l l = a 2 -2 a -3 carl "have real solu tion s f or x if a belongs to (a) ( - » , - 1] u [3, oo) (b) [1 - < 5, 1 + VfT]

sn-

(a) (b) (c) (d) 85. A

(a)

0

+ a-b-

(b) x e (-<*>, 2] (d)x e [- 2, 2]

89. If a , (3 are roo ts of 3 75 JC2 - 25A - 2 = 0 and

(c) [1 - V5", - 1] u [3, 1 + \ 5 ] (d) No ne of these

c)x

+9=0

(2, 4) [ - 5, - 4) u [2, 3) [ - 4, - 3) u [3, 4) None of these

(a) xe [2, oo) ( c ) j c e [ - l , 1]

84. If a, b, c are rational and no two of them are equal then the equations

and a(b-

3JC

1 88. If sin 6 + cos* 8 > 1, 0 < 6 < tt/2, then

(d)4

(b - c)x2 + (c-a)x

+ x + 1= 0

30 where [ A] is the greatest integer less than or equal to x, the number of possible values of x is (a) 34 (b) 33 (c) 32 (d) Non e of the se

po ss es s ? (a) 1

JC2

87. I f 0 < x < 1 000 and

(a) - 2 < a< 0

82. How many roots does the

(d) .v 2 - 3x + 9 = 0

(a) (b) (c) (d)

a for which the equation

2 (log 3 x) 2 - I lo g 3 x I + a= 0

(b) x +

(c)

to x, is

(d)[5,oo)

value s of

(a) x - x + 1= 0

86. Th e soluti on set of (X) + (X+1) =25, where (x) is the nearest integer greater than or equal

(b)( 5,~ )

(c)(—o,-5] 81. The

roots

n nn a +p then

Lim

(b)

116

29 (C) 358

+ b (c - a) x + c (a - b) = 0

Z

n

sr is

n —> oo r= 1

12

(d) None of these

have rational roots will be such that at least one has rational roots have exactly one root common have at least one root common

The equation A2 + ax + b 2 = 0 has two roots each of which exceeds a number c, then

quad ratic

(b) c2 + ac + b 2 >0

equatio n

whos e

roots

(a) a > 4b 2

are

(c) - a/2 > c (d) None of these

whe re a , (3, y are the roots of x + 27 = 0 is

Practice Test MM: 20

Time : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct 1. Le t a , p, y be th e roots of the equ ati on (x - a ) (x - b) (x - c) = d, d0, then the roots of the equation (x - a) (x - p) (x - y) + d = 0 are (a) a, b, d (b) 6, c, d

(c) a, b,c 2. If one

root

answer(s).

[10 x 2 = 20] (d) a + d, b + d, c'+ d of t he eq ua ti on

2

ix - 2 (1 + i) x + 2 - i = 0 is (3 - i), then the other root is (a) 3 + i

(b) 3 +

Theory of Equations

23

(c) - 1 + i (d ) - 1 i > 3. In a quad rat ic equ ation w ith leadin g coefficient 1, a student reads the coefficient 16 of x wrongly as 19 and obtain the roots are -15 and —4 the correct roots are 5 (a) 6, 10 <• (b) -6 , - 1 0 (c) - 7 , - 9 (d) None of th es e 4. The num be r of solu tions of | [x] - 2x \ = 4, where [x] denotes the gre atest integer < x, is (a) In fi ni te (c) 5. The interval of 51/4

logl

X>

(b) 4 * (d) 2 x in which the inequality

g l /5 log

5

soluti on

th e

2

(d)xe

8. If

a, p, y a re

nn + ^ , n e I

(d) 9 = ^ th e

root s

,n e / of t he

2

3

2

(b) - (27q + 4r )

3

(d) - (27r + 4g 2 )

(a) - (27q + 4r ) (c) -,( 27r

2

+ 4^ )

(a) zero (b) one , (c) two (d) th re e 10. The nu mb er of solut ions follow ing ineq ualit y 2 1/sin x 2

2

2

g 1/sin

of

the

2

x3 ^1/s in ,v 4 ^ 1/sin

w he re .v, e (0.

,

for i = 2. 3. .., n is

(a) 1 (b)

2 "

~

1

'

(c) n n (d) infinite number of solutions

Q

7

'• If x 2+px ' + 1 is a factor of

Max. Marks 1. First attempt 2. Second attempt 3. Third att empt

cubic

3

x + qx + r = 0, th en t he va lue of n (a - P) =

2

2

(c) .v el

(c) 0 = 2 nn , ne I

equa tion

(x + L ) + [x - L=] (JC - L ) + [x + L ] whe re and (x) are the gre ate st integer and nearest integer to x, is (a (b)xeN 2

)xeR

of

(b) 0 =

x3 + x2 + 2x + s in x = 0 in [- 2n, 2n] is (are)

I

set

(a) 9 = nn, n e I

9. The number of real roots of the equation

2 v 5 (a) ( 0 , 5 " 2 (b)[5 , ~ ) (e) bo th a an d b (d) Non e of th es e

6. The

2 cos 2 6 X3 + 2x 4- si n 29, t he n

mus t be 100%

An swers

1. (b)

2. (b)

3. (c)

4. (b)

5. (c)

6. (ai

7. (b)

8. (c)

9. (c)

10. (d)

11. (b)

12. (ci

13. (d)

14. (a)

15. (a)

16. (b)

17. (b)

18. (bl

19. (a)

20. (c)

21. (a)

22. (b)

23 . (b)

24. (ci

25. (d)

(a)

27. (a)

28. (d)

29. (d)

30. (ai

31. (b) 37. (d)

32. (c) 38. (b)

33. (d) 39. (b)

34. (c) 40. (c)

35 .( b) 41. (ai

36. (ci 42. (bl

-

43. (c)

44. (b)

45. (a)

46. (a )

47. (hi

48. (.ci

49. (b)

50. (a)

51. (c)

52. (ai

53. (ai

54. (ai

55. an

56. (a)

57. (b)

58. (a)

59. (ci

60. (bl

2

the

xn ^ ^ ,


24

63. (a), (c)

64. (a), (b), (d)

65. (a), (d)

66. (a), (b), (c), (d)

61. (a), (b), (c)

62. (a), (b)

67. (a, b, c)

68. (b)

69. (c)

71. (b), (c)

72. (a), (b)

73. (b)

74. (c)

75. (c)

76. (b), (d)

77. (b)

78. (a), (b), (c)

79. (b), (c), (d)

80. (a)

81. (b)

82. (d)

83. (a), (c)

84. (a), (c)

85. (c)

86. (b)

87. (c)

88. (b)

89. (b)

90. (a), (b), (c)

5. (c)

6. (c)

1. (c)

2. (d)

3. (b)

4. (b)

7. (a)

8. (c)

9. (b)

10. (b)

70. (b)

SEQUENCES AND SERIES § 3.1. (i) If a is th e first ter m and d is the common difference, then A.P. can be written as a, a+ d, a + 2d,..., a + ( n - 1) d, ... Tn = a + (n- 1) d= I (last term) nth term: d=Tn-Tn-1 where Tn' = I-(n - 1) d nth term from last (ii) Su m of first n terms : S n = - [2a + ( n - 1) d ]

n, = 2 (a

in )

+ /

an d Tn=Sn-Sn--\ (iii) Arithmetic me an for any n positive numbers ai, az, a3,.... an is _ a i + a2 + a3 + - . + a n A n A n are inserted between a and b, then (iv) Ifn arithmetic means Ai, A2, A3 ' b-a ' r, 1 < r< n and /Ao = a, /4 n +1 = b. A r= a + n+ 1

a is th e firs t term an d r is the com mon a, ar, ai 2, ai 3, ar A,..., ar" _ 1 ,.... nth term: Tn = ar n~1 = / (last term)

(i) If

where nth term from last

r=

ratio,

then

G.P .

Tn 1

Tn' =

7

(ii) Su m of first n te rm s :

an d

Sn =

a ( 1 ( 1

_ ^ ,

Sn = an

if r < 1 if r = 1

Su m of infinite G.P. whe n I rl < 1. /re. - 1 < r< 1 =

(I rl < 1)

(iii) Geo metr ic me an for any n positive numb er s b\, b 2 , £>3 G.M. = (biiJ 2b3. ...

(iv) If n ge ome tr ic me ans G1, G 2

bn )

bn is 1/n

G n are inserted betwee n a and

b then

can

be

written

a s

26


r_ ,\ b n+ 1 , Go = a, and Gn +1 = b a (v) : Let Xdenote the figure which do not recur, and suppose them x in number; let /denote the recurring period consisting of /figures, let R denote the value of the recurring decimal; then R = OXYYY....; Gr = a

10xxR = X and

10*

+y

YYY....\

XY xfl =

YYY....;

therefore by subtraction

_

n

XY-X ( 1 0 x + y - 10*)

(i) If

a

is

the

first

a, (a + d) r, (a + 2d) i 2

term,

d

the

com mon

d) z " - 1 Tn=(a + (n- 1) d) (a + ( n - 1)

diff ere nce

and

r the

co mmon

ratio

then

is known a s A.G.P.

nthtermof A.G.P.: (ii) Su m of first n terms of A.G.P. is

c; a drQ-/ 1-1) On — , "t" — (1-0 (iii) Su m upt o infinite te rm s of an A.G .P. is =

— +

0 - 0

[a +( n-1 )d] ^ (1-r)

(1 - f)

2

(

I

1)
We shall use capital Greek letter Z (sigma) to denote the sum of series. (i) w (ii)

(iii) (iv)

1 r= 1 + 2 + 3 + 4 + .... + n =

n

r= 1

(n

+ 1

2

)

zn

=

+ 1) 1 ) _ 1-.22 +, 2„22 +, 0322 +, .... + . „2 n( n+ 1) (2( n+ iv ?^ = n2 _= n(n+1) = Zn2 c ^

r= 1

I

r= 1

6

r3 = 1 3 + 2 3 + 3 3 + .... + n 3 =

Z a= a+ a+ a+ ... n terms

n(n+ 1)

2

l2

= [2 n f

= En 3

= na.

Not e : If nth term of a se qu e n ce is given by Tn = an 3 + bn 2 + cn+ d, then

Sn=

n Z r= 1

n

Tr= a

n

r +b Z r +c Z r= 1 r= 1 r=1 I

n

n r + Z d. /-= 1

If the d iffe renc es of the s uc ce ss ive ter ms of a se rie s ar e in A.P. or G.P., we c an find n th term of the series by the following method : Step (I) : Den ote the nt h term and the sum of the ser ies upto n terms of the series by Tn and Sn respectively. Step (II): Rewrite the given ser ies wi th ea ch term shifted by one plac e to the righ t. Step (III): Subtract ing the abo ve two forms of the series , find T n.

Sequences and Series

(i)

27

an are in H.P. then — , — ,.

If th e s eq u en ce a-i, a2, a3,

n th terms:

ai

a2

an

are in A.P.

1

Tn =

+ (n-

ai

a2 ai ai a 2 a 2 + ( n - 1) (ai - a2) (ii) Harm onic me an for n positive numbers ai, a2, a3,..., a

H (iii) Ifn Harmonic mean Hi,

H2, H3,...,

n I ai

a2

""

n

is

an

Hnare inserted betw een a and b, then

1 1 ^ u (a-b) 6- - 1 - — — L - r aHr (n + ^)ab If A, G, H are respectively A.M., G.M., H.M., between a and pos iti ve th en

— = -+rd where

b both being unequal

and

(i) G2 = AH (ii) A> G> H. and every mean must lie between the minimum and the maximum terms. Note that A = G = H iff all terms are equal otherwise A> G> H.

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. If

a, b, c

1 •Jc+Ja

are

in

1 ' -la+Jb

A.P.,

1

then

If the sum of first

+ Vc

'

ar C m

(a) A. P.

(b) G. P.

(c) H.P.

(d) no defi nite seque nce

2. If a, b, c, d, e,f are in A.P ., then ( e — c ) is equal to : (a) 2 (c - a)

2(d-b)

(b )

(6)2

(c)2 (f-d)

(d-c)

X

X

3. If log 3 2, log, (2 - 5) and log, (2 -1/2) in A.P., then the value of

are

JC is :

(a) 2

(b) 3

(c)4

(d)5

4. If the ratio of the sum s of m and n terms of an 2

2

A.P., is m : n , then the ratio of its mth and nth terms is : (a) (m - 1) : ( «- ! ) (b) (2m + 1): (2n+ 1) (c) (2m — 1): (2n — 1) (d) none of these

n positive integers is -j

times the sum of their squares, then (a)5

(b)6

(c)7

(d)8

n equals :

6. The interior angles of a polygon are in A. P. the smallest angle is 120° and the common difference is 5°. Then, the number of sides of po ly go n, i s : (a )5

(b) 7

(c )9

(d) 15

7. If a, b, c are in A.P. then the equation 2 (a-b) x + (c - a) x +(b - c) = 0 has two roots which are : (a) rational and equal (b) rational and distinct (c) irrational conjugates (d) complex conjugates 8. If the sum of first n terms of an A.P. is 2

(Pn + Qn ), where P, Q arc real numbers, then the common dilTcrcncc of the A.P., is


28 )P-Q (a (c) 2Q

( b)P + Q

17. If a, b, c are digits, then the rational number represented by 0 cababab .. . is (a) cab/990

(d)2 P

f

9.

a" + b" is the A.M. n-I a +bn~ a an d b, then the value of n is : (a) - 1 (b) 0 (c) 1/ 2 (d) 1 if

between

10. Given two numbers a and b. Let A denote their single A.M. and S denote the sum of n a and b then ( S / A ) depends A.M.'s between on : (a) «, a, b (b) n, a (c) n, b (d) n only 11. If x, (2x + 2), (3x + 3), ... are in G.P., then the next term of this sequence is : (a) 27 (b) - 27 (c) 13-5 (d) - 13-5 12. If each term of a G.P. is positive and each term is the sum of its two succeeding terms, then the common ratio of the G.P. is : • VT - n (a) 2 (c)-

"VfT+1

(b) (d)

1-V5

2 J I 13. The largest int erval f or which the series 9 1 + ( x - 1 ) + ( x - 1 ) " + ... ad inf . may be summed, is: (a) 0 < x < 1 (b) 0 < x < 2 (c) - 1 < x < 1 (d) - 2 < x < 2 14. Three numbers, the third of which being 12, form decreasing G.P. If the last term were 9 instead of 12, the three numbers would have formed an A.P. The common ratio of the G.P. is : (a) 1/3 (b) 2/3 (c) 3/4 (d) 4/5

15- The coef fici ent of x 49 in the product

(d) None of these = .1-658, then (a) 208-34 (c) 21216

b)/990

(a) 2 (c) 8

(b )4 (d) No ne of the se

19. The H. M. of two numbers is 4 and their A. M.

and

G.

M.

satisfy

the

relation

2 A + G2 - 27, then the numb ers are : (a) -3 and

1

(b) 5 and - 2 5

(c) 5 and 4

(d) 3 and 6

20 . If £ n = 55 then Z n2 is equal to (a) 385 (b) 506 (c) 1115 (d) 302 5 21. The natural numb ers are grouped as foll ows : {1}, {2, 3, 4}, {5, 6, 7, 8, 9}, ... then the first element of the nth group is : (a) n - 1

2

(b) n

+1 2

(c) ( « - l ) - 1 (d) (n — l) + 1 22. A monkey while trying to reach the top of a pole of he ig ht 12 me tr es ta ke s every ti me a jump of 2 me tr es but slips 1 me tr e wh il e holding the pole. The number of jumps required to reach the top of the pole, is : (a) 6 (b) 10 (c) 11 (d) 12 23. The sum of the l . n + 2. ( n - 1) + 3. (« - 2) + .. . + (a)

series 1 is :

» («+!)(;; + 2 ) ... » («+!)(;/ + 2 ) (b) 3

(c)n

(»+l)(2»+l) 6

» (» + 1) (2/; + 1) 3

in A.P., then2 the roots of the quadratic equation : px + qx + r= 0 arc all real for:

(b) 1

(c) -2500 50

(d) (99c + 10a +

18. If log 2 (a + b) + log 2 (c + d)> 4. The n the minimum value of the expression a + b + c + d is

24. If p, q. r are three positive real numbers are

( x - l) ( x - 3 ) . .. (x — 99) is

16- If (1-05)

(c) (99c + 10a +

2

2

(a) - 99

ab)/990 b)/99

(b) (99c +

f

(1-05)" equ als :

n= I (b) 212-12 (d) 213-16

» ' > I?]"7

(c) all p and r

> 4 V3~(b)

-7

(d) no p and r

: 4 VT

Sequences and Series rr

25. If

29

a + bx b + cx c + dx. -——=-—~ = a — bx b — cx c — dx

then

a, b, c, d are in (b) G.P.

(c) H.P.

(d) none of these H.M.s

Hj and H2

are

r

- a

h

ai a2 ay th en — , — , — ,. ..,

—— ar e in

26. Tw o A. M.s A, and A2; two G.M .s G, and G two

.. . are in H.P . and

n

m=Za

(a) A.P.

and

31. If a, , a2,

2

inserted

be tw ee n an y tw o nu mb er s : th en H ]

1

+ H^ '

/(")

(a) A.P .

(b) G.P.

(c) H.P.

(d) No ne of these

32. The sixth term of an A.P. is equal to 2. The equals (b) G^

1

(C)G,G2/{A\+A2)

value of the common difference of the A.P.

]

+

G2

which makes the product aja

(D){AX+A2)/G,G2.

27. The sum of the products of ten numbers ± 1, + 2, + 3, ± 4, ± 5 ta king two at a time is (a) - 5 5 (b) 55 (c) 165

(a) n - m + 1 : m (c) m : n - m + 1

(b) n - m + 1 : n (d ) n : n - m + 1 .

(d) None of these

33. The so lution of lo g^ - x+ l o g x + log

x + ... + log "yf x = 36 is (a) x = 3

(b)x = 4<3

(c)x = 9

(d) x =

34. If a]t a2, a3,.., an are in H.P. then a\

29. One side of an equila teral triangl e is 24 cm. The mid points of its sides are joined to form another triangle whose mid points are in turn jo in te d to for m still an ot he r tr iang le. Th is pr oc es s co nt in ue s in de fi ni te ly . Th e su m of the perimeters of all the triangles is \A

24cm

a

+ ... + a n ' Q] + + ... + a an ..., —ax + a 2 + •.. + an _ j are in

(a) A.P.

(b) G.P.

(c) H.P.

(d) A.G .P.

(c) H.P.

(d) None of these

(b) 169 cm (d) 625 cm

(b) G.P. (d) None of these

'

0>§

24cm

(a) A.P.

n

35. If the arithmetic progression whose common difference is none zero, the sum of first 3n terms is equal to the sum of the next n terms. Then the ratio of the sum of the first 2n terms to the next 2n terms is

24cm

30. If f + ^ , b, ^ +,C are in A.P., then 1 - ab- be 1 b are in

2

a2 +

< 4

(a) 144 cm (c) 400 cm

least is given

(a) J

(d) - 1 6 5

28. Given that n arithmetic means are inserted be tw een two sets of nu mb er s a, 2b and 2a, b, where a,beR. Supp ose fur ther that mth mean between these two sets of numbers is same, then the ratio, a : b equals

4a5

by

36. Let a, b, c be three positive prime numbers.

a, 7-

The progression in whic h Va, \ b , can be three terms (not necessarily consecutive) is (a) A.P.

(b) G.P.

(c) H.P.

(d) No ne of thes e

37. If n is an odd integer greater than or equal to 1

then

n - ( n - l f + (n-

the

value

-. .. + ( - I ) " "

of 1

l 3 is

30


(a)

(b) (c)

43.

(n + 1) ( 2 » - 1)

(n + 1) ( 2 n + 1)

n

terms

of

the

series

(c) n I1 1 + n

(d) None of these

such that

is

(a) A.P .

(b) G.P .

(c) H.P.

(d) No ne of these

cube s

of

jc"

2

( c )-^(»

2

lal< 1,

24

a

£

b\z=

X c

\b\< 1 a n d l c l < 1,

(a) A.P .

(b) G.P .

(c) H.P.

(d) No ne of these

47. Let an be the nth term of the G.P. of positive G.P .

such

that

+ a< = 216. The n a, =

numbers. Let

100

£ a2n= a and n=1

(a) a / p

an

A.P.

16

= 147, then

(a) 96

(b) 98

(c) 100

(d) No ne of these

=

=

(d) Vp /a "

aua2,

a3,.., a„ are real numb ers 48 . If a, = 0 and i then the such that I a, I = la,_! + II for all and

Arithmetic mean of the numbers has value JC where

a, + a 4 + a-j + .. . + a

-1

(b) p / a

(c) Vot/p

(d) None of these is

100

£ n=1

such that a * fi, then the com mon rat io is

(b) 10

/ \ 54 (c) 1 OX — < an >

+ 2) (n2 + 4)

then JC, y, z are in

n (n2 + 1) (3n + 4) (c) 24 (d) None of these

108

+12)

where a, b, c are in A.P. such that

n (n - 1) (3 n + 2)

(a) 12 or

2

n = 0

24

2

2

46. If jc — Z a,y=

- n ) is

2

— = — and a

+16) (n

(d) None of these

n (n + 2) (3/t+l)

{an} be

are

in the polynomial

(j c- 1 ) ( j c - 2 ) (JC- 3) ... (JC

41. Let

numb ers

3

n( n- l )

(c) 2 (n + 1) . (2« + 1) (d) None of these 40. The coefficient of

natural

(a) i n 3 (n2 + 1) (n 2 + 3)

16

(b)

the

the sum of the numbers in the nth group is

(b) T7 n ("

sum is n (n + 1)

(a\ +a]+

grouped as l 3 , (2 3 , 3 3 ), (4 3, 5 3 , 6 3 ) , .. . then

when n is even. When n is odd, the

4?. If

(b) n (n + 1)

45. The of

l 2 + 2.2 2 + 3 2 + 2.4 2 + 5 2 + 2.6 2 + ...

(b)

series,

then ah a2..., a„ me in

' 2

sum

the

... + a2n) 2 < (aâ2 + a2a3 + ... + ...+ a n_ ]an)

(b)V3",|

•45 +

2

of

(a) n

(a] + a\+ ...+a2n_,)

4

39. The

infinity

44. If ah a2,.., an are n non zero real numbers

A.P. then the sines of the acute angles are

(a)

to

(n-\T(2n-l)

38. If the sides of a right angle d triangle fo rm an

(a)

sum

1 f + 3 | 1 - - + ... is n

(d) None of these

(d)

The

(a) jc < - 1

(b) JC< -|

(d) jc = -

1

alt a2,..,

an

Sequences and Series

31

a2, a 3 (aj > 0) are in G.P. with common

49. If

r, then the value of

ratio

inequality 9a, +5a

3

a\

g\g2n

> 14a 2 holds can not lie

is equal to

in the interval

(d)

1

9 '

x 203in the expa nsion of

50. The coefficient of i

-2)(x

- 3 ) ... (x 20 - 20) is

(a) -35 the

(d)15 n terms 1+2 + 3

of

1 +2

of

the

series

3 3 + .. is S. then S„ 1 + 2 + 3 exceeds 199 for all n greater than (a) 99 (b) 50 1

(c) 199

(d) 100 1 , > 0 4 - 2 sin2x j: The numbers 3 , 14, 3 fo rm first three terms of an A.P., its fifth term is equal to (a )- 25 (c) 40

(b) -1 2 (d) 53

71

,n + l n. 2' 22n-l

(d) None of these 55. If a, ah a2, a3,..., a2n, b are in

=

89tt 4 (c)

(d)

90

71

45 / 46

8p 16 128 T 57. Let S = - + — +... + — 5 65 (a) 5 = (b)5 =

, then

1088

545 545 1088 1056 545 545 1056

58. If sin 6, V2 (sin 0 + 1 ) , 6 sin 6 + 6 are in G.P. then the fifth term is (a) 81 (c) 162

(b) 82 -12 (d) No ne of the se

59. If In (a + c), In (c - a), In (a-2b A.P., then

+ c)

are in

(a) a, b, c are in A.P. (b) a2, b2, c2 are in A.P. (c) a, b, c are in G.P. (d) a, b, c are in H.P. 60. Let «], a2, a 3 , ... be in an A.P. with common difference

A.P.

S2n> b are in G.P. and

H. M. of a an d b then

(b)

96

(d) No ne of these

(c) n. 2"

a

(a)

(d ) S =

54. In a sequenc e of ( 4 n + l ) terms the first (2n+ 1) terms are in A.P. whose common difference is 2, and the last (2n + 1) terms are in G.P. whose common ratio is 0-5. If the middle terms of the A.P. and G.P. are equal then the middle term of the sequence is 2n + 1 n. 2 (a)

(b)

11 CA T • U 11 1 5o It is given that - 7 + + - 7 + ... <0

(c ) S =

53. If 0-2 7, x, 0-7 2, and H.P., then x must be (a) rational (b) irrational (c) intege r

(d)'

, 1 1 1 , then — + —r + —r + .. . to °° is equal to

(b) 21

(c) 13 51. If

+ an + 1

gn gn + \

(b) 2 nh

(c) nh

A

\)(x

t an

g2g2n-\

(b)

(c)

(JC-

, a2 + a2n-\

2n

(a) fl.oo) r

+a2n

r for which the

and

h is the

not

a

multiple

of

3.

Then

maximum number of consecutive terms so that all are primes is (a) 2 (c) 5

' (b ) 3 (d) infi nite

32


Each question, in this part, has one or more than one correct answer(s). a, b, c, d corresponding to the correct answer(s). 61. If tan 'JC, tan 1 y, tan 1 Z are in A.P. and x, y, z are also in A.P. (y being not equal to 0, 1 or -1), then


(c) a5+ c> 2

2b5

2

(d) a + c > 2b2 68. The sum of the produc ts taken tw o at a time

(a) x, y, z are in G.P.

of the numbers 1, 2, 2

(b) x, y, z are in H.P. (c)x = y = z

(d) (x-y)2

+ (y - z)

2

+ (z - x) 2 = 0

62. If d, e,f are in G.P. and the two quadr atic 1

ax + 2 bx +=c 0 equations and 9 dx + 2ex + / = 0 have a comm on root , then d e f . n are in H. P. (a) — — ' b \ d e f (b) — , -f , are in G.P . a b c (c) 2 d b f = aef+ cde (d) b df= ace"

p, q, r are in H.P. 63. If three unequal numbers and their squares are in A.P.; then the ratio p : q : r is (a) 1 — V3~: — 2 : 1+V3~ (b) 1 : V2 : - V 3 " (c) l:- V2:V 3 (d) 1 + V3~: - 2 : 1 - V3~ 64. For a positive

intege r n, let a (n) = 1 + 1 / 2

+ 1/ 3 + 1 / 4 + ... + 1/ (2 " - 1) then (a) a (100) < 1 0 0 (c) a (200) < 100

(b) a (100) > 100 (d) a (200) > 100

65. The p\h term Tp of an H.P. is q (p + q) and 17th term Tq is p (p + q) when p > 1 , q > 1 then

(a)Tp

+q

= pq

(b ) Tpq = p + q

(c )Tp + q>Tpq (d) Tpq >Tp + q 66. If the first and (2n - l) th ter ms of an A.P. , a G.P. and a H.P. are equal and their nth terms are a, b and c respectively, then (a)a = b = c (b )a + c = b (c)a>b>c

(d) ac = b

2

67. If a, b, c be three unequal positive quantities in H.P. then , , 100 , 100^ 100 )a +c (a >2b (b) a 3 + c 3 > 2b*

, 2 3 ,.., 2"~ 2 , 2" ~ 1 is

(b)

2 2 "-2'

(d)^

2

!

+ j

"- •2" + f

69. The sum of the infinite term s of the sequ ence 5 9 13 + + ... is : 3 2 . 7 2 I 2. I I 2 l l 2 . 15 2 (a) (c)

70.

2

2

(b)

18

36

54

The sum of 1 1 1.2.3.4 ' 2.3.4.5

n +

terms 1 3.4.5.6

of

the

series

is

w(n +6 n + 11) (a) 18 (n + 1) (n + 2) (n + 3)

(b)

n + 6 18 (n + 1) (n + 2) (n + 3)

1 (c) — w 18 3 (n + 1) (n + 2) (n + 3) 1 6 2 (n + 1) (n + 2) (n + 3)

71. Let a, b, c be positive real numbers, such that

bx2 + (V(a + c) 2 + 4b2) x + (a + c) > 0 V x e R, then a, b, c are in (a) G. P.

(b) A. P.

(c) H. P.

(d) Non e of thes e

72. (1 ^ ) z + (2 j)

2

the sum is : 1390 (a)

9 1990 (c) 9

+ 3 2 + (3 | ) 2 + ... to 10 terms,

(b)

1790 9

(d) None of these

73. The consecutive odd integers whose sum is 45 2 - 21 2 are :

33 80. If a, b, c

an >

74 .

bn >

are in H.P., then the value of

I

i_i

b

c

1

1

- +- - T

1

,IS

a

3_

an = (x) w2 bn

r>

(x) U-

\

ne N

3

2

x+y

7S

-

81. 2

x+y

xY2

(d)

bn

-

2 >,

(a)

bn

If 1.3 + 2. 3 2 + 3.3 3 + .. .+n .3" =

a

x-y

2 ,

(c)

r,

ca

(d) None of these

a2 a 3 ... an (a)

J2__±

T

c"

n( n+ l) ( n + 2)

(c)"C

(2n —I) 3" + b . . ... ^ then (a, b) is :

(b) Z n (d) " +2C3

3

82. If an A.P.,

a7 = 9 if a , a

2

a

is

7

lea

st, the

common difference is (a) (n -2 , 3)

(b )( n- l, 3)

(c ) (n, 3)

(d) (n + 1, 3)

(b)i

33

43 20

76. If x, I x + 1 I, I x — 1 I are t he th ree term s of an A.P. its sum upto 20 terms is : (a) 90 or 175

(b) 180 or 350

(c) 360 or 70 0

(d) 720 or 1400

23

(a) 20

83. If cos cos x and cos (x + y) are in H.P. then cos x se c y/2 is (a) 1

2

77. I f S n , ~ l n

, Z n

3

are in G.P. then the

84. If value of n is : (a) 3

78 . If

£

« (« + 1) (n + 2) (n + 3)

r= 1

8

denotes Lim

the

rth

(a ) z = x

term

of

a

series,

^

85. If

(d )x = y

bl,b2,b3(bl

>0)

are

=z

three

successive

terms of a G.P. with common ratio r, the

then

value

of

r

for

which

the

inequality

b3 > 4b2 - 3by holds is given by (b>i

79. Given that 0 < x < ti/4 and rt/4 < y < tan 2 * x = a,

(a) r > 3

(b) r < 1

(c) r = 3-5

(d)

BG- If log x a, a1/2 and log

(d) 1

k=0

(b) x = y~

(c) z~ 3 = y

X — is : r = 1 tr

( - \f

(d) V5

1, log,.*, log j) 1 , - 15 lo gj Z are in A.P.,

then (d) non existent

(c)2

(b) <2

(c) <3

n/2 and b,

i =0

r=5-2

fc x are in G.P., then

x is

equal to (a) log

a

(log/, a)

(b) log

a

(log, a) - log a (log, b)

(c) - log a (log a b) (d) log a (log, b) - lo g a (log, a) 87. If a, b, c are in H.P., then (b)a + (d)

(c)

a

b

ab

b-ab

ab a + b- 1

(a)

b+c-a'c+a-b'a+b-c b

b-a

b-c

are in H.P.

34


(c)«

-f,f 2

(d) , 7

,c-^ 2

, +cc+aa

are inG.P 2

.

Vm + V ( m - n )

ar e in H. P.

«

if

88 . If the ratio of A. M. between two positive

a an d b to their H.M. is a: equal b then is to

:mn;

real numbers

£

r=1 2

+ cn (a) a = 1 / 2

r

( r + i) (2

r

. , + 3) = an + bn

+ dn + e, then (b )6 = 8 / 3

(e) c = 9 / 2

(d )e = 0

V(w - n) - AIn If l, l

Vn~-

- n)

0g9

(3

1

1

+ 2) and log 3 (4-3" - 1) are in

A.P., then x is equa l to

y[ m+ -l (m -n )

(a) lo3g 4

Vm - V(m - n)

(c)

(b) log

i _ iog3 4

4

3

( d ) i o g 3 (0 75)

Practice Test Tim e : 30 M in.

MM : 20

(A) There are 10 parts in this question. Each part has one or more than one correct n 1 3 + 90

21, where

integral part of

x, then k =

*

1. If X 71

[x] denotes

(a) 84

'(b) 80

(c) 85

(d) non e of th es e

If x e |1, 2, 3, ..., 9} an d

the

answer(s). [10 x 2 = 20]

The series of natural numbers is divided into groups : 1; 2, 3, 4; 5, 6, 7, 8, 9; ... and so on. Then the sum of the numbers in the nth group is 2

fn{x) =xxx ...x

(n

2

digits), then /"„ (3) + fn(2) =

3

2

(a) (2n - 1) (n - n + 1) (b) n - 3n + 3n - 1 (c) n + (n - l ) 3 (d) n + (n + l ) 3 6. The nu mb er s of divi sons of 1029, 1547 an d 122 are in

(a) 2f2n (.1)

(a) A.P.

(b) G. P.

(b)/n(l)

(c) H.P .


(O/hnU)

(d) -

fin

The coefficient of x15 in th e product

(4) whose common difference in non of first 3n terms in equal to n terms. Then the ratio of first 2n terms to the next 2n

(1 - X ) ( 1 -2x)(1

2

-2

X ) ( 1 -23.X)

...

(1 - 2l5.x) is (a) 2 1 0 5 - 2 121 (c)2

120_

m

(b)2

2 104

- 2 1 05

105

(d)2

- 2 104

2

1/2

(b) 173

(c) 174

(d) 1/5

a, b, c are in real numbers abc = 4, then minimum value of b 1

(b) 3 (c) 2

(d) 1/2

8. The roots of equation x + 2 (a - 3) x + 9 = 0 hi, h 2 h20, lie between -6 and 1 and 2, [a] are in H.P., where [a] denotes the inte gra l p ar t of a, a nd 2, a 1 ( a2, ...,a20, [a] are in A.P. then

a3his

=

(a) 6

(b) 12

(c) 3


Sequences and Series 40

r \ n frc-l Z k + 2. Z k kl = k=l J \ - -i \ (

Value of L = lim n -> fr ix — 2 3. Z k + ... k=l

r2

\

1.

\

10. If a , P, y, 5 are in A.P. and where

is

+71.1

f(x)=

-

(a) 1/24 (c) 1/6

(b) 1/12 (d) 1/3

x+a x+p X+Y

x+P X + Y

x+b

Jq / (X) <2*

x+a-y x -1 xp+ 5

d is :

then the common difference (a) 1

(b) - 1

(c) 2

(d) - 2


must be 100%

Answers

l.(a) 7. (a) 13. (b)

2.(d) 8. (c)

3. (b) 9. (d) 15. (c)

4. (d) 10. (d)

5. (c) 11. (d)

16. (c)

17. (d)

6. (c) 12. (a) 18. (c)

19. (d) 25. (b)

14. (b) 20. (a)

21. (d)

22. (c)

23. (a)

24. (a)

26. (d)

27. (a)

28. (c)

29. (a)

30. (c)

31. (c)

32. (c)

33. (d)

34. (c)

35. (a)

36. (d)

37. (a)

38. (a)

39. (a)

40. (b)

42. (b)

43. (a)

44. (b)

45. (a)

46. (c)

41. (a) 47. (a)

49. (b)

50. (c)

52. (d)

53. (a)

54. (a)

55. (a)

56. (a)

51. (c) 57. (a)

58. (c)

59. (d)

60. (b)

63. (a), (d)

48. (c)

61. (a), (b), (c), (d)

62. (a), (c)

65. (a), (b), (c)

66. (c), (d)

67. (a), (b), (c), (d)

68. (d)

64. (a), (d)

70. (a), (c)

71. (b)

72. (d)

73. (d)

74. (b)

75. (d)

76. (b)

77. (a)

78. (c)

79. (c), (d)

80. (a), (b), (c)

81. (a, d)

82. (c)

83. (b)

84. (a), (b), (c), (d)

86. (a), (b)

87. (a), (b), (c), (d)

69. (d)

85. (a), (b), (c), (d)

88. (c)

89. (a), (b), (c), (d)

90. (c, d)

1. (b)

2. (c)

3. (d)

7. (a)

8. (b)

9. (a)

4. (a) 10. (a), (b)

5. (a), (c)

6. (a)

PERMUTATIONS AND COMBINATIONS n distinct objects in a row taking

§ 4.1. The number of ways of arranging

r(0 < r< n) at a time is denoted

by nPror P(n, 0n

and

Pr= n(n- 1) (n- 2) ... (n- r+ 1) /?!

" (n - 0 ! n Note that P0 = 1 , n Pi = n an d nPn~ i = nPn= n \ The number of ways of arranging n distinct objects along a circle is

nbe ad s alo ng a circular wire is

The number of ways of arranging

(n- 1)! ^ '

The number of permutations of n things taken all at a time, p are alike of one kind, ano the r kind and r a r e alike of a third kind an d the res t n - (p + q + r) ar e all diffe rent is

q are alike of

n p! girl '

The number of ways of repeated any number of times is

n distinct objects taking

r of them at a time where any object may be

n r.

The number of ways of selecting

r(o< r< n ) objects out of n distinct objects is denoted by

nCr

C(n, r> and n c _ n( n- 1) (n -2 ) ... (n -r +1 ) r. (r- 1). (r- 2)... 2-1

n r\(n-i)\ if r> n, then

nCr =

0

The number selecting at least The number of combinations of § 4.9. The number of selecting robjectsfrom n~ kCr -k-

n) n

The number

r

_nPr

"n" n

2n n+ 1 r

r= 0,1, 2, ..., n) when k

n

k k

Cr.

q + r+ s)

q n=

m . . . m%*

(p+q+1)

q, r

or

Permutations and Combinations

37

_(P + g+/)!

p\q\r\ p

p 3p I

3 ! (p I) 3

(p!) 3 '

n

n n

n+ r~ 1

Cr-

n~

r

Cr-

,r«

• 1

1

1

1

xf " =

,

n+r~ 1

(

,ni

Cr.

(i.e., 1+ k+ x+ x k) k

I (i.e., 1+ k+

xr X

r\

1 +

T7

x°~

X +

2

2!

+1 )

+

-

jn

+

7!

_

x)-

1+

1

xk

TT + f l + ^ T

_£ y

_ ^y-

.

_

r= 1 n

o

n

(n + n r— 1

(n

r) (p

i).

)-

Cr-

Objective Questions

n r"- rCi

(r-

r

rC2

rCr -

n n'' Ep(n !) =

.P.

n' 1

n' r

y .

n'

V.

41 «»• 4T

n<

.P

s

.

ps+

n an = ( n - 1 )! (ai + a 2 + ... + an)

an ,

HM,

J (1 0" ~ 1)

n Xs a1

(

X, ai

Ci X +

2

ai

C 2 X + ... +

a2

0

C a , X ") X ( Ci X +

n

"C2-n

m

m

n m

(m nC3

n

"C4

"C5

- mC3

K2

2

012

2

C2 X + ...+ Ca2 x" ) X ( a3 Cl X+ a3 C2 X2 + ... + a3 Ca 3 X"3) X ....


39

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. Whe n

simplified , 5

47

c4+ z

52 -n

n=1

52

^ykti- If 7 points out of line

= 10, nCr= 45 and

n

Cr+,

= 120 the n

vX

r equals (d)4

10

C,.,>2.

x .which

(b )8

(c) 9

(d) 10

(a) 30

(b) 45

(c) 90

(d) 120

minimum value of

n ! (21 - n) ! is

(a) 9 2! !

(b) 10 1 1 1 !

(c) "Pr

(a) 16

(b) 24

(c) 28

(d) 56

of

poin ts

of

(a) 3* 4

(C) P 3

The maximum number of points of intersection of 8 straight lines, is (d) 56

v/9. The maxi mum n umbe r of points into which 4 circles and 4 straight lines intersect, is (b) 50 (d) 72

(b) 1023

(d) 10 !

(b) r (d) "C

r

(b) 4 (d)4C3

/ T h e numb er of ways in which the letters of ^

(b) 16

(a) 26

9

15. The number of ways in which three students of a class may be assigned a grade of A, B, C or D so that no two students receive the same grade, is:

(d) 21 ! number

(c) 56

which

(d) 1260

(a )n

intersection of 8 circles, is

28 Ac)

in

different toys can be distributed among three different children so that the youngest gets 4,

The number of ways in which r letters can be po st ed in n letter boxes in a town, is :

6. If n is an integer between 0 and 21, then the

(a ) 8

(d) 205 of ways

pe rs on s ar e not to sit to ge th er , is : (a) 120 (b) 48 0 (c) 600 (d) 720

jo in in g th e an gu la r po in ts of de ca go n, is

max imum

thus

The number of ways in which 7 persons can be se at ed at a ro un d ta bl e if tw o pa rt ic ul ar

(b) 20

(c) 20 !

number

(a) 55

(c) 28 (d) 40 The number of triangles that can be formed

^T^The

(c) 201 The total

(c) 2 1 0

that can be drawn

in an octagon is (a) 16

triangles

\Z. Ever y one of the 10 avail able lam ps can be switched on to illuminate certain Hall. The total number of ways in which the hall can be illuminated, is

Cj is

(a) 7

^J kf The n umber of diagonals

of

(b) 185

(c) 1240

3. The least positive integral value of satisfies the inequality 10

12 lie on the same straight number

the middle gets 3 and the oldest gets 2, is (a) 137 (b) 23 6

(b)2

(c)3

the

(a) 19

( d ) 5 2 C4

(a ) 1

than

formed, is

(b)49c4

C5

2. If "Cr_,

expression

C 3 equals

(a ) 4 7 C 5 (c )

the

the word AR RA NG E can be made such that bo th R ' s do no t com e to ge th er is : (a) 900 (b) 1080 (c) 1260

(d) 1620

Six identical coins are arranged in a row. The total number of ways in which the number of heads is equal to the number of tails, is

40


(a) 9 (c) 40

handshakes is 66. The total number of

(b) 20 (d) 120

pers ons in the ro om is

If 5 parallel straight lines are intersected by 4 parall el st ra ig ht lin es, the n th e nu mb er of pa ra ll el og ra ms th us fo rm ed, is : (a) 20 (b) 60

J

(c) 101

(d) 126

27

(d) 18

20. The

sides AB, BC an d CA of a triangle ABC have 3,4 and 5 interior points respectively on them. The number of triangles that can be constructed using these interior points as vertices, is (a) 205 (b) 208 (c) 220 (d) 380 Total number of words formed by using 2 vowels and 3 consonents taken from 4 vowels and 5 consonents is equal to (a) 60 (b) 120 (c) 720

J

*

Twenty eight matches were played in a football tournament. Each team met its opponent only once. The number of teams that took part in the tournament, is (a ) 7 (b) 8 (c) 14 (d) No ne of these Ever ybod y everybody

in a room else. The

shakes ha nd with total numebr of

(d) 14

(a) 224

(b) 280

(c) 324

(d) 405

The total number of 5-digit telephone numbers that can be composed with distinct digits, is (a) >

2

(O ,0c5

(b) '°P

5

(d) None of these

A car will hold 2 persons in the front seat and 1 in the rear seat. If among 6 persons only 2 can drive, the number of ways, in which the car can be filled, is

J

(a) 10

(b) 18

(c) 20

(d) 40

In an examination there are three multiple choice questions and each question has 4 choices of answers in which only one is correct. The total number of ways in which an examinee can fail to get all answers correct is

*

(d) No ne of these

Ten different letters of an alphabet are given. Words with five letters (not necessarily meaningful or pronounceable) are formed from these letters. The total number of words which have atleast one letter repeat ed, is (a) 216 72 (b) 30240 (c) 697 60 (d) 99748 A 5-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways, this can be done, is (a) 216 (b) 240 (c) 600 (d) 720

(b) 12

(c) 13

The total number of 3-digit even numbers that can be composed from the digits 1, 2, 3, ..., 9, when the repetition of digits is not allowed, is

The total number of numbers that can be formed by using all the digits 1, 2, 3, 4, 3, 2, 1 so that the odd digits always occupy the odd place s, is (a )3 (b) 6 (c)9

(a) 11

30

(a) 11

(b) 12

(c) 27

(d) 63

The sum of the digits in the unit's place of all the numbers formed with the digits 5, 6, 7, 8 when taken all at a time, is (a) 104 (b) 126 (c) 127

(d) 156

Two straight lines intersect at a point O. Points A|, A2,.., An are taken on one line and poin ts B\, B2,..., Bn on the other. If the point O is not to be used, the number of triangles that can be drawn using these points as vertices, is: (a)n (n-l )

(b)n(n-

l)

(c) n (n - 1)

(d) n ( « - l)

2 2

How many different nine digit numbers can be fo rm ed fro m the nu mb er 22 33 55 88 8 by


41

rearranging its digits so that the odd digits occupy even positions ? (a) 16 (b) 36 (c) 60 /

33.

n For 2 < r < n, r (a) (c)2

' n + P r- 1 f n+ , r

V

\

(d) 180 r n + 2

K (b) 2

/

r

\

\

+

- 1

J

r ,

" J

~

2

;

r+

( n^+, 2T A

(d) r v r / 34. The number of positive integers satisfying the inequality - , r\r\ • tl + \ s-i n + \ /-< Cn _ 2 Cn _ ! < 1 0 0 I S (a) Nin e

(b) Eigh t

(c) Fiv e (d) No ne of these 35. A class has 21 student s. The class teacher has be en as ke d to make n groupsof r students each and go to zoo taking one group at a time. The size of group (i.e., the value of r) for which the teacher goes to the maximum num ber of times is (no group can go to the zoo twice) (a) 9 or 10 (b) 10 or 11 (c) 11 or 12 (d) 12 or 13 36. The number of ways in which a score of 11 can be made from a through by three persons, each throwing a single die once, is (a) 45 (b) 18 (c) 27 (d) 68 37. The number of positive integers with the pr operty that th ey can be ex pr es se d as the sum of the cubes of 2 positive integers in two different way is (a) 1 (b) 100 (c) infi nite (d) 0 38. The number of triangles whose vertices are the vertices of an octagon but none of whose sides happen to come from the octagon is (a) 16 (b) 28 (c) 56 (d) 70 p copies of 39. There are n different books and each in a library. The number of ways in which one or more than one book can be selected is (a) p" + 1

(c)(p+\)n-p

p+l)" (b)( -l

(
40. In a plane there are 37 straight lines, of A and 11 which 13 pass through the point pa ss th ro ug h th e po in t B. Besides, no three lines pass through one point, no lines passes through both points A and B, and no two are par allel, the n th e nu mb er of inte rs ec ti on poin ts the lines ha ve is equa l to (a) 535 (b) 601 (c) 728

(d) 963

41. We are required to form dif fere nt word s with the help of the letters of the word INTEGER. Let tii\ be the number of words in which / and N are never together and m2 be the number of words which being with / and end with R, then m]/m2 is given by (a) 42 (c) 6

(b) 30 (d) 1/30

42. If a denotes the number of permutations of x + 2 things taken all at a time, b the number of permutations of JC things taken 11 at a time and c the number of permutations of JC — 11 things taken all at a time such that a = 182 be, then the value of JC is (a) 15

(b) 12

(c) 10 (d) 18 43. There are n points in a plane of which no three arc in a straight line ex ce pt 'm ' which arc all in a straight line. Then the number of different quadrilaterals, that can be formed with the given points as vertices, is n-m+I (a) "C 4 - m C 3 ' Ci — Ca (b)"C (c)

n

4-

C 4 - m C 3 ( m - n C,)-

(d) C 4 + C 3 .

"r

Ci

44. The number of ordered triples of positive integers which are solutions of the equation JC + y + z = 100 is (a) 5081

(b) 600 5

(c) 4851

(d) 4987

45. The number of numbers less than 1000 that can be formed out of the digits 0, 1, 2, 4 and 5, no digit being repeated, is (a) 69 (b) 68 (c) 130 (d) Non e of these

42


46 . A is a set containing

n elements. A subset P,

is chosen, and A is reconstructed by replacing

Px. The same process is

the elements of repeated for subsets The P,,P

number 2,

P\,P2,

of

... Pm with m > 1.

ways

of

...,Pm so that P, u P

2

choosing

u ... uPm

=A

is (a) (2

m

- \) m n

(b)

(2n

— 1 )m

+

(c)'" "Cm (d) Non e of these 47. On a railway there are 20 stations. The number of different tickets required in order that it may be possible to travel from every station to every station is (a) 210

(b) 225

(c) 196

(d) 105

48. A set containing n elements. A subset P of A is chosen. The set A is reconstructed by replacing the element of P. A subset Q of A is again chosen. The number of ways of choosing P and Q so that P n Q = (j) is (a) 2 2 " - 2"C„ (c) 2" - 1

(b) 2" (d) 3"

49. A fat her with 8 childre n takes 3 at a time to the zoological Gardens, as often as he can without taking the same 3 children together more than once. The number of times he will go to the garden is (a) 336 (c) 56

(b) 112 (d) Non e of these

50. If the (n + 1) numb ers a, b, c, d,.., be all different and each of them a prime number, then the number of different factors (other than 1) of am. b. c. d ... is (a ) - m2" (c) (m + 1) 2"- 1

(b)

The letters of the word SURITI are written in all possible orders and these words are written out as in a dictionary. Then the rank of the word SURITI is (a) 236

(b) 24 5

(c) 307

(d) 315

The total number of seven-digi t then sum of whose digits is even is (a) 9 x 106 (c) 81 x 10

(b) 45 5

x 10

(d) 9

x 10

numb ers

5

5

In a steamer there are stalls for 12 animals and there are cows, horses and calves (not less than 12 of each) ready to be shipped; the total number of ways in which the shipload can be made is (a) 3

12

(b) 12

(O ,2P3

(d)

,2

C3

The number of non-negative integral solution of X] + x2 + x3 + 4x 4 = 20 is (a) 530

(b) 532

(c) 534

(d) 536

The number of six digit numbers that can be formed from the digits 1, 2, 3, 4, 5, 6 and so that digits do not repeat and the terminal digits are even is (a) 144

(b) 72

(c) 288

(d) 720

Given that n is the odd, the number of ways in which three numbers in A.P. can be selected from 1, 2, 3,4,.., n is (a)

(w -i r

(b)

(n+lT

(m+ 1) 2" (d) No ne of thes e

A is a set containing

n elements.

A subset P

51. The nume br of selec tions of fou r letters from

of A is chosen. The set

the letters of the word ASSASSINATION is

replacing the elements of

(a) 72 (c) 66

is again chosen. The number of ways of choosing P an d Q so that P r\Q contains

(b) 71 (d) 52

52. The number of divisors a number 38808 can

A is reconstructed by P. A subset Q of A

exactly two elements is

have, exc ludin g 1 and the number itself is :

(a) 9. nC2

(b) 3" - "C

(a) 70

(b) 72

(c) 2. "Cn

(d) Non e of these

(c) 71

(d) No ne of thes e

2


43

60. The num ber of times the digit 5 will be written when listing the integers fr om 1 to 1000 is

(b) 272 (d) None of these

(a) 271 (c) 300

Each question, in this part, has one or more than one correct answers). a, b, c, d corresponding to the correct answer(s).

67. The re are n seats round a tabl e mar ked 1, 2, 3, ..., n. The number of ways in which m (< n) persons can take seats is

61. Eight straight lines are drawn in the plane such that no two lines are parallel and no three lines are concurrent. The number of parts into which these lines divide the plane, is (a) 29

(b) 32

(c) 36

(d) 37

(a )nPm (c)"C

(b) 6

(c) 6 !

(d ) 6 C 6

of the integer 240 is (b )8

(c) 10

(d) 3

numbers are to be formed using only the three digits 2, 5 & 7. The smallest value of n for which this is possible is : (a)6 (b)7 (c)8

(d)9 position

v=x'i+yj

vector

+ zh

of

a

point

P

x,y,zeN

when

is and

a = 1 + j + %. If r . a = 1 0 , The numbe r of po ss ib le positi on of P is (a) 36

9

(c) 66

(d) C 2

66. Sanjay has 10 friends among whom two are married to each other. She wishes to invite 5 of the them for a party. If the married couple refuse to attend separately thencan the invite numberfive of different ways in which she friends is (a) S C 5 (c)

10

(b) 2 x 8

C5 - 2 x C 4

8

C3

(d) Non e of these

d

)

j

(b) 310 (d) 398

In a certai n test, ther e are n ques tio ns. In this test 2 n~' students gave wrong answers to at least i questions, where i= 1, 2, 3, .. , n. If the total number of wrong answers given is 2047, then n is equal to (a) 10 (b) 11 (c) 12 (d) 13

71. Th e exp one nt of 3 in 100 ! is (a) 12 (b) 24 (c) 48 (d) 96 72. The numbe r of integral solutio ns x\+x2 + x?l = 0 with X j > - 5 is (a) 34 (c) 136

(b) 72

(

69. The numebr of rectangles excluding squares from a rectangle of size 15 x 10 is : (a) 394 0 (b) 494 0 (c) 594 0 (d) 6940 70

64. An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit

65. The

mxm!

(a) 165 (c) 295

63. Num ber of divisors of the for m 4 n + 2 (n> 0) (a) 4

(b) nCm x (m - 1) !

68. If a, b, c, d are odd natural numbers such that a + b + c + d = 20 then the num ber of v alues of the ordered quadruplet (a, b, c, d) is

62. The number of ways of painting the faces of a cube with six different glours is (a) 1

for each question, write the letters

of

(b) 68 (d) 50 0

73. The number of ways in which 10 candidates Ai,A2,—,Al0 can be ran ked so that A, is always above A 2 is 10 ! (a) (c) >

W

(b) 8 ! x 2

C2

(d) '°C2

74. If all permutations of the AGAIN are arranged as fiftieth word is (a) NAA GI (b) (c) NAA IG (d)

letters of the word in dictionary, then NAG AI NAI AG


44

80. Seven d iffe ren t lectur ers are to delive r lectures in seven periods of a class on a pa rt ic ul ar da y. A, B an d C are three of the lectures. The number of ways in which a routine for the day can be made such that A delivers his lecture before B, and B before C, is (a) 210 (b) 420 (c) 840 (d) No ne of thes e

75. In a class tournament when the participants were to play one game with another, two class players fell ill, having played 3 games each. If the total numebr of games played is 84, the number of participants at the be gg in in g wa s (a) 15 (C) 6C2

(b) 30 (d) 4 8

76. The n umbe r of ways of distributing

10

diff ere nt books a mon g 4 studen ts (Si - S 4 ) such that

and S2 get 2 books each and

S 3 and S 4 get 3 books each is (a) 12600 (C)

Q

(b) 252 00

82. Th e numbe r of zer os at the end of 100 ! is (a) 54

(b) 58

(c) 24 2 !2!3!3!

77. The number of different ways the letters of the word VECTOR can be placed in the 8 bo xe s of th e give n be lo w su ch that no ro w empty is equal to

(a) 26

81. If 33 ! is divisible by 2" then the m ax imu m value of n = (a) 33 (b) 32 (c )3 1 (d) 30

(b) 26 x 6 !

(c) 6 ! (d) 2 ! x 6 ! 78. In the next world cup of cric ket there will be 12 teams, divided equally in two groups. Teams of each group will play a match against each other. From each group 3 top teams will qualify for the next round. In this round each team will play against others once. Four top teams of this round will qualify for the semifinal round, when each team will play against the others once. Two top teams of this round will go to the final round, where they will play the best of three matches. The minimum number of matches in the next world cup will be (a) 54

(b) 53

(c) 52

(d) Non e of these

O. Points AUA2, ..,A„ 79. Two lines intersect at are taken on one of them and B\, B 2,.., Bn on the other the number of triangles that can be drawn with the help of these (2n + 1) points is (a) n

(b) n 2

(c) n

(d) ;/ 4

(d) 47

83. The maxi mum pe rm ut at io ns of EARTHQUAKE is

number 4 lett er s

(a) 1045

(b) 21 90

(c) 4380

(d) 2348

of of

differ ent th e wo rd

84. In a city no pers ons have identic al set o f teeth and there is no person without a tooth. Also no person has more than 32 teeth. If we disregard the shape and size of tooth and consider only the positioning of the teeth, then the maximum population of the city is (a) 2 32

(b) 2

32

(c) 2- 2

(d) 2

32

- 1

32

- 3

85. Ten persons, amongst whom are A, B & C are speak at a function. The number of ways in which it can be done if A wants to speak be fo re B, and B wants to speak before C is 10 '

(a)— b

(b) 21 87 0

(c)f^

(d) >

7

86. The num ber of ways in whic h a mixe d double game can be arranged from amongst 9 married couples if no husband and wife play in the same game is (a) 756

(b) 1512

(c) 3024

(d) Non e of these

87. In a colleg e examin ation , required to answer 6 out of which are divided into two containing 5 questions, further

a candida te is 10 questions sections each the candidate


45

is not permitted to attempt more than 4 questions from either of the section. The number of ways in which he can make up a choice of 6 questions is (a) 200 (b) 150 (c) 100 (d) 50

If the number of arrangem ents o f ( « - l ) things taken from n different things is k times the number of arrangements of n - 1 things taken from n things in which two things are identical then the value of k is (a) 1 /2 (b) 2 (c) 4 (d) Non e of these

88. The number of ways in which 9 identical

balls can be placed in three 9 (a) 55 (b)o 9! (d) 12 (c) (3 !)3

identical boxes is ! „ ir

The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is 1

(a ) P2 2

5

7

(b) C 2 2

(c) 7 C 2 5 2

5

(d) None of these

Practice Test Time : 30 Min.

M M : 20

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] 1. The nu mbe r of point s (x,y, z) is space, whose each co-ordinate is a negative 6. The nu mb er of way s in whi ch we can choose 2 distinct integers from 1 to 100 such that integer such tha t x+; y+ z + 12 = 0 is difference betwe en t hem is at mos t 10 is (a) 385 (b) 55 (c) 110 (d) Non e of th es e (b) 72 (a) 1 0 C 2 2

3

3

5

The number of divisors of 2 . 3 . 5 . 7 of the form 4n + 1, n e N is (a) 46 (b) 47 (c) 96 (d) 94 3. The number of ways in which 30 coins of one rupee each be given to six persons so that none of them receives less than 4 rupees is (a) 23 1 (b) 462 (c) 693 (d) 924 4. The number of integral solutions of the equation 2x + 2y + z = 20 where x > 0, y > 0 an d z > 0 is (a) 132

(b) 11

(c) 33

(d) 66

5. The number of ways to select 2 numbers from (0, 1, 2, 3, 4) such that the sum of the squares of the selected numbers is divisible by 5 a re (r ep it it io n of di gits is allowed). (a) 9 Ci

(b) 9 P 8

(c)9

(d)7

(c) 1 0 0 C 2 90, (d) None of these 7. Number of points having position vector a i + b j +c % where a, b, c e (1, 2, 3, 4, 5} such tha t 2a + 3 6 + 5° is divisible by 4 is (a) 70 (b) 140 (c) 210 (d) 280 8. If a be an element of the set A = (1, 2, 3, 5, 6, 10, 15, 30| and a, (3,y are integers such that cxpy= a, then the number of positive integral solutions of aPy = a is (a) 32 (b) 48 (c) 64 (d) 80 If n objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is ( a ) (" -2 ) (n- 3)( ra- 4)

(fe)

n-

2Q

6 3

C3+"~3C2

(d) Non e of th es e of positive integral solutions of abc = 30 is (a) 9 (b) 27 (c) 81 (d) 243 (C)"~

46



must be 100%

Answers

BINOMIAL THEOREM n

x, (x+y)" =

"-V +

(n + ).

.-.

=>

7 r + i = nCrx n~ ry r,

where r = 0 , 1, 2

n. n \

i.e., n i.e.,

n+1

( n + 3

= Tn - p + 1

Tp

Tp+ T[P]

V

(x + y)

xn

y/x) n. n

nCn/2

n

"Cn 1 2

C0+C1 + Q1+C3+

Cn = C3

2'n-

1

n

Cn\ 1 '2

n.


48

C%+tf+C%+... (c) CoCr+

+

2n

Cn

Cr+ 1 + Cz Cr+ 2 + ... + Cn-rCn

2n

=

Cn-r n

Cz,

Q) n

n

f<

0 = (Vp + Q ) n - (Vp - Q) n

/+ f -

n

n

( l + f ) f = ( l+ f) f ' = (VP + Q)"(VP-Q)

= (P-Q

2

) " = A".

n (VP + Q)n + (VP -Of

1 Hence

=

l+f+f

•.'

(/+0(1 = (P-Q

2

) " = A".

n

+X2 + X3 + x£2.

Xk"

Xkf

n

az

n, i

~

1

MULTIPLE CHOICE -I /iac/j question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. !• The coefficients of

x1/,

yzt2 and xyztin the

expansion of (x + y + z + t)are in the ratio

3. If the coe ffi cie nt of x

(b) : 2 :14

(a) 2

(c) 2 : 4 : 1

(d) : 4 :12

(c) 4

00 of (x + a) 1 0 0 + (x — a ) 1after simplific ation,

is

in the expansion of

( 1 + a x ) is 32, then a equal s

(a) 4 : 2 : 1

2. The total number of terms in the expansion

3

(b) 3

(d) 6 4. i n t h e expansion of (1 +x) of the

43

, the coefficients

l)t h and the (r + 2)th term s are

equal , then the val ue of r, is

(a) 50

(b) 51

(a) 14

(b) 15

(c) 154

(d) 20 2

(c) 16

(d) 17

49

Binomial Theorem

5. If the three successive coefficients in the

The sum of coefficients of the two middle

JC)" are 28, 56 and

Binomial expansion of (1 +

to

(a) 4

(b )6

(a) ( 2 " ~ 1}C„

(c) 8

(d) 10

(d ) 2 "C„

6. If m and n are any two odd positive integers with n < m then the largest positive integer which divides all numbers of the form 2

Cn then the sum of first of the series aC0 + (a + d) C x + (a + 2d) C 2 + ..., is (n+1)

2

25

terms

(a) 0 (b) [a + nd] 2" (c) [2a + {n-\)d\2n~x (d) [2a + nd] 2""'

(b)6 (d)9

7. The nu mbe r 5 (a) 2 ( c) 5

n

If Cr stands for

(m - n ), is (a) 4 (c)8

JC)2" ~1 is equal

terms in the expansion of (1 +

n equals

70 respectively, then

25

The

- 3

is divi sibl e by (b) 3 ' (d) 7

numbe r 3

n, if the expansion of (2x~1 + x2)" has a term independent of x, then a possible value for n is (a) 10 b) 16

8. For a positive integer

(c) 18 (d) 22 9. The term independent of JC in the expans ion

of

rational

terms

(a) (c) 3

(b)4 (d) 1

If the number of terms in the expansion of (1 + 2JC - 3JC2)" is 36, then n equals : (a) 7 (b )8 (c) 9 (d) non e of thes e If the sum

of the c oeff icie nts in t he 12

expansion of (2 + 3cx+cV) of

(a)

(a) 5/12

(b)l

(c) ' " Q

(d) Non e of these

(c) 70 n

-

l+t (1 -x

If

a2nx

l+t 1 -t

In

1

(b) 56

8

is

l + t

even

- t

+ x 2)n = a0 + alx + a2x2+

(3"-l

of

(c )A 2 + B 2 19 tThe largest term

numbered

...+

+a2n equals )

(d) | + 3"

2

in

ar e 2

the A an d B

2 n

(JC - a ) is : 2

(b)A -B

(d) None of these in

the

expa nsio n

(a) 13 th term

(b) 19th term

(c) 20th term

(d) 26th term

The sum of the series 1 ! (« - 1) ! + 3 ! (« - 3) ! +... + (a)

IS

(b)

JC is equal to (b) 20 (d) 60

terms

(JC + a) n

(2 + Sx)25 where x = 2 is its

expa nsion of f x + —1 is 64, then the term

(a) 10 (c) 40

of

(a)4 Afl

12. If the sum of the binomi al coe ffici ents in the

independent of

sum

respectively then the value of

1 -t

(d) 70

(b)| 3")

]

2

(d) -1 , - 2

expansion

then a0 + a2 + a4+...

(a)^(3"+l)

(b)

If the sum of odd numbered terms and the

JC in the expansion

[(f '-^ jc+ Cr' + iy V (a) 56

,2

(c) 1 ,- 2

10. The term independent of of

vanishes then

c equals

is

3

in

V3~+ 6 V5 ) 10 is

(c)-

2"

n!

+

5 ! (n - 5 ) !

of

50


21. The greatest coefficient in the expansion of

28. The expression "C

, , , -.2n + 2 • (l+x) IS

(2w)i

(a)

w

2

(n!)

numb er

(a) l 2 n

2 {(«+l)!} (2n)! (d)n ! («+ 1) !

n ! (n + 1) !

22. For integer

r = 0

the

coefficients in the expansion of (1 be in g ev en , th en

Bino mial

+ x)",n

(a)2"n

If

is equ al to

(l +^ )

2

(n + 1)

(c)

- 1

-

1

(«+l)

(d) v /

in

the 1

(x sin a + x

cos a)

expre ssion

10 !

040x

("+D

are

,0

(c).('°C

2

then a

(d) None of these

0

+ a 2 + a 4 ... +a

(d) No ne of the se

(a) nil

(b) 30

(c) 31

(d) 32

, the rema inde r is (b) 0 (d) 6

n

cr

r-0 / \

The number of terms in the expansion of (V3~+ 4 VJ ) 1 2 4 which are integers, is equal to

(2 20 - 1)

i?32

C5

(d)(10!r

... +

equals

(b) 2

(a) 1 (c) 4

(' V Q)" - (' "C ,)" + .. .

19

3S

(c) 2 2 0 (2 19- 1) 34. If 7 divides 32

10

(51 r

(a) 2 1 9 (2 20+ 1)

^13

(b) 12 (d) 18

(b)5)

40

of

, a e R is

(1 + x + 2* 2 ) 20 = a 0 + fl) x + a 2x2+

33 . If

_ ( 1 0 C 9 ) 2 + ( 10 C 1 0 ) 2 equals

(a) C5

C2.7C3

expansion 10

(b)

. , 1

2" + ' - 1

the end in the expansion of

26- Th e

th e

(d) No ne of these

(a) 2

If the seventh terms from the beginning and

equal then n equals (a) 9 (c ) 15

9

(b) - 2.

,n + l

2"

in

is

C. is

(b) (H+l)

(a) (n+ 1)

terms

abAc in the expa nsion of

7

x

= C 0 + C 1 ^ + C 2 x 2 + . . . + C„x" ,

3

9

(c) C 7 . C 4

3

1 1

2

rational IS (b )7 (d) 8

31- The coefficient of

9

then the value of 0

of

The greatest value of the term independent of

(d)n.2"~ n

The numb er expansion (1 of (a) 6 (c )5

(b) 3rd ter m (d) 5th ter m

(a) 2. 9 C 7 . 7 C 4

(b) n. 2" ~'

(c) n.2"~ 2

(VT+ 3 V2) 9 , is its (a) 2nd term (c) 4th term

(1 + a + b - c )

C 0 + ( C 0 + C,) + (C 0 + C, + C J + . .. +

(C0 + Ci + C2+ ... + C„ _ 1 )

"

(d) No ne of the se

The first integral term in the expansion of

is

(a) 4 (b) 3 (c)l (d)0 23. If C 0 , C,, C 2 ,.. .. C„ are

3

(b) 2

(c) 5"

n > 1, the digit at unit place in the

X r! + 2

+ 4. "C, + 4 2 . "C 2 + ... +

4". "C n , equals

(2n + 2)!

(b)

0

nC r

+ "Cr+1

is equal to

n

(b)

2

(c)

n(n+1)

The

W

largest

b 2

(d)

2

b \ +

(a )b

2 100

j

m

ter m

in

n+ 1 2 w(w-l) 2( n+ 1) the

e xpan sion

. 1S

h N'OO (b )l l

of

Binomial Theorem

51 100

(O

,00

(a) 39 37. The coe ffi cie nt of x is the polynomi al C* +

(* +

Q>) (X + 2n + ,

c ,)(x+

(d) Non e of these

+ #20 x is equal to 200 and

x > 1, then x is equal to (a) 10 V2 (b) 10 (d) Non e of thes e

(b) 20

(c) 48. If

3n

(d )"C3r

Cr

the

se cond

term

(a) 4

(c) 2" ~ (5n + 4) (d) None of these

x, then

• •

(b) 7/8 (d) None of these

(c) (x

expanded

(a) (n - 1) an

(b) n an

(c) 2

(d) None of these m

m

+1

(b)

)

1/ x in the expansion n

is

(nIn ! («-

n!

(2n - 1) (2n + 1) !

The

53

coeffi cient of x

100 Z

10 0 C Cmm(x-3) (x- 3)

100_m

10 0

(C) -

(b )

C 47

I0 0 C

100

(d)53

in the expansion

. 2 m is

m = 0

(a ) (d)"CB + i

+

1/5 24

2x-,/5)22

(c) (d)

x The coefficient of in -.m + (1 + x) m + (1 + x) m + + . .. + (1 + x)", m < n is

+ 2x~

3/5

n\ (a)

(d) None of these

3/ 5

— 2 x~ 1/ 5 ) 2 3 (d)(x

in

n " 1 r If a. = Z then Z r=0— equ al s r — 0 Cr Cr

(c)"C

2

(l +x )" (l + l/x)

19

expansion of ( 1 + x ) when ascending powers of x is ,19 18 (a) 2 (b) 2

3/5

Coeff icie nt of

41. The sum of the last ten coefficients in the

Cm

expans ion

Which of the following expansion will have term containing x (a) (x- 1 / 5 + 2x 37 5 ) 25 (b) (x

n e Wis

n+l

the

, then the value of

(b) 3 (d) 6

(c) 12

If f x) denotes the fractional part of

(a)

5/ 2

" C 3 / n C 2 is

2

°n

in

is 14a

(b) 2" ~ 2 (n 2 + 5« + 4)

n

"Cr

(b) 3.

(a) 2" ~ 3 (n 2 + 5n + 4)

(c) 2 I 8 - 1 9 C I O

+...

(c) 210 (d) 42 0 47. In the expansio n of (1 +x ) n (1 +y) n (1 +z) n, the sum of the coefficients of the terms of degree r is (a)("C r ) 3

... + Cnx",

n 2 then the valu e of Z (k + 1) Ck is k=0

(a) 3/8 (c) 1/8

2

th en a t equals

(a) 10

= C0 + C,x+C2x2+

2x

20

n6

(l+x)n

( l + 2 x + 3x 2 ) 10 = a 0 + a , x + a

46. If

38. If the four th ter m of

rvs

+ 11 1983 - 7 1983 is

(b) 2 (d)0

(c)3

(c) 2 2 "

If

1983

(a)l (b ) 2 2 " + 1 — 1

(c) 10 4

(d) 43

The unit digit of 17

C n ) is

are

(b) 29

01(c)

C2)...

(a) 2"

4 00

The last two digits of the number 3

(d) None of these

C50

c53 m

C

m

52


(a) 2k where ke /

52. The value of x, for which the 6th term in the expansion of 2 log 2 V(9*

'+ 7)

_

+

is 84 is

,(1/5) log, (3'"' + l)

The

(a) 4

numbe r of irrational ,,,1/5 ,

e

terms

01/10 N55

+3

(a) 47 (c) 50

(b)3

where k e /

(d) 8"

expans ion ot (2

equal to

2k+\,

(b)

(c) 4"

in

the

•

)

is

(b) 56 (d) 48

(c) 2 1

' r- 1 I nC/C. r

p

2"

r = 1 p =0 (a) 4" - 3" + 1 (l+x)n

then

(d) 4" - 3 "

= C 0 + C ]x+C2x2+

Z Z

C,C,

... +

Cnxn,

dx then k =

(a)

(b) 2

z

s

(C)

is (b)2

2« -.

2. 2n !

- M l + ]

—

The +

2n

(d)2

2.n !

(97)

50

2

n+ 1 , ( n2 . n. (n !) (2n+ 1) ! the expres sion

(d) 2

value

of

99

- 99-98

(a) Z n

(b) Z n

(c) Z n

(d)

+ ... + 99 is

(a )- l

n

c„

"-1

of

n> 3, then

Z (- 1/

( n - r) (n - r + 1)

(n -r +2 ) C r = .

(b) - 2

(c)-3

cn

2H

" + 1 C 2 + 2 [ 2 C 2 + 3 C 2 + 4 C 2 + ... + "C 2 ] is

n In !

value 1.2

1

= £_[ x (1 -x )"~

The 2

(c) 2ln

2n + 1

5 2

(b) 4" - 3" - 1

(c) 4" - 3" + 2 If

3

is equal to

(d)0

If x = {-^3 + 1)", then [JC] is (where [JC] denotes x) the greatest integer less than or equal to

(a )4

(b) 3

(c )0

(d) 1

MULTIPLE CHOICE -II Each question, in this part, has one or more than one correct answers). a, b, c, d corresponding

to the correct

n

The coe ffic ient of x in the polyno mial

{x + nC0) (x+3"C,)

(X+

(a) .n2"

(b) n . 2"

+

If '

(d) n . 2" + 1 n

C (a)£ r=0

2

+3

(b)£

C

(d)£

r

r=0 +x + x 2)n = a0 + a^x + a2x2 + ... +

(1 2

a 2 n x " then

(&)a0-a2

in the expansion of

[(1 + A.) (1 + |x) (X+|i)]" is 2

n C

r=0

is

The coefficient of X

( c) £

5"C 2)...

(x + (2n+\)"Cn)

(c) (n + 1) . 2"

For each question write the letters

answer(s).

+ a4-a6 +.... =

0,

if n is odd. (b) a\ - a 3 + a5 - an + .... = 0,

C2+2

(c)

if n is even. - a2 + - a6 + .... = 0, if n = 4p, p

e /+

(d ) aj - a 3 + a 5 - a1 + .... = 0,

ifn

=

4p+\,ps

Binomial Theorem

53

of C 02 + 3C, 2 + 5C22 + ...

64. The v alue

to

(n + i) terms, is

(a ) 2, 1 ~ 1 C„ _ i 2

"71C„ 2

(c) 2(n + 1) . " ~ X 'Cn ( d ) 2 " - 1 ^ + (2«/+ 1) .

2n_

'c„_ |

65. If n is even p osi tiv e inte ger, the n t he condition that the greatest term in the expansion of (1 +

x)n may have the greatest

(b) The number of irrational terms = 19 (c) The middle term is irrational (d) the number of irrational terms = 17 If fl), a2, a3, a4 are the coefficients of any

coefficient also is

four consecutive terms in the expansion of

, . n n +2 (a)—— < x < n+2 n n+1 n (b) < X < n+ 1 n+4 < X < (c) n+4 n

(1 + x)n, then

(c)

(d) None of these

(c)" (d) 67. If

C4

+ 4 C„ (n + 1) (n + 2) (n + 3) (n + 4)

is

a

pos itiv e

integ er

where

and

is an integer

(a) a is an even integer 2

"Cr + " _ , C r + "~2Cr (a) "Cr+ [

+ rCr is

+ (b)"

+ ,

Cr

2a,

a2 + a3 1

1)) in the

+ (x + 3)"'

2

(x + 2) + ... + (x + 2)"~

(a) "Cr (3r - 2")

(b) nCr

(c) "C r ( 3 r + 2"~ r )

(d) None of these

a„

(1000)"

greatest when (a) /(= 998 there

+1

68. For 1 < r < n, the value of

«2

xr(0
(x + 3)"" 3

is equal to

4 2 — a2 r+ — a3

(d)

The coefficient of

If

is divisible by 2 2 "

(c) The integer just below (3 V3~+ divisible by 3 (d) a is divisible by 10

is

2 1

Cr+1

(d) No ne of these

- If (8 + 3 V7)" = P+F, where P is an integer an d F is a proper fraction then (a ) P is an odd integer (b) P is an even integer (c)F.(P+F) = 1

(x + 2) is

(3"~r-2n~r)

fo r n G N. Then

is

(b) n = 999 (d) n .= 1001 a

term

containing

in

J V ~3

,v + —

then

v" (a) n - 2r is a positive integral multiple of 3 (b) /I - 2r is ev en (c) 7i - 2r is odd (d) None of these The value of the sum of the seri es 3. " C - 8. "C, + 13. "C\ - 18. "C , + ...

+1

(c)"

2a2

(c) n = 1000

an d 0 < (3 < 1 then (b) (a + P)

^

a2 + a3

Let

24 n

1

2

+ (x + 3)"~

(a ) n + 1 + 4

a3 a3 + a4

2

a2 + a3

expansion of

of (x + 2y - 3z + 5w — hi)" is (b)"

a, aj + a

a

(a)

66. The n umber of disti nct terms in the expansion

69

10 3

when divided by 25 is (b) 25 (d) 9 _1_ ,20 3 In the exp ans ion of V4 + 'V6 (a) The number of rational terms = 4

(given that Cr = "Cr)

(b) (2n+l).

-F) (P+F ) = 1

(d)(l

The remainder of 7 (a) 7 (c )1 8

0

(71 + 1) ter ms is (a) 0

(b) 3"

(c) 5"

(d) No ne of the se

up to

54


77. If C 0 , Cj, C 2 ,..., Cn are coefficients in the bi no mi al

ex pa ns io n

(1 + JC)",

of

83. If

su m

then

C 0 C 2 + Q C 3 + C 2 C 4 + ... + C n_2Cn is equa l

greatest

to

(1 +x)n is (a) 35

(a)

(b)

2n ! (n - 2) ! (n + 2) !

of

the

2n !

c oef fic ien ts

in

th e

(x-2y + 3z)" = 128, t hen the

coefficient

in

the

expansion

of

(b) 20

(c) 10

(d) 5

84. The

(x + V(x

expressio n

3

- l))3

2

((n-2)!)

- 1) is a polyn omial of degre e

2n !

(c) {{n + 2)

!)z

(d) 2 "C n _ 2 78.

the

expansion of

(a) 15

(b) 6

(c )7

(d) 8 25

85. In the expans ion of (JC + Y + Z) (a) every term is of the form

If a + £> = 1 then "L r C rab

equals

(b)

it

Ck = nCh

then

2

in the expansion of

JC50

\9 99 , », 2 ,, ,

+JT)

,9 98

a—b

is

(d)

,000

c51

(c) 25"

3

] is (b) 28 (d) 56

(1 + 2JC +

]0

3

in the expansion of

is (b) 60 (d) 24 0

3JC 2 ) 10

= a0+ ai x +a2x2 + ... +

l L

#20 JC, then (a) a , = 20

(b) a 2 = 210

(c) a 4= 8085

(d) No ne of thes e 2

(d) none of these

20

+1

82. The number of terms in the expansion of

(b) 0

(c)2"

89 . If

= m and if / i s the fractional

(b) 20"

(a) 3" — 1

(be + ca + ab) (a) 30 (c) 120

part of m, then / m is eq ua l to

[(a + 3b) ( 3 a-b) (a) 14 (c) 32

is

"C 1; JCh i < j is

88. The co efficient of a ' V c

+ 3JC ( 1 + JC)

1000

+, .... 1000+ ... +1001 x (a) c50 , , 1002^ (c) c50

2

(b- n)

(d) 0

87. T he valu e of I X ,=07=1

.1 00 0 . / , / , .

(a) 15 n + 1

value of n (n - 1)

2

80. The coeffi cient of

81 . If (6 V6~+ 14)"

the

1.2

(b)

(c ) {ab)

+1

k

1)" (a -n)

n

+ 2* (1

.Z

then

-...+(-

(a) a" + b"

w2 (n + l)2 (n + (d) 144

( 1 +JC)

n > 3 and a,beR,

(a - 2 )(b -2)

(n + 2)

l)(n + 2) 12

r-k

ab - n (a - 1) (b - 1) +

12

(c) n(n+

,, ,

86. If

1) {n + 2) 12

n(n+ l)

.y

(b) the coefficient of x y z is 0 (c) the number of terms is 351 (d) none of these

equals

n(n+

25 — r

i 9

If « is a positive integer and

(a)

Cr. Ck.x

(b ) n (d) nb

(a) 1 (c) na

I

25/^

90

(d) a ^ = 2

2

. 3 7. 7

- ^ coef ficie nt of the midd le term in the expansion of (1 + JC)In is ( b ) 1.3.5....(2n-l) 2n n! (c) 2.6

(4n - 2)

(d) No ne of the se

Binomial Theorem

55

Practice Test MM : 20

Time: 30Min.

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] 2 20

1. If

( l + x + 2x ) i0

OS40x

=a

Q+

then a j + a 3 +

(a) 2 1 9 ( 2 2 0 - 21)

2

aiX +a2 X + ...+

2

. 2

cos x 2 1 1 +, COS

. 2

2 COS

sin x sin x

n + 1 *

+ ... + 037 eq ua ls

(b) 2 2 0 (2 19 - 19)

(c) 2 19 (2 20 + 21) (d) None of these 2. If maximum and minimum values of the determinant 1 + sin x

(b)

w i f

sin 2x X

X

6. Let a 0 , a 1 ( a 2 , ... be the coefficients in the 2n expa nsio n of (1 + x + x ) ar ra ng ed ord er of

sin 2x

x. The value of ar - nCi ar_ 1 + "Cj ar_ 2 - ...

1 + sin 2x

+ (- l) r nCra0 = ... where r is not divisible

are a and p then (a) a 3 - p 17 = 26 (b) a + P 97 = 4 2n (c) (a - P ) is alwa ys an even int ege r for neN. (d) a triangle can be constructed having its sides as a, P and a - p

3. The coefficient of x in (1 +* ) is

(d)

(1-JC+JC )

(a) 1 (b) 2 (c) 3 (d) 0 4. The number of terms in the expansion of ( N 9 + VtT) which ar e int ege rs is given by (a) 501 (b) 251 (c) 60 1 (d) 45 1 5. If n is an even integer and a, b, c are distinct, the number of distinct terms in the expansion of (a + b + c) n + (a + b - c) n is

by 3. (a) 5 (b) 3 (c) 1 (d )0 7. In the expansi on of (3 x / i + 3 5174 )" the sum of the binomial coefficients is 64 and the term with the greatest binomial coefficients exceeds th e t hi rd by (re - 1) th e va lue of x must be (a) 0 (b) 1 (c) 2 (d) 3 8. The last digit o f 3 (a) 1 (c) 3

+lis (b) 2 (d) 4

1 1 1 and f(x 1! 11 ! 3 ! 9 ! 5 ! 7 !. ml = f(.x).f(y)Vx,y,fa)=l,f '(0) = 10 th en (a) f\n)m = (b)f'(m)=n (c)f\n) * f\m) (d) None of th es e

9. If:

10. The number 101 (a) 100 (c) 10000

10 0

- 1 is divisibl e by (b) 1000 (d) 10000 0


56

Answers

DETERMINANTS

an 321

a2i 331

322

a 13 323 333

ai2 a22

311

312

332

and

an a2i

ai2

331 341

332 a42

322

ai3 a23 333 a43

ai4 a2 4 334 344

a,y e C~Yi, j

a(y, a,y A =

a,y Ay

312 322

311 321 331

332

C,y.

313 323 333

= an Cn + ai2 C12 + ai3 C13.

k, k k x = a, 32 + X2

b

(ix)

bi

C1

b2

C2

bs

C3

bs = C3

ai

bi

Ci

32

b2

C2

33

b3

C3

X

=

C1

32

b2

C2

a3

b3

C3

t>2 C3 = bi ci ai <*2 «3

x

bi

P1 p2 p3

71 Y2 73

bi 02

Xi +

x3

C3

bi

Ci

bz

C2

bs

C3

Obj ecti ve Mathematics

58

aiai +i?i(3i + ciyi a 2 a-| + £>2PI + C2 Y1 a3ai + b3pi + C3Y1

aia.2 + £>1^2+ C1Y2 aia3 + bip3 + C1Y3 a2CC2 + + C2Y2 a 2 a3 + £>2p3+ C2Y3 a3a2 + fc*3p2+ C3Y2 a3«3 + teP3 + C3Y3

Note tha t we can al so multiply r ows by column s or column s by rows or columns by columns .

The system of homogeneous linear equations aix+ b\y+ ciz = 0 a2*+ bzy+ C2Z = 0 and 33X + b^y + C3Z = 0 has a non trivial solution (i.e. at least one of x, y, z is non zero) if ai £>1 ci a = 32 £>2 c2 = 0 a3 te <5 and if a ^ 0, the n x = y = z = 0 is th e only solution of ab ove sys te m (Trivial solution). : Let us consider a system of equations a-ix+ toiy+ ciz = di a2X+ b>2y+ C2Z = cfc a3X+i>3y+ C3Z = cfe d^ C1 ai C1 bi 61 a = a2 ai = cfe b2 Here C2 b2 C2 d3 as bs c3 c3

ai a2 a3

a2 =

By Cramer's rule, we have.

d^ ch. cf e

c1 C2 c3

A3 =

a2 x = Al -y = and a a

ai a2 a3 z =

bi

<* cf e cfe

to

a3 a '

(i) A / 0, the n syste m will hav e unique finite solution , and so equa tion s are c onsis tent . i.e., equations (ii) Ifa = 0, and at least one of a1, a2, a3be non zero, then the system has no solution are inconsistent. (iii) If a = a1 = a2 = a3= 0 then equations will have infinite number of solutions, and at least one cofactor of a is non zero, i.e. equations are consistent.

ft If

F (x) =

92 93 hi /12 h3 where ft, h, h\ flu, 52, g3; hi, te, hx are th e functions o f x, then F'(x)

=

ft'

fe'

h'

91

92 /72

93 h3

+

ft

f2

91'

92'

/71

ft 93'

+

91

/n'

92

fits /13'

Determinants

59

MULTIPLE CHOICE -I

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. If the value of the a 1 b 1 1 1

3 + 2? 5 - i 7- 3 i 2i -3i 5+i 3-2i 7 + 3i (a) z is purely real (b) z is purely imaginary

deter minan t 1 1 is positiv e, then c

(a) abc > 1

(b)abc > - 8

(c) abc < —

(d) abc > - 2.

If z =i

Given that q* - pr < 0, p > 0 the value of

p <7 px+qy q r qx+ ry is px +qy qx + ry 0 (b) positive (a) zero q 2 + pr (c) negative

(c) z is mixed complex number, with imaginary part positive (d) None of these 8. In a third order determinant a,-, denotes the eleme nt in the ith row and the yth colu mn

i = j i>i - l ; i
3. I f / ( n ) = a" + P" and

(a) 0 (c) - 1 1 +/( 2)

1 +/(3)

= k (1 - a ) (a) 1 (c) txp

2

1 b b2

1 a 2 a

1 c 2 C

, A2 =

1 1 1

be ca ab

(a) A, + A2 = 0

(b) A, + 2A 2 = 0

(c) A,=A

(d) A, = 2 A2

2

6. If a , p are non

If 1 +xz A(x) =

real numbers

a b c

then

(c)

(b) V • + 1

(d)

- 1

e

x

x sin x

11. If V- 1 = i, and to is a non real cube root of unity then the value of

satisfying

x — 1 = 0 then the value of X+ 1 a P a k+p 1 is equal to X+ a P

log (1 + x")

• 2x cos X tan x sin then (a) A (x) is divisible by x (b) A (x) = 0 (c) A' (x) = 0 (d) None of these the determinant The value of log„ (x/y) log a (y/z) log fl (z/x) log,, (y/z) log,, (z/ x) log h (x/y) logc(z/x) log, (x/y) log c (y/z) (a) 1 (b) — 1 (d) None of these (c) log a xyz

1

3

(a) 0

(b )l (d) None of these

1 +/( 4)

(1 - P) 2 (1 - Y)2, then k = (b) - 1 (d) a p y

If x, y, z are integers in A.P., lying between 1 and 9, and x 51, y 41 and z 31 are three-digits numbers then the value of 5 4 3 y 41 z 31 x51 x y z (a) x + y + z (b) x - y + z (d) x + 2y + z (c)0

A,=

, then

—( 1 -i (a) 1 (c)

CO

(0 - 1 co2 — 1

1 + i + to - 1 - (' + to is equal to - 1 (b) i (d) 0

60

Obj ective Mathematics

12. If

the sin 2 x co sl r cos 2 x

cos 2x si n 2 * cos 4x

deter minant cos 4x cos 2 * cos 2x

is expanded in

powers of sin x then the constant term in the expansion is (a) I (b) 2 (c)-l

b + c,c + a and a + b are three factors of the -2a a+b a+c The determinant b +a -2b b+c c +a c+b —2c other factor in the value of the determinant is (b)2 (a) 4 (c) a + b + c The

(d) None of these

value

of in/3

- in/3

e

18. If the system of equations 2 x - y + z = 0, x-2y + z = 0, tx-y + 2z = 0 has infinitely many solutions and f(x) be a cont inuous +f{x) = 2 then function, such that f(5+x) 2 r 'f(x)dx=

(d)-2

13. Using the factor theorem it is found that

14

(c) A ABC is an isoscel es tria ngle (d) None of these

e

determinant

in/A

2/71/3 IS

1

• in/4

the

- 2m/3

(c) - 2 +

19. If

2x x(x-l) x(x+ 1) (x — 1) xC*-I)(or2) .v (A2 — 1) then/(200) is equal to (a) 1

(b)0 (d) -200

(c) 200 16. If 1

n

2k

n2 + i;+ 1 i n'

2k- 1 L Dk *=1

56

2 n +n

n~ + n + 1

then n equal s

(a)4

(b)6

(c) 8

(d) Non e of these

17. If a, b, c arc sides of a triangle and a 1/

(a+ l) 2

X

Y

X

X

X

X

p

X

(b+ l) 2

(c+1 r

= f(x)-xf'(x)

then

(a)(x-ot)(*-P)(*-7)(x-S) (b) (x + a)(x + P) (x + y) (x + 5) (c)2(x-a)(x-P)(x-y)(x-8) (d) None of these

(«-1) (b-iy (c1) (a) A ABC is an equilate ral tri angle (b) A ABC is a right angled isoscelcs triangle

vz

is

5

(a) 0 (b) log* logy logz (c)l (d)8 21. If f(x) = ax + bx + c, a, b, c e R and equation f(x) -x = 0 has imaginary roots a and P and "{, 8 be the roots of f(f(x)) - x= 0 2 a 5 then p 0 a is Y

P

1

(a)0 (b) purely real (c) purely imaginary (d) none of these 22. If n is a positive integer, then

n + 2r = 0, then

log lo g ; y

log, A-

x+ 1

x

f(x)=

and

X

log,, A; 3 1

=

a x x

/ (A) is equal to

15. If

(d )t

(c)5

20. For positive numbers x, y, z the numerical value of the determinant 1 log^y logxz

1 (b)-(2 + V2) (d)2-<3

(a) 2 +

(b) — 2t

(a)0

n+ (a) 3 (c)-5

'-n+l 4^ l » +2

H + 3L ~ n+ 1 +4n L n +2 n + 5,-, ( -n + 3

n

(b) - 1 (d) - 9

n + C\ Hr + 2 n+ ti + 3 n + (l>n h +4

61

Determinants

(x-2) 23. Le t A(x ) =

28. I f A + fl+C = 7t, then

(x-i r 2

(x-1)

(x+1)

X

(x + 2)"

Then the coefficient of x in A (x) is (a)-3 (c)-l

(a) 1

(b)-2 (d) 0 X

if/M =

sinx

sec x

tan x

1

2

then

- sin x cos x 0

a

equals

( - 1 )"b (_1)« + 2 c


26. If a , (5, y ar e the root s of the x + px + q= 0,

then

a determinant

the

val ue

of

the

y

a

31. If 3" is a fact or of the dete rmin ant 1 "C,

p

t-2

(a) 4 (b )2 (c )0 (d) - 2 27. If a, b, c > 0 andx,y,ze R, then the value of the determinant -x.2 {a + a x ) 2 •a ) y is (b> + b~y)2 (b ~b->)2

(cz + c~z)2 (a) 6 (c)2

(b)l (d) None of these

(c)-l

is

a

(cz-c-f

n+1 (-1)' 1 +c 1 -c

a+ 1 a- 1 is equal to b+ 1 1 -b 1+c 1 -c

(a) 3

equa tio n

p Y

p Y

(b)6 (d) 2

(-1 )" + lb 1 -b b+ 1

Dna a+ 1 a- 1

(-

sin 2x = 1,

(a) 3 (c) 1

cos x 2 cos 2x 3 cos 3x

30. The value of

(b)3 (d)0

cos x 0 sin x

sin x sin 2x sin 3x

(c)4

-n/2

0 sin x cos x

cos x cos 2x cos 3x

Then/'(71/2) = (a) 8

,1/2 then the value of J /(x) dx is equal to

25. If

(d)2

29. Let f ( x )

e

is equal to

(b) 0

(c)-l

cos X 2 X

(a) 5 (c)l

sin (A + B + Q sin B cos C A - sin B 0 tan cos (A + B) - tan A 0

1 n + 3/

n + 3 si

1 « + 6c n ri + 6, TVs-i <-2

then the maximum value of (a )7

(b) 5

(c)3

(d)l

n is

32. If a, b, c are non zero real numbers and if the equations {a - l ) x = y + z, {b - 1) y = z + x, {c - 1) z = x + y has a non trivial solution then ab + bc + ca equals (a ) a + b + c (b) abc (c) 1 (d) Non e of these

(b)4 (d)0

MULTIPLE CHOICE -II Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). is divisible by

33. The determinant 2

a +x A =

ab ac

2

ab 12 , 2 b +x be

C

ac

(a) x

be

(c)x

2 ,

+ X

2

3

(b) x (d) x 4

62

Ob j ective Mathematics

34. If

39. The roots o f the equation

-,k- 1

y

Dk =

the n

2

2

9 sin 1 0

2

x +* s i n 9 + si n 6

1 +sin•

3 cos 2 9

1+

9

2

8 + co s 6

sin 29

2 6

3 sin 2 6

are

sin 9/2

(b) sin 2 0, cos 2 0

(a) sin 0, cos 0 2

(c) sin 0, cos 0

£ Z)^ is equ al to k= 1

ABC

cos

(a) Com ple x (b) Real (c) Irrat iona (d) Ration al The determinant a b aa + b b c ba+c aa + b ba + c 0

0, cos 0

the

value

of sin

. B sin —

cos

tan (A + B + C)

sin

2

'A+B+C

the

. B sin —

sin (A + B + Q

is

2

(d) sin

40. In a triangle determinant . A sm-

(a)0 (b) independent of n (c) independent of 0 (d) independent of x, y and z The value of the determinant V6 2i 3 + V6
2

x +x sin 9 + si n

2

+ x cos

X +X COS 0 + COS 0

'n+1

n+1

x

3x

z sm

2" — 1

2

1 sin kQ k(k + 1)

is less than or equal to (a) 1/2

(b) 1/4

(c) 1/8

(d) Non e of these

If

At4 + Bp + Ct 2 + Dt + E =

t +3t t - 1 f — 3 f+1 2 —t t-3 t-3 t +4 31

is equal to zero, if

then E equals

(a) a, b, c are in A.P. (b) a, b, c are in G.P.

(a) 33 (c) 27

(c) a, b, c are in H.P.

If a, (3, y are real num ber s, t hen 1 (cos ((3 - a ) cos (y - a ) co s (Y - P) A = cos (a - P) 1 1 cos (p - y) cos (a - y) is equal to (a)-l (b) cos a cos P cos y (c) cos a + cos P + cos y (d) None of these

(d) a is a root of ax + b x + c = 0 (e) (x - a ) is a fac tor of ax + 2 bx + c The digits A, B, C are such that the three digit numb ers A88, 6B8, 86C are di visibl e by 72 then the determinant A 6 8 B 6 is divisi ble by 8 8 8 C (a) 72 (b) 144 (c) 288 (d) 216 If a> b> c and the system of equations ax + by + cz = 0, bx+cy + az = 0,

cx + ay + bz = 0 has a non trivial solution then both the roots of the quadratic equation at2+ bt + c = 0 are (a) real (c) posit ive

(b) of opposi te sign (d) compl ex

(b) - 3 9 (d) 24

If all elements of a third order determinant are equal to 1 or -1, then the determinant itself is (a) an odd numbe r (b) an even numb er (c) an imaginary number (d) a real number The

coeffici ent of x i n the 1*1 b

(l+x)" (1 +xp <

(l+x)

(1 +xpb'-

(l+x)"

fll&2

A

(a) 4

(b)2

(c)0

(d) - 2

determinan t

(l+x)a^ (1 +xf^ (1

+x)a^

is

Determinants

63

45. If a , P be

the

roots

of

the

s1 (b2 - Aac)

equation

(a)

ax2 + bx + c = 0 Let S„ = a" + (3" fo r n > 1 Let

A=

3 1 +5, 1 +S2

1 +5,

1 +S2

1 +S2 1+S3

1 +S3

(b)

, sb2 - Aac (c) 4 a

(a + b + c) 2 (b2 - Aac) (a + b + c)

1 +S4

then A =

Practice Test 20

M.M :

Tim e : 30 Min

(A) There are 10 parts in this question. Each part has one or more than one correct k +2

k X

1. If

X kk++2

/k y

y

y

2

k+2 Z

k Z

k+ 3 X k *+ + 3 3

(a) -11

y

y

k+3 Z

1 1 — • then x y k =- 1 (a) = &- 2 (b) (c) = 0k (c) k =1 2. If fr (x), g r (x), hr (x), r= 1, 2, 3 ar e po ly no mi al s in x such that fr — Sr (a) - K ( a ). r = L 2, 3 and = ( x - y) (y - z ) ( z -x )

— +

fl (x) f(x) f 3 (:x) 2 g! (x) g2(x) g 3 (x) h(x) h h (x) (x) x 2 3 then F' (x) at x = a is (a) 1 (b) 2 (c) 3 (d) None of th es e 3. The l arge st value of a thi rd order determinant whose elements are equal to 1 or 0 is (a)0 (b)2 (c) 4 (d) 6

answer(s). [10 x 2 = 20] then the value of 5a + 4(3 + 3y + 28 + X = (b)0 (c) - 1 6 (d) 16 6. Let 1 si n 0 1 A= - s i n 6 1 sine ;0< 6<2r ct hen - 1 - si n 6 1 (a)= A 0 (c) Ae [ - 1, 2]

(b) A e (0, oo) (d) A e [2, 4] n +2 n n +1 re + 2 p n+lp 1 "n + 2 n "re + 1 n + 2r l re + ly-, c c +2 re + l

7. Let fix) =

F (x) =

4. Let f(x) =

x 6 P

a constant. Then (a )p

cos x 0 3 P

where p is

d.3 — j \f (x)| at x = 0 is dx (b

3

(c) p +p x 2 5. Let x2 x x x

si n x -1 2 P

.. 6 6

(a) n + +n 1 (c)! n 8. Eliminating

a b-c 1 (a) 1 1 1 (b) 1 1 X

(c)

)p+p2

(d) indep enden t o f

where the symbols have their meanings . The f(x) is divisible by

p

= a x 4 + p x 3 + "p? + 8 x + X

=

1 y -z

usual

(b) (TI + 1)! (d) non e of th es e a, b, c, from

c

b c x y = 0 z

,y-X

-y -z -x 1 z

• we get

= 0

- X

1 z

= 0

(d) None of these 9. The syste m of equat ions x-y cos G + z cos 26 = 0, - x cos 0 + y - z cos 0 = 0, x cos 20 ~y cos 0 + z = 0, has non trivial solution for 0 equals to

Obj ectiv e Mathematics

64

(a) TI /3 (c) 2 TT /3

10. Let A(x) =

(b) TC/6 (d) TC/12 x +a x +b x + a - c x + b x +c x- 1 and x + c x +d x-b +d

A(x) dx = - 16 where a, b, c, d are in A.P., o then the common difference of the A.P. is equal to (a) + 1

(b) ± 2

(c) ± 3

(d) ± 4


must be 100%

Answers Multiple

Choice-I

l.(b) 7. (a) 13. (a)

2. (c) 8. (a) 14. (b) 20. (d)

19. (a) 25. (d) 31. (c)

Multiple

26. (c) 32. (b)

4. (c) 10. (d)

5. (a)

6. (b) 12. (c)

9. (a) 15. (b) 21. (b)

16. (d)

11. (d) 17. (c)

22. (b)

23. (b)

18. (b) 24. (d)

27. (d)

28. (b)

29. (c)

30. (d)

35. (a), (b) 41. (b)

36. (b), (e) 42. (d)

Choice • -II

33. (a), (b), (c), (d) 37. (a), (b), (c) 38. (a), (b) 44. (c) 43. (b)

Practice

3. (a)

34.(a), (b), (c),(d) 39. (a) 40. (c) 45. (b)

Test

1. (b) 7. (a),.(c)

2. (d)

3. (b)

8. (b), (c)

9. (a), (b), (c), (d)

4. (d)

5. (a) 10. (b)

6. (d)

PROBABILITY The probability of an event to occur is the ratio of the number of cases in its favour to the total number of cases (equally likely). p _ _ n (£ ) _ number of favourable ca se s n(S) total numbe r of ca se s : If a is the number of cases favourable to the event £, b is the number of cases favourable to the event £", then odds in favour of £ ar e a : b a n d odds against of E a r e b: a a In this ca se P( £) = a+ b b and P(£) = a+b P (E)+ P( £ ) = 1. 0 < P (E) < 1 therefore maximum value of P (E) = 1 and the minimum value o f P (£) = 0.

(i) Equ all y likely E ve nt s : The giv en even ts ar e said to be equally li kely, if non e of the m is expe cted to occur in preference to the other. (ii) : Two eve nts a re said to be i ndep enden t if the o ccur ren ce of one do es not E-\, E2,..., En for Independent Events. depend upon the other. If a set of events P (£1 n

E2 n £3 n . .. . n

£

n

) =

P (£1)

P (£2)....

P

(£„)

: A set of events is said to be mutually exclusive if occurrence of one of them precludes the occurrence of any of the remaining events. If a set of events £1, £2 En for mutually exclusive events. Here P (£1n £20 n En) = <|> then P (£1 u £2 u .... u £„) = P (£1) + P (£2) + P (£3) + .... +P (£7) (iv) : A set of events is said to be Exhaustive if the performance of the experiment results in the occurrence of at least one of them If a set of Events £1, £2, ..., £ n then for Exhaustive Events P(£1 u £2 u . . . . u En) = 1 (v) : A set of events is said to be mutually exclusive and exhaustive if above two conditions are satisfied. If a set of Events £1, £2 En then for mutually exclusive and exhaustive events P(£ 1 u £2 u .... u En) = P(£i) + P(£ 2 ) + ... + P(£ n ) = 1 (vi) Co mp ou nd Ev en ts : If £1, £2, En are mutually exclusive and exhaustive events then, if £is any event (iii)

P(£) = £ P ( £ n £ / ) =.£ P(£ /) /= 1 /=1

• P( E

if P(£;) > 0

§ 7.3.

£2 has already occured is called the conditional ( F \ £2 ha s already occure d. It is den ote d by P — 1

The probability of occurrence of an event £1, given that probability of oc cu rr en ce of £1 on the condition that

66


/. e..

P(£i n E P{Ez)

P | #

2)

, £2 * $

§ 7.4. Baye's Theor em or Inverse Pro babi lity : If E1, £2

£n are n mutually exclus ive and exh aust ive eve nts suc h that P( £/ )> 0 (0 < /< n) and Eis any event, then for 1 < k< n, P(E *) P| | E j

n f

k= 1

k

Remark : We can visulise a tree structure here P(A) = p; P(S) = q; 7' f?' p \ T l = PI ; p i ^ l = <71;

R ' B If we are to find f A } v

=

P2;

p

q

Ifl.

=

<72;

we go

/ P

p f ^ U p^jp(A)

+

p ^|j p(S )

If a die has m faces marked 1, 2, 3, ..., m and if such n dices are thrown. Then the probability that the sum of the numbers on the upper faces is equal to ris given by .. .. , , r. th e coeffic ient of x in

m n

... +X

)

m

Suppose a binomial experiment has probability of success p and that of failure probability of r s u c c e s s in a se ri es of n independent trials is given by p + q = 1 and r = 0, 1,2, 3 P(r) = nCr ff (f~ rwhere n

q(p+q

=

1), then

(i) Th e probabi lity of gett ing at lea st k suc ces s is P( r> k) = l ^ n C r p r q " ~ r (ii) Th e probab ility of gett ing at mo st k success is l< P (0 < r< k) = ZQ nCr.p r.

(iii) Th e mea n, the vari anc e and the stan da rd deviation o f binomial distribution ar e np, npq, -Inpq. : If £1 an d £2 are two events, then (i) £1 u £2 stands for occurrence of at least one of £1, £2 (ii) £1 n £2 st an ds for the simulta neou s occur renc e of £1, £2. (iii) E or E or £° st an ds for non occur rence . (iv) £1' n £2' stands for non occurrence of both £1 and

£2.

Probability

67

§ 7.7. Expectation If p be the probability of success of a person in any venture and M be the sum of money which he will receive in case of success, the sum of money denoted by pM is called expectation. § 7.8. Imp ortant R esu lts (i) If Ei and £2 are arbitrary events, then (a) P (Ei u Ez) = P (£i ) + P (Ez) - P (Ei n Ez). (b) P (Exactly one of E-\, Ezoccu rs) = P (Ei n Ez') + P (EV n E2 ) = P(Ei)-P(Ein

E

z)

+ P(EZ) - P (Ei n Ez)

= P (£i) + P (Ez) - 2P( £i n Ez) = P(Ei u E 2 )-P(EI n£ 2) (c) P (Neither Ei nor E 2) = P (Ei ' n Ez') = 1 - P (Ei u E 2) (d) P(Ei'uE 2 ' ) = 1 - P( Ei n Ez) (ii) If Ei, E2, E 3 are three events then (a) P (Ei u Ez u E3 ) = P (Ei ) + P (E 2 ) + P (E3) - P (Ei n E 2 ) - P(E2 n E3) - P( E 3 n Ei ) + P ( E m E 2 n E3 ) (b) P (At lea st two of £1, £2, E 3 occur) = P (£ 1 n £ 2) + P ( £ 2 n £3 ) + P ( £ 3 n £ 1 ) - 2 P (£ 1 n £ 2 n £3) (c) P (Exactly two of Ei, Ez, £3 occur) = P (£ 1 n £ 2) + P (£ 2 n £3 ) + P (E 3 n £ 1 ) - 3 P(Ei n £ 2 n E3 ). (d) P (Exactly on e of £1, £2, £3occur) = P (£1) + P (£2) + P (£3) - 2 P ( £1 n £2 ) - 2P (E 2 n £3 ) - 2 P ( £ 3 n £ 1 ) + 3 P ( £ 1 n Ez n £3). (iii) If £1, £2 , £3 En ar e n events then (a) P (£1 u £2u .... u En) 1 - P (£1) -

P ( E 2 ) - .... -

P (£„).

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. For n indep enden t events Aj's, P (A,) = 1/(1 + i ) , / = 1,2,..., n. The proba bility tha t atl eas t one of the event s occurs is (a) 1 / n (b) \ / ( n + 1) (c) n/ (n + 1 ) (d) none of these 2. The probabilities that a student will obtain grades A, B, C or D are 0.30, 0.35, 0.20 and 0.15 respectively. The probability that he will receive atleast C grade, is (a) 0.65 (b) 0.85 (c) 0.80 (d) 0.20 3. The probabilitity that a teacher will give an, unanounced test during any class meeting is 1/5. If a student is absent twice, the probability tha t he will miss at least one test, is :

(a) 7/25 (c) 16/25

(b) 9/25 (d) 24/25

If the probability for A to fail in an examination is 0.2 and that for B is 0.3 then the probability that either A or B fails, is (a) 0.38 (b) 0.44 (c) 0.50 (d) 0.94 A box contains 15 transistor s, 5 of which are defective. An inspector takes out one transistor at random, examines it for defects, and replaces it. After it has been replaced another inspector does the same thing, and then so does a third inspector. The pro bab ility that at least one of the inspectors finds a defective transistor, is equal to (a) 1/27 (b) 8/27 (c) 19/27 (d) 26/ 27

68


pro babili ty that the largest number appe aring

6. There are 5 dupl icat e and 10 srcinal items in an automobile shop and 3 items are brought at random by a customer. The probability that none of the items is duplicate, is (a) 20/91 (b) 22/91 (c) 24/91 (d) 89/91

on a selected coupon is atmost 9, is

addresses, the letters are kept in these envelopes. The probability that all the letters are not placed into their right envelopes is

8. A man is known to speak truth i n 75% cases. If he throws an unbiased die and tells his friend that it is a six, then the probability that it is actually a six, is fa) 1/6 (b) 1/8

(d) 3/8

A single letter is selected at random from the word 'PROBABILITY'. The probability that it is a vowel, is (a) 2/11 (b) 3/11 (c) 4/11 (d) Non e of these

C4

a sample of 100 bulbs company are defective. 3 out of 4 bulbs, bought not be de fe ctiv e, is : (b)

90

C /96C 3

(d) (

C 3x

4

C,)/

C

(b)l-(l '

-p) n

(d) - [1 - (1 -p)"]

n letters are written to n different persons and addreses on the n envelopes are also written. If the letters are placed in the envelopes at random, the probability that atleast one letter is not placed in the right envelope, is (a) 1 (d) i- rr

Three athletes A, B and C participate in a race. Both A and B have the same probability of winning the race and each is twice as likely to win as C. The probability that B or

(c) 8/81 (d) 35/81 A bag contains 5 red, 3 white and 2 black balls. If a ball is pic ked at ra ndom, the pro bability that it is r ed, is (a) 1/5 (b) 1/2 (c) 3/10 Cd) 9/10

(a) C 3 /

(a) 1 /n

(c) 1 -

A bag contains 7 red and 2 white balls and another bag contains 5 red and 4 white balls. Two balls are drawn, one from each bag. The pro bability tha t both the balls are white, is (a) 2/9 (b) 2/3

12. 10 bulbs out of manufactured by a The probability that by a cu st om er will , , ,100r

7

(c) ( 8/15 ) (d) Non e of these The probability that a man aged x years will die in a year is p. The probability that out of n men M,,M 2 ..., Mn, each aged x, M\ will die and be the first to die, is :

(c) - j

(b) 1/3 (d) 5/6

(c) 3/4

(b) ( 3 / 5 )

7

7. Three letters are written to three different per sons and addresses on the three env elopes are also written. Without looking at the

(a) 1/2 (c) 1/6

7

(a) (1/15 )

4

Fif teen coup ons are numb ered 1, 2, 3, ..., 15 respectively. Seven coupons are selected at random one at a time with replacement. The

C wins the race is (a) 2/3 (b) 3/5 (c) 3/4 (d) 13/25 A number is chosen at random from among the first 30 natural numbers. The probability of the number chosen being a prime, is (a) 1/3 (b) 3/1 0 (c) 1/30 (d) 11/30 Out of 13 applic ants f or a job , ther e are 8 men and 5 women. It is desired to select 2 persons for the jo b. The proba bility tha t atleast one of the selected persons will be a woman, is (a) 5/13 (b) 10/13 (c) 14/39 (d) 15/39 Two athletes A and B participate in a race A along with other athletes. If the chance of winning the race is 1/6 and that of B winning the same race is 1/8, then the chance that neither wins the race, is (a) 1/4 (b) 7/24 (c) 17/24 (d) 35/4 8

Probability

69

20. Three players A , B, C in this order, cut a pack of cards, and the whole pack is reshuffled after each cut. If the winner is one who first draws a diamond then C's chance of winning is (a) 9/28 (b) 9/37 (c) 9/64 (d) 27/64 21. 6 girls and 5 boys sit together random ly in a row, the probability that no two boys sit together, is : (a)

6 !5 ! 11 !

6 ! 7 ! (c) 2 ! 11 !

(b)

6 ! 6! 11 !

5 ! 7! (d) '2 ! 11 !

A mapping is selected at rando m from the set of all mappings of the set A = {1, 2 ..., ;i} into itself. The probability that the mapping selected is bijective, is

W-7 n !

n

n

Three letters are written to three different 2" and add resses on the three envel ope s persons are also written. Without looking at the addresses, the letters are kept in these envelopes. The probability that the letters go into the right envelopes, is (a) 1/3 " (b) 1/6 (c) 1/9 (d) 1/27 24. An unbia sed die with face s mar ked 1, 2, 3, 4, 5 and 6 is rolled four times. Out of the four face values obtained, the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5, is (a) 1/81 (b) 16/81 (c) 65/81 (d) 80/81 The probability of guessing correctly alleast 8 out of 10 ans wer s on a true -fal se examination, is (a) 7/64 (b) 7/128 (c) 45/ 102 4 (d) 175/1024 The proba bil ity that the 13th day of a randomly chosen month is a second Saturdav. is (a) 1/7 ' (b) 1/12 (c) 1/84 (d) 19/84

A box contains cards nu mbe red 1 to 100. A card is drawn at random from the box. The pro bab ility of dra wing a number which is a square, is (a) 1/5 (b) 2/5 (c) 1/10 (d) Non e of thes e 28. An integer is chosen at random from the numbers 1,2, .., 25. The probability that the chosen number is divisible by 3 or 4, is (a) 2/2 5 (b) 11/25 •(c) 12/25 (d) 14/25 Two players A and B throw a die alternately for a prize of Rs. 11/- which is to be won by a player who first throws a six. If A starts the game, their respective expectations are (a) Rs. 6; Rs. 5 (b) Rs. 7; Rs. 4 (c) Rs. 5.50; Rs. 5.50 (d) Rs. 5.75; Rs. 5.25 30. A three-digit number is selected at random from the set of all three-digit numbers. The prob abi lity that the number sel ected has all the three digits same, is (a) 1/9 (b) 1/10 (c) 1/50 (d) 1/100 31. In a college 20% student s fail in mathematics, 25% in Physics, and 12% in bot h sub jects. A student of this college is setected at random. The probability that this student who has failed in Mathematics would have failed in Physics too, is : (£01/ 20 (b) 3/25 (d) 12/25 (d) 3/5 A purse contains 4 copper and silver coins, and a second purse contains 6 copper and 2 silver coins. A coin is taken out from any purse, the pro bab ility that it is a co pp er coin, is (a) 3/7 (b) 4/7 (c) 3/4 (d) 37/5 6 33. Three of th e six vertices of a regula r he xagon are chosen at random. The probability that the triangle with three vertices is equilateral, is

<4 J_ (d) 20


70

34. T he probabil itie s of two events A and B are 0-3 and 0-6 respectively. The probability that both A an &B occur simultane ous ly is 0 1 8 . A nor B Then the probability that neither occurs is (a) 0 1 0 (b) 0-28 (c) 0-42 (d) 0-72 35. One mapping is selected at random from all the mappings of the set A = {1, 2, 3,.., n] into itself. The probability that the mapping selected is one to one is given by

(a) A (C)

(»-l) n-I

(d) None of these

Two persons each makes a single throw with a pair of dice. The probability that the throws are unequal is given by 73 (aA
If the mean and va rianc e of a binomial variate X are 7/3 and 14/9 respectively. Then X proba bility that takes value 6 or 7 is equal to ^ (C)

729

(b)

729 13

729

bag contains a white and b black balls. Two players A and B alternately draw a ball from the bag, replacing the ball each time after the draw. A begins the game. If the pro bability of A winning (that is drawing a B white ball) is twice the probability of winning, then the ratio a: bis equal to (a) 1 : 2 (c) 1 : 1

(b) 2 : 1 (d) Non e of thes e at random from 100

X

Y

(a) 2/3 (c) 1/50

00, 01, 02, ..., 99. Suppose product of the digit P (X = 7/Y=0) is given (b) 2/19 (d) Non e of these

Let X be a set containing n elements. Two subsets A and B of X are chosen at random the probability that A u B = X is (a)

2n

(b) 1/ 2 "C„

Cn/2^

(c) 1.3.5. ... (2n - l)/2 " n ! (d) (3/4)" A natural number x is chosen at random from the first 100 natural numbers. The probability 100

that x +

cn-is > 50

x (a) 1/10 (b) 11/50 (c) 1/20 (d) Non e of thes e If X and Y are independent binomial variates B (5, 1/2) and B (7, 1/ 2) then P(X+Y= 3) is (a) 55/1024 (b) 55/4098 (c) 55/2048 (d) None of these A die is rolled three times, the probability of getting a larger number than the previous number each time is (a) 15/216 (b) 5/54 (c) 13/216 (d) 1/18 A sum of money is rounded off to the nearest rupee; the probability that round off error is at least ten paise is (a) 19/101 (b) 19/100 (c) 82/101 (d) 81/1 00 Eight coins are tossed at a time, the probability of getting atl east 6 heads up, is 7 57 (a) 77 (b )7 7 64 64 37 229 (d) ^ 256 256 10% bulbs manufactured by a company are defective. The probability that out of a sample of 5 bulbs, none is defective, is (a) (1/2)

5

(c) (9/ 10)

(b) (1/1 0) (d) (9/ 10)

5 5

Of the 25 questions in a unit, a student has worked out only 20. In a sessional test of that unit, two questions were asked by the teacher. The probability that the student can solve both the questions correctly, is (a) 8/25 (b) 17/25 (c) 9/1 0 (d) 19/30

Probability

71

48. The probabili ty that at least one of the events A and B occur is 0-6. If A and B occur simultaneously with probability 0\2, then P(A) + P(B), where AandB are complements of A and B respectively, is equal to (a) 0-4 (b) 0-8 (c) 1-2 (d) 1-4 49. Let /I = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8}. An element (a, b) of their cartesian product

A x B is chosen at random. The probability that a + b = 9. is (a) 1/5 (b) 2/5 (c) 3/5 (d) 4/5 50. Dialing a telephone numbe r, a man forgot the last two digits and remembering only that they are different, dialled them at random. The probability of the number being dialed correctly is (a) 1/2 (b) 1/45 (c) 1/72 (d) 1/90 51. If A and B are any two events, then the pro bab ility that exa ctl y one of the m occurs, is (a) P ( A n B ) + B(A nB) (b) P [A u B) + P (A u B) (c) P(A) + P (B) - P (A n B) (d) P (A) + P (B) + IP (A n B) 52. A speaks truth in 609c truth in 70% cases. The will say the same thing single event is (a) 0:56 (b) (c) 0-38 (d)

cases and B speaks probability that they while describing a 0-54 0-94

53. If the integers X and (i are chosen at random between 1 to 100 then the probability that a number of the form 7X + 7U is dirisible by 5 is (a) 1/ 4 (b) 1/ 7 (c) 1/ 8 (d) 1/ 49 54. If two events A and B are _such that P (A) > 0 and B (B) * 1, then P ( A/B ) is equal to

(a) I — P (A/B)

P(B)

n times. Let X= the head occurs. If

1

- P ( A/B)

P(B)

55. A three digit number, which is multiple of 11, is chosen at random. Probability that the number so chosen is also a multiple of 9 is equal to (a) 1/9 (b) 2/9 (c) 1/100 (d) 9/10 0 \ - p 1-2/7 T , l+4p so. If —— 4 ' - , ——^ 4 , —— 2 '- are probabilities ot three mutually exclusive events, then 1 (b )- J< p< J (a )\
(C)j:
(d1) None of these

57. A box contai ns ticke ts number ed 1 to 20. 3 tickets are drawn from the box with replacement. The probability that the largest number on the tickets is 7 is (a) 7/20 (c) 2/1 0

(b) 1 - ( 7/ 20 f (d) Non e of thes e

58. Tw o nu mbe rs .v and y are c hosen at ra ndom from the set {1. 2. 3 30}. The probabilit y that .v" - y" is divis ible by 3 is (a) 3/29 (c) 5/29


59. A fair die is thro wn until a score of less than five points is obtained. The probability of obtaining less than three points on the last throw is (a) 3/4 (b) 4/5 (c) 5/6 (d) 1/2 60. Seven di gits fr om the digit s 1. 2. 3. 4. 5, 6, 7. 8. 9 are written in a random order. The probabili ty that this seven digi t nu mb er is divisible by 9 is (a) 2/9 (b) 1/5 (c) 1/3 (c) 1/9

Each question, in this part, has one or more than one correct answers). a. b. c. d corresponding to the correct answerfs). 61. A fair coin is tossed number of times

( b)


p (X = 4) . P (X = 5) and P (X = 6) are in A.P. then the value of n can be

72


(a) 7 (c) 12

(b) 10 (d) 14

62. A and B are two events, the probability that exactly one of them occurs is given by (a) P (A) + P ( B ) -2 (A nf i) (b) P (A u B) - P (A nB) (c) P (A') + P{B') - 2 P (A' n B') (d) P (A r\ B") + P (A' n B) 63. A wire of length I is cut into three pieces. What is the probability that the three pieces form a triangle ? (a) 1/2 (b) 1/4 (c) 2/3 (d) Non e of these 64. Suppose X is a binomial variate B (5, p) and P (X=2) = P (X= 3), then p is equal to (a) 1/2 (b) 1/3 (c) 1/4 (d) 1/5 65. A bag conta ins four tickets marked with 112, 121, 211, 222 one ticket is drawn at random from the bag. Let E, {i = 1, 2, 3} deno te the event that ith digit on the ticket is 2. Then (a)

and E2 are independent

(b) E2 and E3 are indepndent (c) E 3and

are independent

(d) £], E2, E3 are independent 66. A bag conta ins fou r tickets numbere d 00, 01, 10, 11. Four tickets are chosen at random with replacement, 'the probability that sum of the numbers on the tickets is 23 is (a) 3/32 (b) 1/64 (c) 5/256 (d) 7/256 67. A natural number is selected at random from the set X= (x/1
{x/0 < x < 10, x is an integer} the probability that Ix — > I < 5 is 87 89 (a) (b)

, ^ 91

121

121

(d)

101 121

The adjoining Fig. gives the road plan of lines connecting two parallel ro^ds AB and A XBX. A man walking on the road AB A jB,. takes a turn at random to reach the road It is known that he reaches the road A XB\ from O by taking a straight line. The chance that he moves on a straight line from the road AB to theAjB, is (a) 0-25

(b) 0 04

(c) 0-2 (d) Non e of these If a e [ - 20, 0], then the probabilit y that the grap h of the function y= 1 6 x + 8 ( a + 5) x - la - 5 is strictly above the x-axis is (a) 1/2 (c) 17/20

(b) 1/17 (d) Non e of thes e

Two distinct numbers are selected at random from the first twelve natural numbers. The prob abi lit y that the sum will be div isibl e by 3 is (a) 1/3 (c) 1/2

(b) 23/ 66 (d) No ne of the se

If A and B are independent events such that 0
(b) A, B are exclusive

(c) P (A) = P (B) (d) P (B) < P (A) The probability that out of 10 persons, all born in April, at least tw o have the same birthday is 30, 30, (a)-

'10

(30)'°(C)

(b) 1 —

(30) 10 30

10

C 10

'1 0

30 !

(d) None of these

(30) A pair of fair dice is rolled toge ther till a sum of either 5 or 7 is obtained, the probability that 5 comes before 7 is (a) 0-2 (b) 0-3 (c) 0-4 (d) 0-5

Probability

73

76. A second order determinant is writeen down at rand om using the number s 1, - 1 as elements. The probability that the value of the determinant is non zero is (a) 1/2 (b) 3/8 (c) 5/8 (d) 1/3 77. A five digit num ber is chosen at rand om. Th e probability tha t all the digit s are dis tin ct and digits at odd place are odd and digits at even place s are even is (a) 1/ 25 (b) 1/ 75 (c) 1/ 37 (d) 1/ 74 A natural numb er is selected fro m 1 to 1000 at random, then the probability that a particular non-ze ro dig it app ear s at most once is (a) 3/ 25 0 (b) 143/250 (c) 243/25 0 (d) 7/250 79. If A and B are events at the same experiments with P(A) = 0-2, P (B) = 0-5, then maxi mum value of P(A' nB) is (a) 1/4 (b) 1/2 (c) 1/8 (d) 1/16 80. xx,x2, Jt3, ..., xi0 are fif ty real numbers such

xr
that

fo r

r= 1, 2, 3, ..,49.

Fiv e

numbers out of these are picked up at random. The probability that the five numbers have x 20 as the middle number is 2Q

(a)

50 19

30

C2x30C2

(b)

C

C2 x

31

C3

c<

If A and B are two events such P (A) = 1/2 and P (B) = 2/ 3, then (a) P (A yj B)> 2/3 (b) P (A n B') < 1/3 (c) \/6
two

events

A and B,

P(A) = P(A/B)=\/A and P (B/A) then (a) A and B are independent (b) A and B are mutually exclusive (c) P ( A ' / B ) = 3/4 (d) P (B'/A') = 1/ 2

A bag contains 14 balls of two colours , the number of balls of each colour being the same. 7 balls are drawn at random one by one. The ball in hand is returned to the bag be fo re each new drawn. If the proba bility that at least 3 balls of each colour are drawn is p then 1 (a) p > ^

J_ (b ) P =; : 2

(c)p<\

(d )p<

2

All the spades are taken out from a pack of cards. From these cards; cards are drawn one by one wit hou t re plac ement till the ace of spades comes. The probability that the ace comes in the 4th draw is (a) 1/13 (b) 12/13 (c) 4/13 (d) Non e of these

2nC,

(d) None of these

50-

( a ) p = 1, 0 = 0 (b ) p = 2/3, q= 1/2 (c) p = 3/5, q = 2/3 (d) there are infinitely many values of p and q.

86. Let X be a set containing n elements. If two subsets A and B of X are picked at random, the probability that A and B have the same number of elements is

C 2 X 19 C 2 50

A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The proba bilit ies of the student passing in tes ts I, II, III are p, q and 1/2 respectively. If the proba bility tha t the student is succ es sful is 1/2 then

n

that

if = 1/2,

1_ (b) 2 ris-,

1.3.5. ..•(2n-i) 3" I a" 2 .n ! 4 A die is thrown 2 n + 1 times, n e N. The pro bab ility that faces with even numbers show 2 nodd + 1 number of times is 1 (b) less than (a) 2 2n + 3 (c) greater than 1/2 (d) Non e of these 88. If E and F arc the complementary events of the events E an d fjc spc cti vcl y th en (a) P (E/F) + P ( E/F) = 1 (b) P (E/F) + P (E/F) = 1


74

(c) P ( E/F) + P (E/F) = 1 (d) P (E/F) + P ( E/F) = 1 89. A natural number x is chosen at random from the first one hundred natural numbers. The (x - 20) (x - 40) pr obab il it y that < 0 is (x-30) (a) 1/50 (b) 3/50 (c) 7/25 (d) 9/50

90. Three six faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is k (3 < k < 8) is (k-\)(k-2) (a) (b) 432 432 (C)*"'C

2

216


M.M. : 20

(A) There are 10 parts in this question. Each part has one or more than one correct A four digit number (numered from 0000 to 9999) is said to be lucky if sum of its first two digits is equal to sum of its last two digits. If a four digit number is picked up at random, the probability that it is lucky number is (a) 1 67 (b) 2-37 (c) 0 067 (d) 0-37 2. A number is chosen at random from the numbers 10 to 99. By seing the number a man will laugh if product of the digits is 12. If he chosen three numbers with replacement then the probability that he will laugh at least once is s3 , t o s3 31 43 (a) 1 (b) 1 — 45 45 41 42 (d) 1 - 7c 45 43 3. If X follows a binomial distribution with p a r a m et er s n =8 and p= 1/ 2 the n p ( | x-4 | < 2) = (a) 121 /12 8 (b) 119/ 128 (c) 117 /12 8 (d) 115/ 128 4. Two numbers b and c are chosen at random (with replacement from the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9). The probability that (c) 1

2

x + bx + c > 0 for all x e R is (a) 17/123 (b) 32/81 (c) 82/ 125 '(d) 45/ 143 5. Suppose n boys and m girls take their seats randomly round a circle. The probability of their sitting is ( 2 n >Cn ) 1 when

answer(s). [10 x 2 = 20]

(a) no two boys sit together (b) no two girls sit together (c) boys and girls sit alternatively (d) all the boys sit together 6. The probabilities of different faces of a bi as ed dic e to appear ar e as follows : Face number Probability

1 01

2 3 0-32 0-21

4 015

5 005

6 017

The dice is thrown a nd it i s known th at either the face number 1 or 2 will appear. Then the probability of the face number 1 to appear is (a) 5/21 (b) 5/13 (c) 7/23 (d) 3/10 7. A card is selected at random from cards numbered as 00, 01, 02, ...,99. An event is said to have occured. If product of digits of the card number is 16. If card is selected 5 times with replacement each time, then the pr ob ab il it y t h a t the eve nt occu rs ex ac tl y three times is ,3 97 3 (a) 5 C 3 100 100 (b) 5 C a (c) 5 C 3

3

97

100

100

•3

97

100

100

(d) 20 (0 03) 3 (0- 97)2' Let A, B, C be three mutually independent events. Consider the two statements Sj and S 2 .

Probability

75

S j : A and B u C are independ ent S 2 : A and B n C are independent (a) Both Si and S

2

the two players P eight winners is (a) 4/15

then

are true

(b) Only Si is true (c) Only S 2 is true (d) Neither Si nor S

2

is true

9. Sixteen players ^16 Pây i n a Pi, P 2 - •••» tournament. They are divided into eight pa ir s at ran do m. Fr om ea ch pai r a win ne r

t

and P

2

is among the

(b) 7/15

(c) 8/15 (d) 17/30 10. The probabi lities t ha t a stu de nt in Mathematics, Physics and Chemistry are a, p and y respective ly. Of the se subjects, a student has a 75% chance of pa ss in g in a t lea st one, a 50% ch an ce of pa ss in g in at le ast two , an d a 40% ch an ce of

is tw decided on the of eras game playedpa ir. be een the tw obasis pl ay of the Assuming that all the players are of equal strength, the probability that exactly one of

pa ss in g in ex ac tl y tw o su bj ec ts . Whic h of the following relations are true ? (a) a + p + y = 19/20 (b)a + P+ y =2 7/ 20 (c) ap y= 1/ 10 (d) aPy = 1/ 4


must be 100%

Answers Multiple Choice l.(c)

2. (b)

3. (b)

7. (b) 13. (b)

8. (d)

9. (c) 15. (d)

19. (c) 25. (b) 31. (d) 37. (b) 49. (a) 55. (a)

Multiple

Choice

5. (c)

6. (c)

11. (b)

12. (d)

16. (b)

17. (a)

18. (d)

22. (d)

23. (b)

24. (b)

26. (c)

21. (c) 27. (c)

28. (c)

29. (a)

30. (d)

32. (d)

33. (c)

34. (b)

35. (c)

36. (d)

38. (c) 44. (d)

39. (b)

40. (d)

41. (d)

42. (a)

45. (c)

46. (d)

47. (d)

48. (c)

50. (d) 56. (d)

51. (a) 57. (d)

52. (b)

53. (c)

54. (c)

58. fd)

59. (d)

60. (d)

14. (d) 20. (b)

43. (b)

4. (b) 10. (b)

-II

61. (a), fd)

62. (a), (b), (c), (d)

63. (b)

64. (a)

66. (a)

67. (b)

68. (c)

69. (c)

70. (d)

65. (a), (b), (c) 71. (a)

72. (b), (c), (d)

73. (a)

74. (c)

75. (c)

76. fa)

77. (b)

78. (c)

79. (b)

80. (b)

81. (a,, (b), (c), (d)

82. (a), (c), (d)

83. (a), (b), (c), (d)

84. (a)

85. (a)

87. (d)

88. (a, d)

90. (a, c)

Practice

89. (d)

86. (a, c)

Test

l.(c) 7. (b), (d)

2. (b) 8. (a)

3. (b) 9. (c)

4. (b) 10. (b), (c)

5. (a), (b), (c)

6. (a)

LOGARITHMS AND THEIR PROPERTIES e is the base of natural logarithm (Napier logarithm). In x= lo g e x

i.e., i.e.,

and logio eis known as Napierian constant l o g 1 0 e= 0-43429448 ... Since lnx= logio x. loge 10 =

Inx= 2 303 logio x

(i)

a k = x<^> l o g e x = /c; a > 0, a * 1, x > 0

(ii) a * = e * '

(iii) n! (x) = 2nn /'+ ln(x) ; x * 0,i=^T, x> 0 (iv) log (v) log a ( m/n ) = log a m - l og a n;a> 0, a* 1, m, n> 0 (vi) log a (1 ) = 0; a > 0 , a * 1 (vii) log a a= 1; a > 0, a * 1 (viii) loga (mf = n log a (m); a > 0, a * 1, m> 0 (x)

a'°

9s w

=x; 4 > 0, a * 1 , x > 0

(ix) a'

(xi) log/, (x) =

logi

a

i(x) ; x > 0,

(xv) log a (x 2) * 2 log (x)a

na

,a> 0

(mn) = log a m+ log a n; a > 0, a * 1, m, n> 0

096 (x)

= (x)

loga (x) lo g a (b)

logt,{a)

; b * 1; a, b, xan positive numbers.

; a * 1, bit 1; a, b, xare positive numbers.

(xiii) loga {x) 2k = 2k \oga I x\; x> 0,

(xii) loga* (x) = j lo g a (x); a> 0, a * 1, x > 0 (xiv) loga2ic (x) = ^

a

= 2-30258509 .. logio e

a > 0, a * 1

0; a > 0, a * 1 Sin ce domain of log

a

(x 2) is R ~ {0}

and domain of log a (x) is (0, °<>) ar e not s a m e and for x < 0, lo g a x is imaginary. (xvi) log a x > 0 if either x > 1, a > 1 o r 0 < x <1 , 0 1, 0 < a < 1 or 0 < x < 1, a > 1 (xviii) If loga (x) > log a (y) then x > y if a > 1 and x 1, loga (x) < k o 0 < x < a k and log

a

(x) > k <=> x > a k

(xx) If 0 < a < 1, loga (x) < k o x > a k

a

(x) > k o 0 < x < a k

and log

(xxi) Ifa" = a y then the following case hold : (i) x an d y ca n be an y inte ger if a = 1 (iii) x a n d y e a n be any real num ber if a = 0

(ii) x a n d y ca n be an y ev en int ege r if a = - 1 (iv) x = y i f a * 0, - 1, + 1.

MULTIPLE CHOICE—I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d which ever is appropriate. 1. If A = lo g 2 lo g 2 lo g 4 256 + 2 logvy 2 then (a) 2 (b) 3 (c )5 71og|ifl

(d) 7

+ 51ogD fffV3 15 J ' |24 | ' is equal to

A=

(a )0 (c) log 2

(b) 1 (d) log 3

3. For y = log n x to be defined 'a' must be il 1og ° | 80

(a) any positive real number (b) any num ber (c) > e (d) any positive real number ^ I

Loga rithm s and Th ei r Properties

77

If log 10 3 = 0-477, the number of digit in 3 (a) 18

(b) 19

(c) 20

(d) 21

If x = log 3 5 , y = log

I7 25,

40

(a) xyz (c)0 1

is

- +

whic h on e of the

6. Th e dom ain of the fun ctio n V(log (a) (1, oo) (b) (0, oo)

0

If (4

+ 7

1

lo g 3 n

+

...

1

0

is

1

ab

c

iog

(9 )

4=

2

(]

]og, 83

())

t h e n

x

=

(b) 3 (d) 30 z

(c) log fl c = \ogb a (d) log 16

x(y + z-x)

=

a

b = 2 log a c

y(z + x-y)

lo g*

z(x+y-z)

=

logy

logz

!o g

1

3-a 3 +a 3 -a 3+aj

(d) 5

I0

tan 89° =

=^ c— a

(d) xxyy 3

(2^ - 5), log 3 (2* - 7 / 2 ) are in

(b) 2 (d) 4

If y = a' ~ l0«»x and z = a 1' l

lo

s« >' then x = l

2+,

(a) a' +'°s„z l (c)

=

7

(a) 2

(b) 8

(c) 32

(d) 64

If log 3 (x - 1) < log

2

(d) 1 - 3 log

2

z

(b) a °g,°g„z

10

[3+a

(b) 3 log

(c) 1 - 3 log 72

(c)//

(c )3

3 —a 3+a 3-a

(b) 3

(b) xzyx

(a) 1

lo g 43 n !

6 16

(a )ZV

A.P. then x =

(b) 1 (d) 81

(a) 3 log72

b-c

3+

17- If log 3 2, log

lo g 43 n

11- log 7 log 7

12.

1

"

1 + log

y

tan 1° + lo g 10 tan 2° + ... + log

If log 12 27 = a then

(c)4

log 43 n

(d)

(a )0 (c) 27

(a) 2-

+

(b)

lo g 42 n

log|

+

then x"yx = z*'y l = 1

1 (a) log 43 ! n (c)

ca

15. If x , y, z are in G.P. and a = b = c then (a) log fc a = lo g c b (b) log c b = log a c

8. The value of lo g 2 n

y°g*

fc

(b) 1 (d) 3

(a )2 (c) 10

5 x) is

(a) an integer (b) a rational num ber (c) an irrational number (d) a prime number 1

:

1 + log

(a) 0 (c) 2

(d) (0-5, 1) is

2

-

1 + log a be

following is correct ? (a )x y (d) Non e of these

(c) (0, 1] The number log

(b) abc (d)l 1

7

=

0 09(

x - 1) then x lies in

the interval (a )( -o o, i)

(b) (1,2)

(c) (2, » )

(d) No ne of these

a-b

Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). Th e value o f 3

,og4 5

— 5 ,og4 3 is

(a)0

(b)l

(c) 2 (d) No ne of the se The value of \ogh a. lo g c b. log ;/ c. lo g a d is (a) 0 (c) log 1

(b) log abed (d) 1

If x 1 8 =/'=z

28

then

3, 3 lo g

7 log^ are in (a) A.P . (c) H.P.

(b) G.P . (d) A.G .P.

If — - — + — - — > x then x be lo g 37i lo g 4 n

v x,

3 log z y,


78

(a) 2 (c) n


(a )- 2 ( c) o

]o x Sl LOG2)' '0 G2 ^ , 3 2 , , TU 25. I JCf — - = — = ^ - a n d x ; y z = 1then/: =

(a)-8

(b)-4 v

(c)0

(d)l0g2

a

(b) 3

a

I f

.b

\Qaa J^£L (b -c) b+c

.c

=

10S.b LOSC ° = l u £ ^ then (c - a) (a - b)

ic + a

-

a+h

)1

(256j

+ M ( Ina + lnb \ , 26. If—-— In = 2 then

(a)l (c )5

2 8.

(b )- l (d) 1

(b

b — b + a— -

(c)a + b + c

(a) 2

(d) 7

(d) log,,

(b) 4 (c )8

If n = 1983 ! then the value of 1 ' 1 - + + ; —+ .. log 2 n log 3 n log 4 n " ' ' log 1983n

a. log c b log a c

29. If log 3 (5 + 4 log 3 (x — 1)) = 2, then x = (d) log 2 16

30. if 2x' og4 3 + 3' 084 x = 27 then x: (a) 2 (b) 4

is equal to

(c) 8

(d) 16

Practice Test M.M. : 10

Tim e 15 Min.

(A) There are 5 parts in this question. Each part has one or more than one correct answer(s). [5x2= 10] (a) 2 (b) 4 1. Th e inte rv al of .r in which the inequality (c )6 (d) 8 gl/4 log* x > 5 i 1 / 5 log5 x 4. The solution of the equat ion (a) (0, 5" 2 (b)[5 2 V r ,~) log 7 log 5 (Vx + 5 + -Jx) = 0 is (d) None of these (c) both (a) & (b) 2. The solution set of the equa tion log x 2 loga* 2 = log^ 2 is

(a) {2 -^ , 2 ^)

( b ) {1 / 2 , 2 }

(a) 1 (b )3 (c) 4 (d) 5 5. The num ber of solutions of log 4 (x - 1) = log 2 (x - 3) is (a) 3 (c)2

,2

(c) {1 /4 , 2 j (d) None of th es e 3. The least value o f the expressi on 2 log] 0 x - \ogx 0 01 is

(b) 1 (d)0

Answers Multiple Choice -I 1. (c)

2. (c)

(c)

(a)

(b) (c)

(c) (c)

3. (d) (a) (a)

(c) 10. (c) 16. (a)

.(c) (c)

6. (c) 12. (a) 18. (c)

(a), (d)

26. (d)

11. (c)

Multiple Choice -II 21. (a)

22. (a), (c)

(a)

(a)

(d)

(b), (d)

(b)

(d)

(b)

(c)

Practice 1. (c)

Test 2. (a)

(b)

MATRICES

An m x n matrix is usually written a s an 321 ami

ai2 a2 2

•• •••

a m2 •

am a2n

amn

where the symbols ai/ represent any numbers (a/y lies in the Ah row (from top) and /th column (from left)). Matrices rep res ent ed by [ ], (), IIII : If two mat ric es and are of the same order, then only their addition and subtraction is possible and the se matrices are said to be for addition or subtraction. On the other hand if the matrices an d are of different orders then their addition and subtraction is not possible and the se matri ces ar e called for addition an d subtracti on.

: A square matrix be relation matrix

of

is called idempotent provided it satisfies the relation

: A square matrix is the least positive integer for which A k+ then k is said to For k= 1, we get and we called it to be : A square matrix is called Nilpotent matrix of order m provided it satisfies the 1 a nd * 0, where k is positive integer and is null matrix and k is the order of the nilpotent : A square matrix

is called involutory provided it satisfies the relation

where I is identity matrix. : A square matrix will be called symmetric if for all values of / and

j, i.e., a,j= aji or

: A square matrix is called skew symmetric matrix if (i) a,y=-a// for all val ues of / and j. (ii) All diagonal elements are zero, or Note : Every sq ua re matrix can be uniquely ex pr es se d as the su m of symmet ric and sk ew symmet ric matrix.

i.e., are symmetric and skew symmetric parts of : A square matrix and its transpose is an identity matrix. i.e., If = / then If an d are orthogonal then

where ^

and ^

is called an orthogonal matrix if the product of the matrix AA' = I

is also orthogonal.

80


: if a matrix is having complex numbers as its elements, the matrix obtained from by replacing each element of by its conjugate (a ± ib= a+ ib) is called the conjugate of matrix and is denoted by : A square matrix

such that

is called Hermitian matrix, provided a,/ = ajifor

all values of /' an d j or : A square matrix

such that

is called skew-hermitian matrix,

provide d afy = - ay for all values of i an d j or : A square matrix and

is called a unitary matrix if

where

is an identity matrix

is the transposed conjugate of Properties of Unitary Matrix (i) If is unitary matrix, then is also unitary. 1 (ii) If is unitary matrix, then is also unitary. (iii) If and are unitary matrices then is also unitary.

be n rowed square matrix then (adj (adj i.e., the product of a matrix and its adjoint is commutative. If

If

I

I In

rowed square singular matrix then (adj (adj = 0 (null matrix) since for singular matrix, I I If is rowed square non-singular matrix, then I ad j I I since for singular matrix, I I Adj = (Adj • (Adj (Adj = Adj adj (adj

I

is

l"~ 2

where

is a non-singular matrix.

.2

I {Adj (Adj) I I l tn ' , where is a non-singular matrix. Adjoint of a diagonal matrix is a diagonal matrix. d et (nA) = n" det (A) "a

0

0" such that I 0

I * 0

0

I° 0

1

C

(1) IfI I * 0, then the system of equations is consistent and has a unique solution give by (ii) IfI I = 0 and (adj * 0, then the system of equations is inconsistent and has no solutions. (iii) If I I = 0 and (adj then the system of equations is consistent and has an infinite number of solutions.

Matrices

81

(i) If I I * 0, the system of equations have only trivial solution and it has one solution. (ii) IfI = 0, the sy ste m of equ ati on s h as non-trivial solution and it ha s infini te solutions . (iii) If No. of equations < No. of unknowns, then it has non trivial solution. : Non-homogeneous linear equations also solved by Cramer's rule this method has been discussed in the chapter on determinants. § 9.5 Rank of Matrix The rank of a matrix is said to be r if (i) If has at lea st m inors of order ri s different from zer o. (ii) All minors of of order higher than rare zero. The rank of is denoted by p . The rank of a zero matrix is zero and the rank of an identity matrix of order n is n. The rank of a matrix in echelon form is equal to the number of non-zero rows of the matrix. 3. The rank of a non-singuiar matrix (I I * 0) of order n is n.

: If Rank of = Rand of : Rank of = Rank of n n = number of unknowns : Rank of = Rank of = r

(1) where (ii) where

r< n i.e., no solutions. Rank of # Ra nk of

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. Let A and B be two matri ces , then (a) AB = BA (c) AB < BA

(b) AB * BA (d) AB > BA

2. Let A and B be two matrices such that A = 0, AB = 0, then equation always implies that (a) B = 0 (b) B * 0 (c) B = - A (d) B = A ' 3. In mat ric es : (a) (A + B) 2 = A 2 + 2AB + B (b) (A + B)

2

"3 6. 2

2

(d) (A + B) 2 = A 2 + 2BA + B

2

(a) x = 3,y = — (c) x — l,y = 7. Give n A =

*

4" -3 (b) x = 2,y = 5 (d) .x = - l , y = 1 , which of the following

result is true ? (b) A (d) No ne of these

of which of the following matrices exists ? (a)

(b) A • A " = I (d) A • A' = I 0 0" 3 0 is equ al to 0 5 (b) 3

1 co CO

1

L

y

8. Wi th 1 co, co2 as cube roots of unity, inverse

4. The characteristic of an orthogonal matrix A is

(a) 4

X

(c) A"=1 21

(c) (A + B) * A 2 + 2AB + B

(a) A ' A = I (c) A' A " 1 = I '4 5. Th e rank of 0 0

(d) 1 - r 5

(a) A 2 = I

2

= A2 + B2

2

(c)5

(c)

•>

CO"

CO (b)

2

CO

CO 1

1 CO

2

CO 1

(d) None of these

If A is an orthogonal matrix, then A

1

equals

82


(a) A

(b) A'

- 2

18. If A =

(c) A 2 (d) None of these

3 -3 2

10. If A :

, then trace of A is,

16

(a) 17 (c) 8

(b) 25 (d) 15

- 2

1

(a) A

(b) A

(c) 3 A

(d) 3 \

19. The matrix

n x n, then

11. If A is a square matrix of order adj (adj A) is equal to

then adj A=

- 2

- 2

T

is the matrix reflection in

the line

(a)x=\

(b)x + y= 1

1

I A I" ~ A I A l" ~ 3 A ,* T T If A is a squa re matr ix, then ad j A - (ad j A) is equal to (a) 2 I A I (b) 2 I A 11 (c) Null (d) Unit matrix

(c) y - 1 (d) x = y 20. If I„ is the identity matrix of order n, then s-1 (U"

13. If A =

21. If A =

(a) I A I" '(c) IA I"A- 2

(b) (d)

„ is a matrix of rank r, then

x

(a) r = min (m, n)

(b) r > min (m, n)

(c) r < min (m, n)

(d) None of these

14. If A is an ort hogo nal ma tri x, then (a) I A I = 0 (b) IA I = ± 1 (d) None of these (c) IA I = ± 2 1 1 - 1

15. The matri x

2 2 -2

3' 3 is -3

(a)

(c)

0

b 0

0 b 0

0" 0 c

0

1 - 2

is an idempotent matrix

x

then x = (a) - 5 (c)-3

(b) - 1 (d)-4

22. If A is non-sin gular matr ix, then Det (A

23. The matrix

(b);

1 - 1 1

1 Det (A z)

(d) Det1(A) - 3 —4 5 4 is a nilpotent -3 -4

matrix of index (a) 1 (c)3

(d) Cacuchy-Schwar 0 0

(d) n I„ 2 -2 - 4 -1 3 4

(c) Det

(a) Cauchy-Riemann (b) Caley-Hamilton (c) Newton

a

(b) I„

(c) 0

(a) Det

(a) idemp oten t (b) nilpote nt (c) involut ory (d) orthogonal 16. Matrix theory was introduced by

17. If A =

(a) does not exis t

1

then A

(b)

1 /a

0

0

0 0

\/b 0

0 1/c

a 0 0

= 0 ab 0

0 0 ac

- a (d)

0 0

0

-b

0

(b) 2 (d) 4

24. If A is a skew-symmetric matrix, then trace of A is (a)-5 (b)0 (c) 24 (d) 9 1 1 is unitary, then 25. If the matri x A = 12 - i

a= (a)-2 (c)0 26. If 3 A =

(b) —1 (d) 1 and AA' = I, then

Matrices

83

(a)-5 (c)-3

(b)-4 (d)-2

(c) 3

27. The sum of two idepotent matrices A and B is dempotent if AB = BA = (a) 4 (b)3 (c)2 (d) 0

=

(a) A" 1 B~1 C

1

(c) A

-1

C~

1

1

n x n matrices, B~' A - 1

(b)

(d j r ' c ' A " '

30- If A 2 + A - 1 = 0, then A~

is equal to

(a) 1

1

then (ABC)

3 28. The rank of

(d^) No ne of the se

29. If A, B, C are non singular

1

=

(a) A - I

(b

-A

(c) I + A

(d) Non e of these

(b)2

Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s).

31. If A =

then

(a) A"' = 9 A

32. A =

4 2x - 3

(a)f(-x)

(b) A" = 27 A ,„ .-! (d) A does not exist

.2 (c) A + A = A"

x +2 x+\

(a )3 (c )2

2

2

= ,2

i

2

'

a

2

2

+ — = 1 has c (a) no solution (b) unique solution (c) infinitely many solutions (d) finitely many solutions 8 - 6 34. If the matr ix A = - 6 7 2 -4 then X (a) 3

i +

2

2

(b) 6

2

_ 7 = i > - f L2+ y ,2 bL

2 -4 X

is singular,

5 8 X- 6

r

&X + 4 2X + 21

(a) for X = 2, p (A) = 1 (b) for X = - 1, p (A) = 2 (c)fo rA,?t 2,-l ,p(A) = 3 (d) None of these 38. If A and B are square matrices of order 3 such that IA I = - 1,1 B \ = 3, then I 3 AB I equals (a)-9 (c) -27

(b)4

(d) 27

1 X

A =

(c)2 (d)5 35. If A is a 3 x 3 matrix and det (3 A) = k {det (A ) } , k = (a )9 (c) 1

(b )/ (* )

is symmetric, then x

,~> Zr

0 0 =/ (* ), the n A ' = 1

(d)- f( -x ) 37. For all values of X, the rank of the matrix

33. Let a, b, c be positive real numbers. The following system of equations in x, y and z 2

sin JC cos x 0

(c )-f{x)

(b) 5 (d) 4

—2

cos x - sin x 0

36. If A =

(b) —91 (d) 81

39. The equatio ns 2x + y = 5, x + 3y = 5, x - 2y = 0 have (a) no solution (b) one solution (c) two solutions (d) infinity many solutions If A is 3 x 4 matrix B is a matrix such A and BA' are both defined. Then is of the type (a) 3 (b) 3 (c) 4 (d) 4

x x x x

4 3 4 3

84


Practice Test Time : 30 Min

M.M: 20

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] a1 (a) A (b) - A 1. If A= [o f e] , B= [ - 6- o ] an d C = -a ' ( c )N ul l mat rix (d) I then the coorect statement is 3 -4 , the value of X* is 7. I f X = (a) A = - B 1 -1 (b) A + B = A - B 2n + n 5 - n 3n - 4n (c) AC = BC (b) (a) n -re n - n (d) CA = C B 5 2 3 ( - 4) (d) None of these ,then A 1 = (c) 2. If A = 3 1 1" (-1)". (a) r

(c) 3. If

1 -3

-2" 5

(b)

-l -3

-2" -5

(d)

A=

then (a) A + B exists (c) BA exists

an d

1 3 1 3

2 -5 2 5

B=

5 4 -5

(b) AB exists (d) None of these

1 - 2 3 2 - 1 4 is a 3k 4 1 (a) recta ngula r m atr i (b) singul ar matr ix (c) square matrix (d) non singular mat rix A and B be 3 x 3 matrices. Then | A - B [ = 0 i mpli es (a) A = or B = (b) | A | = 0 an d | B | = 0 (c) | A | = 0 or | B | = 0 (d) A = and B = 1 0 0 2 , th en A is equa l to 6.If A= 0 1 0 a b -1

Matrix A such that A

2

= 2 A - I, where I is

the identity matrix. Then for (a) reA- (r e - 1) 1 (b) reA - 1

n > 2, A" =

(c) 2 n ~ 1 A - (re - 1) IIJ (d) 2 n _ 1 A - 1 9. For the equati ons x + 2y + 3z = 1, 2x + y + 3z = 2, 5x + 5y + 9z = 4, (a) there is only one solution (b) there exists infinitely many solutions (c) there is no solution (d) None of these 10. Consider the system of equat ions aix + b-\y + c-\Z = 0, a^x + b$y + c
c c

l 2

=0 c <*3 b3 3 then the system has (a) more than two solutions (b) one trivial and one non-trivial solutions (c) no solution (d) only trivial solution (0, 0, 0)

be 100%

Matrices 90

Answers Multiple Choice -1 1. (b) 7. (b) 13. (c)

2. (b) 8. (d) 14. (b)

9. (b) 15. (b)

19. (d) 25. (b)

20. (b) 26. (c)

Multiple

1. (c) 7. (d)

4. (d) 10. (d)

5. (b) 11. (c)

21. (c) 27. (d)

16. (b) 22. (d) 28. (a)

17. (c) 23. (b) 29. (b)

30. (c)

32. (b) 38. (a)

33. (b) 39. (b)

34. (a) 40. (a)

35. (d)

36. (a)

2. (b) 8. (a)

3. (c)

5. (d)

6. (d)

6. (a) 12. (c) 18. (d) 24. (b)

choice-ll

31. (a), (c), (d) 37. (a), (b), (c)

Practice

3. (c)

Test 9. (a)

4. (c), (d) 10. (a)

10 FUNCTIONS 1. Domain ( f (x) ± g (x)) = Domain f(x) n Domain g{x). 2. Domain (f(x).g(x)) = Domain f(x) n Domain g(x ). 3. Domain ^

j

=

Domain f (x) n Domain g (x) n {x: g (x) * 0}

4. Domain Vf(x) = Domain f(x) n {x: f (x) > 0}. 5. Domain (fog) = Domain [g (x), whe re fog is defined by fog(x) =

f[g(x)}.

Domain and Range of Inverse Trigonometric Functions

(i) (ii) (iii)

y = sin~ 1 x 1

y = cos" y = tan

- 1

x

-1
1

(iv) (v)

y = cot" x y = sec _ 1 x

(vi)

y -- co se c" 1 x

R

- 7t/2 < y< TI/2 0 < y< 7i

x<-1 x> 1 x<-1 x> 1

7t/2 < y < 7t 0 < y < 7t/2 ' - 7t/2 < y< 0 0< / < 7t/2

R

x r

- tc/2 < y < 7C/2 0 < y< 7i

(i) A function is an odd function if f (- x ) = - f(x) for all x. (ii) A function is an even function if f(- x) = f(x) for all x.

If a function f(x) is defined on the interval [0,a], it can be extend on [- a, a] so that f(x) is either even or odd function on the interval [- a, a], : If a function f(x) is defined on the interval [0, a] 0 < x < a = > - a < - x < 0 .•. - x e [-a, 0] We define f (x) in the interval [- a, 0] such that f(x) = f(- x). Let gbe the even extension, then f(x,); x e [0, a] SW = f( - x) ; x e [- a, 0] : If a function f(x) is defined on the interval [0, a] 0 < x < a => - a< -x< 0 - x e [- a, 0] We define f (x) in the interval [- a, 0] such that f(x) = - f (- x). Let gbe the odd extension, then . _ J f(x,) I X E [ 0 , a] g (x) = \-< f( -x ); x e [- a, 0]

Functions

87

A function f{x) is said to be a periodic functi on of x, if the re exist a positive real nu mb er 7" suc h that f(x+T) = f(x) for all x. The smallest value of Tis called the period of the function. If positive value of Tindepen dent o f xt he n f ( X ) is periodic function and if the value of T dep end s upon x, then /(x) is non-periodic.

(i) sin" x, co s" x, se c" x, co se c" x are periodic function s with period

2n an d k according as

n be odd or

even. (ii) ta n" x, co t" x are periodi c funct ion s with period rc, n even or odd. (iii) I sin x I, i co s x I, I tan x I, I cot x I, I sec x I, I cosec x I are periodic functions with period n. (iv) I sin x I + I cos x I, I tan x I +1 cot x I, I sec x I + I cosec x I are periodic functions with period (v)

If f(x) is a perio dic with peri od T, then the function f{ax+ b) is periodic with period

(vi)

If f (x), g (x) and h (x) an periodic with periods Ti, 72, F(x) = ar'(x) ± bg (x).+ ch (x) wh er e a, = { L.C.M. of (7 i. T 2, 7-3}, \ a b c ) L.C.M. of fa1, b, c} : L.C.M. — , — , — = — ' ' \ ci} [ ai £>1ci J H.C. F. of {ai,

...

Let f : A

la I

T3 respectively then period of b, c are constant

B be a One-One and Onto function then their exists a unique function g\B

A

<=> g(y) = x,

f{x) = y

such that

{9

(y) = r 1 (y)}

V x e A and ye B Then g is said to be inverse of and

for 1

n/2.

= r 1of

f.

= I, I is an identity function (fof

: If A and Bare two sets having

1

) X =

I(X )

=

X

f {r \x )}= x m a nd n elemen ts such that 1

(i) Number of functions from A to B = n m (ii) Number of onto (or surjection) functions from

then

A to B

77 = Z (- 1) " '."Cr/ r= 1 (iii) Number of one-one onto mapping or bijection = n! (If A a nd B have same number of elements say n)

y = 1, x>0 The signum function

f is defined as

Sgn f (x) = as x shown in the Fig 10.1

1 if x > 0 0 if x = 0 - 1 if x < 0

0 y = — 1, x<0

>

i Fig. 10.1

88


i.e. [x] < x.. [x] denotes the greatest integer less than or equal to x. Thu s [3-5778] = 3, [0 87] = 0, [5] = 5 [-8-9728] = - 9 , [-0-6] = -1 . n and (n + 1) In general if n is an integer and x is any real number between i.e., n
(i) Iff(x) = [ x+n ], where ne /a nd [ . ] denotes the greatest integer function, then /(x) = n + [x] (ii) x = [x] + {x}, [. ] and {} de no te the integral an d fractional part of x respectively (x) or [x] denotes the least integer function which is greater than or equal to x. It is also known as Thu s,

(3-578) = 4, (0-87) = 1, (4) = 4, [ - 8 239] = - 8, [ - 0 7] = 0

(ii) x = x + {x}- 1, {x} de no te s the fracti onal part of x (iv) ( - x) = - (x) + 1, x g / (vi) (x) >n => x > n, n e I (viii) ( x ) < n = > x < n - 1 , n e I (x) (x + y) > (x) + (y) - 1

(i) (x+ n) = (x) + n, n e I (iii )(- x) = - ( x ) , x e I (v) (x) > n => x > n - 1, n e I (vii) (x) < n=> x < n, n e I (ix) n 2 < (x) < n-i => n2 - 1 < x < ny, n-i,n 2 e I (xi) [ M n ' n+ 1 (xii) (xiii) (x)+

, n e N n +2

x+-

n+4 8

+ x+

n+8

2n, ne N

16

+ ... + x +

n-

1

= (nx) + n - 1, n e N

It is denoted as f (x) = {x} and defined as (i) {x} = fif x= n+ fwhere n e /and0
are R-l and (0, 1) respecti vely

It is given by "y= | xl ={ [ - x, x < 0 it is shown in the Fig

(i)lxl < a => - a < x < a ; ( a > 0) (ii) Ix l >a x < - a or x > a ; (a > 0)

Functions

89

(iii) I x + y I = Ix l + lyl « x> 0 &y > 0 o r x <0 &y < 0 (iv) I x - y I = I xl - I yl => x > 0 & I xl > I yl or x<0 & y < 0 & I xl > I y l (v) I x ± y I < Ixl + lyl (vi) I x ± y I > l lx l- ly ll

By considering a general nth degree polynomial and writting the expression.

it can be proved by comparing the coefficients of x", x"

1

consta nt that the polynomial satisfying the

above equation is either of the form f (x) = x" + 1 or - x n + 1. Now f(3) = 28 => 3^+ 1 = 28 => n = 3

1 (3) = - 3 n + 1 is not possible as - 3 n = 27 is Not true for any valu e of n. 7( 4) = 4 3 + 1 = 65.

Hence

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. If /( x) = - —-—j ^ e n / 1+ x

-1 ' (*) equa ls

(a) [ - 2, VlT]

(b) [ - 2, 3]

(c) [3, VT3~]

(d) No ne of the se

6. The domain of the functio n

(d) None of these

( c )((In

then (a)/(x) (b)/(x) (c)/(x) (d)/(x)

I)71 ,2«JI )

If g(x) = \x ] - [x] 2, where | . ] denotes the greate st integ er fu ncti on, and x e |(), 2], then the set of values of g (x) is (a) {- 1, 0} (b) {- 1, 0, 1}

is one-one but not onto is neither one-one nor onto is many one but onto is one-one and onto

The function f(x) = X I sin x I + +g has period equal to n/2 if X is (a) 2 (c) 3

-

(d) None of these

cos x I


If / i s decreas ing odd functi on th en / (a) Odd and decreasing (b) Odd and increasing (c) Even and decreasing (d) Even and increasing 5. The range of the function f(x) = 3 I sin x I - 2 I cos x I is

(c) {0}

(d) {0, 1,2}

8. Which of the following functions is periodic with period n ?

is

(a)/ (x) = sin 3x (b )/ (x )- =' fc os x I (c)f(x) = |x + 7t] (d )/ (x ) = x c o s x where |x] means the greatest integer not greater than x. 9. The domain of definition of

(a) (-« »,« .) - | - 2 , 2 |


90

(b) (-0 0, » ) - [ - 1, 1]

17. The value of b and c for which the identity / ( x + 1) — /( x) = 8x + 3 is satisf ied, where f (x) = bx" + cx + d are

(c) [ - 1 , 1] u (— 00 , — 2) u (2, (d) None of these 10. Let f-.R—^R be a given A a R and B czR then

= f (A)

(a)/(AuB) (b )f(AnB)

(c )f(Ac)

=

= [f(A)]

(d) /(A \B) 11. The

f(A)

funct ion

and

vjf(B)

18. The value of the parameter a, for which the fu nc ti on /( x) = 1 + ax, a * 0 is the inver se of itself, is

nf (B )

c

\f(B)

= /(A)

domai n

of y = logio logio logio ... log

function

the

(b )( 10 "" ', oo )

2

(c) (10"" , 00) 12. If

[x]

(a.)/(x)-

(d) No ne of thes e {JC} represe nt

and

(a) - 2 (c) 1

integral

a — 1 X

and

(c)/(x)=-

fractional parts of x, then the value of

a - a

flA

~ 1 a +1

(b)/(x)=x

-

(d)/(x) = sin x

a +a

20. If S is the set of all real x for which

2000 x+ r}

r= 1 2000

(b) - 1 (d )2

19. Which of the following functions is even function

10-

n times (a) [10". 00)

(a) b = 2,c=\ (b) b = 4, c = - 1 (c)b = - 1, c = 4 (A) b-—\,c=\

IS

l - e (lA)

- ' > 0 , then S =

(a) (- <*>, 0) u (1,

(a) x

(b) [x]

(c) {x}

(d) x + 200 1

(b) ( (d) Non e of these

(c)(-o=,0]u[l,oo)

13. If [.] denotes the greatest integer function

/( x) = sin 2 x + sin 2 |^x + Y j +

21. If

100

then the valu e of E r= 1 2 (a) 49 (c) 51

+

100

is

(d) 50 (d) 52

14. If /( x ) is a polynomial satisfying /( x) ./ (l /x ) =/(x ) +/ (l / x) and / (3) = 28, then/(4) = (a) 63 (b) 65 (c) 17 (d) Non e of these 15. If f ( x + y) =/ (x) + /( y) - xy - 1 for all x, y and/(l) = 1 then the number of solutions of/(« ) = n, n e N is (b) two (a) one (d) None of these (c) three 16. The function

/( x) = s i n [ ^

cos x. cos ^x + y j

g (5 /4 ) = 1

and

then

(gof) x is (a) a polynomial of the first degree in sin x, cos x (b) a constant function (a) a polynomial of the second degree in sin x, cos x (d) None of these 22. If the function / : [1,

—>[1,

b y / ( x ) = 2 l < A ' " l ) then/" 1 I)

1

is defin ed

(x) is

(a)'

(b)|(l +Vl +41og

2x)

TLX

(11+I)!

(a) not periodic (b) periodic, with period 2 (11 !) (c) periodic, with period (/; + 1) (d) None of these

(c)^(l

-4\+4\og2x)

(d) not defined

23. Let/be a function satisfying 2/(vv) = [f(x)}v+ then Z f(r) = r= 1'

{./'(y)}v and f( l ) =

k*l.

Functions

91

(a) k" - 1

x, if x e Q 1 - x , if xi Q Then for all x e [0, 1 ]/ /( x) is

(b) k"

(c) k" + 1

/«=

(d) None of these

24. Which one of the following functions arc

periodi c (a)/(x) = x

— [x], where [x] < x

(1 /x ) for * * 0, / (0) = 0 (c)f(x) = xc os x (d) None of these domain

of

the

function

33. Let /: /?— > Q be a continuous function such that/(2) = 3 then (a)/(x) is always an even function (b)/(x) is always an odd function (c) Nothing can be said about f(x) being even or odd (d)/(x) is an increasing function

(b) [0, 1], excluding 0-5 (c) [ - 2, 1], excl udin g 0 (d) None of these 26. The graph of the f unct ion y= / (x) symmetrical about the line x = 2. Then

is

(a)/( x + 2) = / ( x - 2)

34. The greatest value o f the func tion

2-x)

(c)/W =/(-*) (d) None of these The range of the function/(x) = 6

A

+ 3l + 6

1

+ 3~'1 + 2 is (b) ( - 2, oo)

(c) (6, oo)

(d) [6, oo)

29. i f

(b) [2, 5] (d) [2, 6)

fix)

(x -x2)

+

I XI

range 2

of function / : [0, - 1

= x -x +4 x + 2s in x is (a) [ -7 1- 2, 0] fb)| 2, 3| (c) [0,4 + 71] (d) f0, 2 + ji| 31. Let fix) be a funct ion defined on |0, that

([.] denotes the greatest integer function) (a )2 (b) 1 (c) 0

(d) - 1

36. I f /( x) = sin~'. { 4 - ( X - 7 ) 3 } 1 / 5 , then its inverse is 1/3 (a) (4 - si n 5 x) (b) 7 — (4 - sin 5 x) 1/3 , , , 5 ,2/3 (c) (4 - sin x) (d) 7 + (4 - sin 5 x ) 1/3

is (a) 2k (c) Jt/2 38. Given

11 ->

f.t]+ COS Tt( is

I sin x I I cosx I + —: 2 [ cos .1 sinx

[x - 1] is the greatest integer) 1 +V5"" 2 1 +V 5 Cd) None of these (c) \2, .2 3

2

nx + x -

„„ „ . . . r ,, . I f 37.The period o f / ( x ) = - ^

then domain of/(x) is (where

30. The

(d)3 4

T-l_ , r cos 35. The period of e

28. If f:X~>Y defi ned by / ( x ) = V3~sin x + cos x + 4 is one-one and onto, then Y is

(c) [5]1,

f{x) = cos {x e W + 2x 2 - x}, x e ( - 1, where [x] denotes the greatest integer less than or equal to x is (a) 0 (b) 1 (c)2

(a) [ - 2, oo)

(a) [1 ,4 ]

(d) Non e of thes e

R —> R is a function such that

/( x) = x 3 + x 2 /' (1) + xf" (2) + / " ' (3) for a ll xe /?, th en /( 2) -/ (l ) (a)/(0) (b)-/(0) (c) /'( 0) (d) - / ' (0)

1 0 ( l - x ) + V(x + 2) is /(x)=l/log (a) [ - 3, - 2] excluding ( - 2-5)

(b )/ (2 + j c) = / (

(b) 1 + x

(c) x 32. If/:

(b)/(x) =x sin

25. The

(a) Const ant

R,f(x)

11 such

(b)7t (d) n/4

x)

f(

1 • R (x) =f{f(x)l (1-x) Then the value of

and h(x) —f{f{f(x))}. fix), a (x). hix) is (a) 0 (b) - I (c) 1 (d)2 39. The inverse o f the functi on

y = log,, (x + V x 2 + I ) (a > 0, a * I) is

92

Ob j ective Mathematic s

(a ) \ ( a - a

45. The

c)

(b) not defined for all JC (d) defined fo r only positive (d) None of these -1

(log 2 JC )

+ Vcos (sin x)

(b) {2, 3, 4}

3 _3_ l l ' <2

defined

by

(d) None of these

f(x) = ^x-4-2-[(x-5)

1+JT

,

(d) None of these then

/ / ( JC)

48.

The period of (a)

1

2 + x, 0 R , g : R - * R be two given fun ctio ns susch that / is injective and g is surjective then which of the following is injective (b) =

(a) g,f

(b )f0g

(c )g„g

(d)// cos X I+

(a) [ - 2 nn, 2nt\ (b) (2 nn, 2n + \ 7t) (4n+ l)7i (4n + 3)n (C)

(d)

^(4n-l)7t

sin x I + I cos x I is . . I sin JC — c os JC I (b) n

2 TI

(c) ti /2 (d) n/4 49. If [.] denotes the greatest integer function

1
44. The domain of/(x) =

2
(c) [5, - j , (- oo ,- 2]

0 < J C < 2

< JC <

+

(b) ( - « , + 2]

3 - JC, 2 < J C < 3 0

-^x-4

is (a)[-5,oc)

x +x + 1 one-one and into one-one and onto many one and onto many one and into

(a) =

then its range is

(b)

(c)

x

, x +2x+ 5 . f ( x— )= "is

2 + x, 2-x, 4-x,

n2 2 T7 — x 16

47. The domain of

f:R—>R be a function

42. Le t/ (* ) =

the

(d) No ne of the se

(a)f

(d) {-1,1}

(a) (b) (c) (d)

func tion

(c) {1, 2, 3, 4, 5}

(a) { JC: 1 < J C <2}

(b)(1) (c) Not defined for any value

the wher e

(a) {2, 3}

46. If /( je ) = 3 sin

2 > 1+x 2x

, -1 + sin

41. Let

of

symbols have their usual meanings, is the set

x

40. The domain of the function

fix) = Vsin

domain

f ( x ) = l6~xC2x_]+20~3xP4x_s,

(4n+\)n

then the domain of•2theI isreal valued function l°g[x+l/2] 3 00 , (a) 2 ' |u(2,oo)

(b) (01

1 J.2

u(2,~)

(d) None of these 50. Let

/(JC) = sin 2 JC/2 + cos 2 JC/2 and g (J C)

= sec 2 JC - tan 2 JC. The two functions are equal •is

COS

X

over the set (a)* (b)R (c) R - 1

JC : JC =

(2n + 1) - , n e /

(d) None of these

Functions

93

MULTIPLE CHOICE-II

Each queston, in this part, has one or more than one correct answer(s). For each question, write the letters a, h, c, d corresponding to the correct answer(s).

51. The domain of/(x) is (0, 1) therefore domain

58. Le t/ (x ) = 2x - sin x and g (x) = 3 Vx, then

off(e x ) + f(ln I A' I) is (a )( -l ,e ) (b) (1, e) (c)(-e,-l) (d) (- e, 1) 52. If / : [ - 4, 0] -> R is defined by ex+ sin JC, its even extension to [-4, 4] is given by (a) — e ~ lj r ' — sin Ix I (b) e~ Lvl - sin I x I (c) e~

ul

+ sin Ix l

53. If f(x) =

(d) - e~ andf(A) =

lvi

59. Let

(a)£(x)=±V(l -x

(b)g(x)

2

= - ^ x

y : - ^
_l

)

(c) g (x) = V(1 -X 2 ) 2

(d)£(x)=Vl +x

55. The period of the function 2

2

rr\ Sin X + sin (JC + It/3) + COS X COS f.t + It/3)• /(x) = a IS (where a is constant) (a) 1 (b) n/2 (c) n (d) can not be determined 56. The domain of the func tion /(x) = sin

• Ix I ^ + cos + tan

_i( 2 - Ix I -1/ 2-1 x1 —

lis

(a) [0, 3] (b) [- 6, 6 ] (c) [- 1, 1] (d) [-3, 3] 57. Let / be a real valued functio n defined by I XI then range of/is / « = V ^ r e+e (a) R (c) [0, 1)

Cb) 10, 1] (d) [0, 1/2)

0 2 • I n x sin x

/«=

+ sin I x I

x +1 then set A is (a) [- 1, 0) (b )( -~ >,- l] (e) ( 0) (d) ( oo) 54. If g (x) be a function defined on [-1, 1] if the area of the equilateral triangle with two of its vertices at (0, 0) and (x, g (x)) is V3/4, then the function is 2

(a) range of gofis, R (b) gofis one-one (c) both/ and g are one-one (d) both/and g are onto

X I X I

for

x= 0

for - 1 < x < 1 (x * 0) for x > 1 or x < - 1

then (a)/(x) is an odd function (b)f(x)

is an even function

(c)f(x) is neither odd nor even ( d ) / ' (x) is an even function 60. Which of the following func tion is perio dic

(a)Sgn (e x) (b) sin x + I sin x I (c) min (sin x, I x I) 1 1 + (d) X+ 2 + 2 ; — x] 2 ([* denotes the greatest integer function) 61. Of the following functions defined from [- 1, 1] to [-1, 1] select those which are not bijec tiv e 2 (a) sin (sin x) (b) - sin (sin x) 7C (c )(Sgnx)ln(ex)

(d)x(Sgnx)

62. If [x] denote s the greatest integer less than or equal to x, the extreme values of the function /(x) = [1 + sin x] 4- [1 + sin 2x] + [1 + sin 3x] + ... + [ 1 + sin nx], e n (a) n - 1

(b) n

(c) n-l-l

(d) n + 2

63. I f / ( x ) is a polynomial function of degree such

(0,

n) are

the second that

/ ( - 3) = 6, /( 0) = 6 an d/ (2 ) = 11 then the graph of t he function / ( x ) cuts the ordi nate x = 1 at the point (a) (1 ,8 ) (b) (1, - 2 ) (c)l,4 (d) none of thes e

94


64. If f ( x + _y, x -y) = xy mean o f f ( x , y) and/(y, (a) x (b) (c) 0 (d) 65. Under

fi

+

then the arithmetic x) is y Non e of these

the condition

the domain

of

f i ' s equal to

do m/| u do m/ 2 . (a) dom/| ^ dom/

2

(b) do m/j = do m/

(c) dom/| > dom/

2

(d)dom/! < dom /

2 2

*2

(c) 1/2 (d)where 1/4 72. Let / (x ) = cos k = [m] = the greatest integer < in, if the period of/(x) is 7t then (a) m e [4, 5) (b) m = 4, 5 (c) ine [4, 5] (d) Non e of thes e 73. Let fix) = [x] 2 + [x + 1 ]- 3, whe re [x] < x. Then (a)/(x) is a many-one and into function (b )/ (x ) = 0 for infinit e number of values of x (c)/(x) = 0 for only two real values (d) None of these 74. Domain of sin ~' [sec x] ([.] is greatest in teger less than or equal to x) is

(0,

(b)

— -^f- 0 2

(c)

-f,0

U

(d) 67

w If e + e'J'W. = e then for /( x)

(a) domain = (— 1) (b) range = (-<», 1) (c) domain = (— 0] (d) range = (-<*>, 1] 68. Let

f(x)

2

= sec '[ l+ co s

x],

where

1

(d) th e range o f / i s j scc~ the

func tion

f:R->R

i, see" be

1

2}

such

that

/( x) = x — [x], where [.] denotes the greatest integer functi on, th en / (a)

x - [x]

1

(x) is

(a) {(2n + 1) 7t, (2/i +9 ) 7t] u {[(2/7! - 1) 7i, 2nm + n/ 3), m e /} (b) {2/!7i, n e /} u {[2 inn, (2m + 1) n), m e /} (c) {(2n + 1 )7t, n e /} u { [2 mn, 2mn, + n/ 3), 111E / } (d) None of these

[ .]

denotes the greatest integer function, then (a) the domain o f f is R (b) the domain of/is [1,2] (c) the range of/is [1,2] 69. If

(d) Non e of these

2

1

66. Domain oi'f'(x) = sin" [2 - 4x ] is ([.] denotes the greatest integer function) (a)

(c) not defin ed

70. The domain of the functi on /(x) = V(2 - I x I) + 1 +1x1) (a) [2, 6] (b) (- 2, 6] (c) [8, 1 2 ] (d) None of these 71. Let f : R - > [0,7i/2) be a functio n define d —1 2 b y / ( x ) = tan (x + x + a). If/is onto then a equals (a )0 (b) 1

75. Let fix) = (x 12 - x 9 + x 4 - x + 1)" domain of the function is (a) (- oo , - 1) (b) (-1, 1) (c)(l,°°) (d) ( - <=

1/ 2

.

The

(b) [x] - x

Practice Test

M.M. : 20

Tim e : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] 1. The function ion /(x ) = (a) an even fun ctio n (b) an odd func tion

log, ^

jc/x is

(c) a periodic func tion (d) None of these 2. If f: R —» R, g : R —> R func tion s th en fix) = 2 min {fix) - g (x), 0) equa ls

be

two

given

Functions

95

(a)f(x)+g{x)(b)f(x)+g(x)

\g(x)-f{x) | | g(x)-f{x)

|

(a)

|

(c)f{x)-g{x)+ | g(x)-f(x) | (d)f(x)-g(.x)~ | g(x)-f(x) | 3. The domai n of th e

(b) (c)

func tion

f (x) = In ^ In ~~ j is (where (.) denotes the (a) (0, «> )- / (c) R-1

(d) None of these

+ x + sin x - cos x 8. Let the function f(x)=x + log (1 + | x |) be defined over the interval [0, 1], The odd extensions of f(x) to interva l

fractional part function) (b) (l, =o)-7 (d)(2,~>)-/

4. If domain of f is Dx and domain of g is D2 then domain oif+g is

[-1, 1] is (a) x +x + sin x + cos x - log (1 + ] x |)

(a)Dl \ Do

(b)-x

+x + sin x + cos x - log ( 1 + | x |) 2 (c) - x + x + sin x - cos x + log (1 + | x |) (d) None of these

(b) Dx - (D1 \ D2)

)D2\(D2\D1)

(c

(d) Di n Z>2 5. sin ax + cos ax and | sin x | + | cos x| ar e pe riodic of sam e f un d ame n ta l period if a equals (a) 0

(b) 1

(c) 2

(d) 4

6. If (x) g is a polynomia l sat isfy ing g (*) giy) = g(x) +£ty) + g (xy) - 2 for all real x an d y and g (2) = 5 then g (3) is equal to (a) 10 (b) 24

where [x] Vx - [x] ' denotes the greatest integer less than or equal to x is defined for all x belonging to (a )R (b )R - (( - 1, 1) u {n i n e 711

9. The

function

f (x)

(c)7e + -(0, 1) (d) R+ - In : n e N) 10. The period o f th e functi on

f (x) = [sin 3x] + | cos 6x | is

(c) 21 (d) Non e of th es e 7. The inter val into which the function (*-l) transforms the entire real y =(X2 - 3x + 3) line is

([.] denotes the greatest integer less then or equal to x) (a) 7i (b) 27t/3 (c) 2TC (d) 71/2 (e) None of these Max. Marks

1. First attempt 2. Second attempt 3. Third attem pt

must be 100%

Answ ers Multiple

Choice

-I

l.(b)

2. (b)

3. (b)

4. (a)

5. (a)

6. (b)

7. (d) 13. (c)

8. (b)

9. (c)

10. (d)

11. (d)

14. (b)

15. (a)

16. (d)

17. (b)

12. (c) 18. (b)

19. (b)

20. (a)

21. (b)

22. (b)

23. (d)

24. (a)

25. (d)

26. (b)

27. (d)

28. (d)

29. (c)

30. (c)


96

31. (c) 37. (c)

32. (b) 38. (b)

33. (a) 39. (a)

34. (b) 40. (b)

35. (b) 41. (d)

36. (d) 42. (a)

43 . (d) 49 . (b)

44 . (d) 50. (c)

45. (a)

46. (b)

47. (c)

48. (b)

53. (a), (b), (c)

56. (b)

Multiple

Choice

-II

51. (c)

52. (b)

54. (d)

55. (d)

57. (d)

58. (a), (b), (c), (d)

59. (a), (d)

60. (a), (b), (c), (d)

61. (b), (c), (d)

62. (b), (c)

63. (a)

64. (c)

65. (b)

67. (a), (b)

68. (a), (d)

69. (c)

70. (d)

71. (d)

72. (a)

73. (a), (b)

74. (c)

75. (d)

Practice

66. (a), (c)

Test

1. (a)

2. (d)

3. (b)

7. (b)

8. (b)

9. (b)

4. (b), (c), (d) 10. (b)

5. (d)

6. (a)

11 LIMITS Let f{x) be a function of x. If for every positive number e, however, small it may be, there exists a positive number 5 su ch that when ev er 0 a

In the definition of the limit we say that I is the limit of f (x) i.e. f(x) -> /, when x-^ a. When x-> a, from the values of x greater than a, then the corresponding limit is called the of f (x) an d is written as Lim /(x) or f ( a + 0). x-> a + The working rule for finding the right hand limit is : "Put a + h for x in f (x) and make h approach zero." In short we have f(a + 0) = Lim f(a+h) /7-> 0

Similarly, when x-> a from the values of x smaller (or less) than a, then the corresponding limit is called the of f (x) an d is written a s Lim f(x) or f(a- 0) x-> a The working rule for finding the left hand limit is : "Put a - h for x in f (x) and make h approach zero." In short we have f { a- 0) = Lim f ( a - h ) h — > 0 If both these limits f(a + 0) and f(a - 0) limit of the function f{x) at x = a, i.e. I is the limit of f(x) as x-> a if f( a + 0) = I = f{a - 0) The limit of f(x) as x-> a then also the limit of f(x) as x a do es not exist.

(i)

Lim n —>

1

\n 1//j + - I =- e = - Lim L.IIII (1 ll -+ r / h) ll n 0 \n

(ii) (iii) (iv)

n

Lim

n

y Lim (1 + a/ ?) 1/ " = e a

x" - a" Lim = na " x-> a x- a

where n e O .

then their

the

98


, . (v)

.. sin 6 C . ... sin x° Lim — — = 1 and Lim x->0 x

e->o ec Lim x-» 0

(vii)

Lim x—• 0

(viii)

Um ~ = 0 (m > 0) x-*oo y n

e x-

1

k )an d

Lim e->o

tan e c = 1 e c

= 1

x

m

.. . .. (1 + x ) (ix) x-» Lim0 x (x)

Lim loga(1+x) x-»0 x

(xi)

Lim{f(x)} x—> a

(xii)

t (180' =

a x- 1 = In a (a > 0) x

(vi)

•

180

-1 = m = log

9( J

x

° = e

Provided g(x) ->

a

e (a > 0, a * 1)

Lim{/(x)-1}fir(x) ~*a

f (x)

1 tor x

a.

o=, (s o no exits) if a > 1 1, if a = 1 0, if - 1 < a < 1 not exi st, if a < - 1

Lim a " =

§ 11.4. Some Important Expansions (i)

tan x =

(iii)

(sin

o

+

ID

1

. -1 (iv)x tan (vi)

x® + .

(II) sin

2

x) 2 = ^

x

4

2

2

x +

3 x x5 x = x- — + —5 3

(1 + x) 1/x=

4

'

.

x =x+

—

1

x2

^.x + 3!

3

l^.x —

5

5!

+

12.32.52 7 — x+. 7!

6

x (v)sec-

x=1+|

T +

5x 4 61 x 6 ^ r + ^ y + ...

11 J? e *1 - ~* + — x 2 24

§ 11.6. indeterminate Forms If a function f (x) takes any of the following forms at x = a. 0 >, 0 X OO, 0', oo", 1° 0' then f (x) is said to be indeterminate at x = a.

§ 11.7. L'Hospital's Rule Let f ( x) a n d g(x) be two functions, such that then

f(a) = 0 an d g (a) = 0.

Lim - ^ r r = Lim -3 -^7 , *->a£f(*) x->ag(x)

Provided f ' (a) and cf (a) are not both zero. Note : For other indeterminate terms we have to convert to

— or — and then apply o

0 oo So me tim es we hav e to repe at the pr oce ss if the form is - or — again.

OO

L' Hospital's Rule.

Limit

99

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of letters a, b, c, d whichever is appropriate : 1. If 0 < a < b, then Lim (b" + a") " is equa l to

9.

(b ) a (d) None of these

(a)e 2. The value of L im

Vl/2(1 - cos

2x) .

:^y2-(y-x)2 Lim r^ is equal to 0 * (V8xy - 4X 2 + Vfoy) 3 (a) 1/4 (b) 1/2 (c) 1 / 2 V2~ (d) No ne of thes e

X

10.

(b) - 1 (c^ jio ne of these »

(c) 0

3- Thevalue of Lim

^

JC->0

+

JS where { JC}

{x}

denotes the fractional part of

x.

4. Let L i m

*->3

(a) 4 (c) 1

(

(a) 0 (c) -1

(a)/(x)» (c) 0

12. The value of the limit lWx Jx . T a -a , .

1

em + 1

(b)2 (d) 0

(c) -1

x sin (1/x ) - x 1 — 1x1

a •2

is

13.

Lim

,

n sm n ! , 0 < a < 1, is equ al t o n+ 1

(a)0«


(C)oo

4 9 + ... + + + 1 «3+l n +1 (b) 2/3 (d) 0 bounded function, then

, . _. 14. Lim xÔ

sin

1

x - tan x1 . , ~ is equ al to 2 x

(a) 1/2

(b) -1/2 (d)=o (c)0. 15. Lim {log{„ _ i) («) log„ (n +1) log„ + 1 (n + 2) ... log„ l _](«)} is equal to

is

n (a) (c) oo

(b) g(x) (d) None of these

16.

8. The integer n for which l

j . (cosx-1) (cosx-e ) Lim x" non-zero number is (a) 1 (b) 2 (c)3, (d)4

+a

(a) 4


n + 1 n is equal to (a) 1 (c) 1/3 7. If x > 0 and g is Li m " 00

of

(b)2 (d) None of these

a

5- - The value of Lim

Lim n —> <*>

(d)l

SJ . sin x - (sin x) . JIII, LI. ^Lim :—-——L~.is equa l to * -»rc/2 1 - sin x + In sin x

(b) -1/9 (d) 1/9 ,

(a)0 (c) -1/3

6.

valu e

(b)7t

r„

(d) does not exist »

f(x) = 1/V(18 -x2), the f £ M ) IS

sin (7t cos x) equal r =2

(a)-7t (c) 71/2

(b) 0 (b) 2

Lim *->0

is

a

_ . finite

(b)k

Lim £ cof n —» oo r= 1

1

" (d) No ne of the se

( r 2 +3/4)is (b) tan

(a) 0

•l

1

-

(c^ tan 17. If Lim

, 1

a

(a) a = l,b = 2

+

b

(d) None ol' these n2 * 2 = e then

(b)a =

2,b=\


100

(d) None of these (c)a=l,be. R Ci , l/n l/n (n - 1 )/n 1 e e - + + 18. Lim ... +" n n n n equals (a)0 (c 1ft

24.The

(d)

. 2 V2~ -( co sx + sinx)" . Lim :— is equal to k/4 1 - sin 2x

, . 3V2 (a)—— 2 * 4-12 (c)

IA TF T 20. If Lim

tan x) sinnx j

( l + " 3 ) + 8e' / X = 2 1 + (1 — b ) e

{a)a=l,b

=2

(c)a =

2,b-3

22

4

th gn

]

^

2

Lim

c

2

k =l

$n+ 1 - 5„

Vyk=z1

n -»00 is equal to

*

»

~.a

v

(b)a (d)2a

26.

Lim * s ' n {* M l ^ where [.] denotes the x- 1

x —^ 1

greatest integer function, is (a) 0_•* (b)-l (c) not existe nt (d) Non e of thes e •

denotes the greatest integer function (a) does not exist (b) is equal to zero (c) - 1 (d) None of these 28. Lim (2 - tan x) 1 /In (tan x) equals to jr —> It/4 (a)«

(a)a= I, b = -2 (b)a=\,b=\ (c) a = 1, b = - 1/ 2 (d) None of these

n

(b) x 2 - l.Ox+^21 = 0

2

- If XLim - x + 1) - ax - b) = 0 then —» 00( V ( x

sn=

are

27. The value of Lim [x + x + si nx] where [.]

(d) None of these

2

roots

(c) x - 14x + 49 = 0 (d) None of t hese

, 0' where

(b) a = 1, b = - 3 1/3

the

2

=_

(d) n + — n

21. If Lim *->0

iif 0 < x < 2

I-2x + 3, if 2 < x < 3

(a) x 2 - 6x + 9 = 0

n-is non zero real number then a is equal to n+ 1 (a)0 (b) n (c) n

{x2-l,

1

quadratic equation whose Lim fix)and Lim is * —> 2 -0 x —> 2 + 0

(d) does not exist

i( ~n)nxz —

f(x) =

25. Let

(b) 2 V2"

a

is

"" 5

T AT

cos (sin x) - cos x

Lim

equal to


)e-}

va lue of

(c) 0

then

(b)l

f

(d)
n! 29. Lim n —> ° ° (mnf (a) 1/em • (c) em

l/n

30. If y = 2

then Lim y x-> 1 +

(a) -1 (c)0

(me N) is equal to (b) m/e (d) e/m is

(b) 1. (d) 1/2

MULTIPLE CHOICE -II Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). x 2 "-l 31. Le t/ (x ) = Lim , then « 00 x 2n + 1 (a) f(x) = 1 fo r I x I > 1

(b) f{x)~- 1 for I x I < 1 (c) fix) is not def ined for any value of x (d) / (x) = 1 fo r I x I = 1

T. LOW Limit

101

y=f(x) has a 32. The graph of the function uniq ue ta ngent at the point (a , 0) thro ugh which the graph passes. Then


f{2h + 2 + h~) —f(2) Lim A->o fih -h z+ 1)-/(1) / ' (2) = 6 a nd /' (l ) = 4 (a) does not exist (c) is equal to 3/2

34. If Lim

:(A-P)

(c) e'

(a) 0 (c)2

„ ,. . i sin A , 40. I f / W H 2 ;

2iven

that

1 then

35.

36.

(b) is equal to ( - 1)'

(a) Lim .v + In (1 - x) .v ->0

J'lf-

42.

Lim

I

1 I dt • is equal to ( A-

1)

(b) 1

(c) - 1

(d) Non e of these log,

*>

[ A]

X

, where [.] denotes the greatest

integer function is

X

(a )0 (c) — 1

-

(b) 1 (d) non exis tent

44. If [A] denotes the greatest

1

Lim \

37. If [ A] denote s the greate st integer less than or x, equal to then the value of Lim (1 - , v + [x- 1] + I 1 -x I) is v-» l (a)0 (c)-l

if

2 ...A n ),

l

( a) 0

.V

(c) Lim 0 VJ + V.v2 + 2.v CO S A"

sin

1 + 0

43. Lim A

m+

(c) is equa l to ( - 1)' (d) does not exist

,V->

x —> oo

(d) Lim A' —> 0

and

1< m
L i m (V(.v~ + .y) - A") equal s

(b) Lim x-> 0

n (A] A

(a) is equal to ( - 1)'

Lim is non zero finite then n x -» 0 x" _ sin x must be equal to (a) 1 (b)2 (c)3 (d) None of these

+

x- a , , / = 1, 2, 3 I x - a,-1

a}
a = - 5/2, b = - 1/2 a = — 3/2, b = — 1/ 2 a = - 3/2 , b = - 5/2 a = - 5/ 2, /; = - 3 / 2

,

an

(b) 6 (d) 1 A; = -

41. If

/d

E

otherwise

(a) 5 (c)7

.0

(a) (b) (c) (d)

* tin, n

A

x +1, x* 0,2 then Lim g {/( A)} is 4 , A = 0 JT -> 0 5 , A = 2

g (x) =

(b) is equal to - 3/ 2 (d) is equal to 3

,v (1 + a co s x) - b sin x

(b) In I a ( a - P) I ala-(3l (d) e

(a) a (a - p)

is

3fix)

Lim (1 + ax2 + bx + c)1/(x ~ a ) is x —»a

then

log, {1 +6 /( A ) } ,

.

Lim

33.

39. If a and (3 be the roo ts of ax" + bx + c = 0,


x+y jr-» 0 JT 'j. to (where c is a constant) 2 si n y (a) € (b) sin 2y e s i n ' y (c)0 (d) None of these

f) -

{[1' x] + [2 A ] + [3

2

A] +

... + [ H

2

equal (a) A / 2 (c) A / 6

(b) A / 3 (d) 0

45. The value of the limit 2

is equal

38.

n — » oo

integer < x, then

2

2

Lim {l x —^ 0

IAin

(a)< n (»+ 1) (c)

*+

2

1/sin

"t+. . + n

(b) 0 (d) n

2

1/sin .risin x

Obj ective Mathe matics

102


M.M.: 20

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] lle r ^ 11 X — Sill X (b)1. Lim 7— is non zero finite th en n 24 24 6 x — si n x (d) None of these (c) must be equal to 24 (b) 2 (a) 1 " 7. Li m ~Jx (Vx + 1 - \5T) equal s (c) 3 2. Lim x -> o

(d) None of these tog* M [x]

ê

denote s

N

greatest integer less than or equal to x) (a) ha s val ue - 1 (b) ha s val ue 0 (c) ha s va lu e 1 (d) Does no t exis t C0S

3. Lim gHL; ([ ] denotes x -> 0 1 + icos x\ integer function) (a) eq ual to 1 (c) Does no t exi st

the

(b) Lim x->0

V( 1 + x ) - 1

(c) Lim x->0

(b)|(a-p)

2

(d)-y(a-P)

ta n ([- it ] x )- ta n ([- n ]) x . — equa ls sin x denotes the greater integer where [ function (a )0 (b) 1 (c) t an 10 -1 0 (d) oo T.

x

2

(c)y(a-P)

-x

2

x 1 - cos x

(d) Lim x-> 0 ^x + V(x 2 + 2x) (b) equ al to 0 (d) Non e of th es e

4. If a and 3 be th e roots of ax + bx + c = 0, then 2 1 - cos (ax + bx+ c) . ,, T. Lim 5 is equ al to x a (x-a)

2

In (1+x)

(a) Lim x->0

grate st

2

(a) 0

oo

X

2

5. If x is a rea l num be r is [0, 1], th en th e value of 2m f(x) =Lim Lim [1 + cos' (n ! rct)l m —» co n —» oo is given by (a) 2 or 1 according as x is rational or irrational (b) 1 or 2 according as x is rational or irrational

Lim 0

2 9. Let a = min {x + 2x + 3, x € R} b = Lim 1 - cos 9 The value 6 —»0 0 v ar .o un Z r=0 (a)

2n +

1

-l

(b)

2n + 1 + l 3.2"

3.2" (c)

4n +

1

-l

(d) None of these

3.2 (c) 1 for all x (d) 2 or 1 for all x

10- Lim (1+ x)

6. Th e val ue of Lim x-» 0

- e + — ex • is

x -> 0 (a) 1 (c) e -

sinx

equals (b) e (d) e

- 2

and of

Limit

103


must be 100%

An swers Multiple

Choice

-I

1. (b) 7. (a)

2. fd) 8. (c)

3. (d) 9. (d)

4. (d) 10. (b)

5. (a) 11. (b)

6. (c) 12. (c)

13. (a) 19. (a)

14. (c) 20. (d)

15. (b) 21. (b)

16. (c) 22. (c)

17. (c) 23. (a)

18. (c) 24. (b)

25. (b)

26. (c)

27. (a)

28. (d)

29. (a)

30. (b)

Multiple

Choice

-II

31. (a), (b), (c)

32. (c)

33. (d)

34. (d)

35. (b)

36. (b)

37. (d)

38. (c)

39. (c)

40. (d)

41. (d)

42. (a)

43. (a)

44. (t )

45. (d)

4. (c) 10. (c)

5. (a)

6. (a)

Practice

Test

1. (a)

2. (a)

3. (b)

7. (b), (c)

8. (c)

9. (c)

12 CONTINUITY AND DIFFERENTIABILITY Continuity of a function f(x) can be discussed in two ways (1) at a point (2) in an interval. x = a if (1) The function f(x) is said to be continuous at the point Lim f(x) exist = f(a) x-> a

i

Lim f(x) = Lim f(x) = [f(x)] x-» a x-> a +

x = a

or

L.H.L. = R.H.L. = value of funct ion at x = a. (2) The function f(x) is said to be continuous in an interval [a, b], if f(x) be continuous at every point of the interval. In other words, the function f (x) is continuous in interval [a, b]. It is exist (be not inde ter mina te ) and be finite ( * for all va lu es of xi n interval [a, b].

The discontinuity of a function f (x) at x = a ca n ari se in two ways (1) If Lim f(x) exist but / f(a) or Lim f(x) exist but * f(a), then the function x-»ax->a + (2) The function f (x) is said to have an i

f(x) is said to have a

Lim f(x) does not exist. x-> a

Lim f(x) * Lim f(x). x-» a x-» a +

The progressive derivative or right hand derivative of f(x) at x = a is given by .. f{a + h)- f (a) , n LIM — R — ,h > 0 h-> 0 h if it exists finitely and is denoted by Rf' (a) or by f (a +) or by ' (a). The regressive or left hand derivative of f (x) at x = a is given by Lim /7-»0 if it exists finitely and denoted by

(a) or by

r — — , /7>0 -h (a - ) or by

' (a).

The function is said to be at x = a fi and (a) both exist finitely and are equal, and their common value is called the at the point x = a. Th e function f(x ) is sai d to be non di fferenti able at x = a if (i) both an d exist but ar e not equa l (ii) either or both (a) and (a) are not finite (iii) either or both (a) and (a) do not exist.

K /

Continuity and Differentiability

105

(i) IfRf (a) and Lf (a) exist finitely (both may or may not be equal) the n f{x) is continuous at x = a (ii) If /(x) is different iable at every point of its domai n, then it must b e conti nuou s in that domai n. (iii) The c onv er se of the ab ove result (ii) is not true, i.e., If f (x) is cont inuo us at x = a then it may or may not be differe ntiab le at x = a. i.e., there should be no break or corner. (iv) Iff (x) is differentiable then its graph must be smooth (v) For a function f (x): (a) Differe ntiable => Conti nuous (from (ii)) (b) Con tinuo us *=> Differentiable (from (iii)) (c) Not cont inu ous Not different iable

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letter a, b, c, d whichever is appropriate : 1. The value of th e/ (0 ), so that the function V( a 2 — ax + x 2 ) — V(g 2 + ax + x2 ) beco mes continuous fo r all x, is given by (a) a (b)Ja (c) - ^ a (d ) - a ^ l a 2. Let / (x) = tan (n/4 - x)/c ot 2x (x * n/4). The val ue which shoul d be assigned to / at x = n/4. so that it is conti nuous every where, is (a) 1/2 (c) 2

V-

th en

(c)2/'(l) X 8. Let f-.R—tR

f o r /

(5x + 32) - 2 to be continuous every where, / ( 0 ) is equal to (a) - 1

4

is (a)8/'(l)

f

3

1


6. Let fix+y) =/ (x )/ (y ) for all xa nd y. Suppose that / ( 3 ) = 3 and / ' ( 0 ) = 11 then /'(3) is given by (a) 22 (b) 44 (c) 28 • (d) Non e of thes e Let/: R —> R be a differentiable function and r / w 2t dt / (1) = 4. Then the value of Lim J X-* 1 4 x - 1

(c)2 Let

(b) 1

6

(d) Non e of these * f"(x) be conti nuous x = 0 a n d / " (0) = 4 the value of 2/(x) — 3/(2x) +/(4x) . Lim is (a) 11 (c) 12 -

at


5. Let / be a funct ion satis fying fix + y) =/(x) +f(y) and fix) =x 2g (x) for all x an d y, where g(x) is a continuo us function then/' (x) is equal to (x) (a) g' (b) g (0) (c )s (0 )+ *' (* ) (d) 0

^ym±m

(b)4/'(l) (d)/'(l) a funct ion

be J

(

0

)

=

such

3

that and

/ ' (0) = 3, then fix) (a)^^— is differentiable i n R (b) fix) is continuou s but not diffe rent iabl e in R (c) fix) is contin uous in R t (d ) fix) is bound ed in R tan 71 [(2ti - 37T>3] , r , . ([.] denote s the 1 + [2x - 3 nf greatest integer function), then (a)/(x) is continuous in R (b)/(x) is continuous in R but not differentiable in R (c)/' (x) exists everywhere but/' (x) does not exist at some x e R (d) None of these rational 10. Iffix) , =1„| j, x. is . .. , then 2, x is irrational (a) fix) is contin uous in R ~ „ , If =fix)

,

106

\


JC + 1 > 0 (b)/is continuous but/' is not so for (c)/and/' are continuous atx = 0 (d) / i s continuous at JC = 0 but/' is not so.

(b)/(jc) is continuous in R ~ Q ( c ) f ( x ) is continuous in R but not differentiable in R (d)f(x) is neither conti nuous not differentiable in R 11. If f{x ) is a twice differenti able function, then bet ween tw o consecutive roots of the equation/' (JC) = 0 there exists (a) at least one root of/(jc) = 0 (b) at most one root of/(jr) = 0 (c) exactly one root of f{x) = 0 (d) at most on e root o f / " {x) = 0 12. If

the

deriva tive

of

+

1,

JC*

0,

JC =

the

function

then Lim g (/ (x )) x->0

JC =

0

then f{x)

is

- ax + JC ) - \{a2 + qjc + x~) V(a + JC) -V(a- jc) JCÔ then the value of/(0) such that/(jc) is continuous at JC = 0 is a -la IaI -la

,. (a)

I -la IaI

(b)-

(d )- ^r

( O T T T

19, Let [.] represe nt the greatest integer functio n and fix) = [tan 2 JC] then (a) Lim /(JC) does not exist

si n x I

tan 2,t/ian 3

JT

, for 2n < x < 2n + 1, n e N where [JC] = Integral part of JC < JC

[JC]

1 2 '

for 2n

+

1 < JC <

2n + 2

(b) is periodic with period 1 (c) is periodic with period 2

6

is n

r, 0 < x< — 6

(d) f f{x) dx exists o 21. A function

continuous at JC = 0 then

f{x)=x

(a) a = log e b, a = 2/3

/(0) = 0

(b) b = log, a, a = 2/3 (c) a = log, b,b = 2

T

1 1+-sin(logA:)

2

[.] = Integral part, The function (a) is continuous at JC = 0 (b) is monotonic (c) is derivable at x = 0

(d) None of these

J

\t\dt, J C > - 1, then -1 (a)/and/' arc continuous lor a +

0

the function

—
x = 0

fix)

JC =

(a) is discontinuous at JC = 1 , 2

15. If the function (1 + I sin x I)a/\

(b)/(jc) is continuous at JC = 0 (C)/(JC ) is non-differentiable at (d)/'(0) = 1 20. f ( x ) =

(d) no discontinuity at x = 0

-) =

0

=

JC -

(b) 6 (d) 1

14. The function fix) = I 2 Sgn 2x\ + 2 has (a) jum p discontinuity (b) removal discontinuity (c) infinite discontinuity

I f /(A

JC *

2

is

16.

;

/

2 0

x = 2

(a )5 (c )7

V

continuous but not differenti able at JC = 0 if (a) « e (0, 1 ] (b) n e [1, oo) (c) n e ( - oo,0) (d) n = 0

- If f(x)

is everywhere 2 ax +b ; x <- 1 continuous, then {b)a = 3,b = 2 (a) a = 2, b = 3 (c) a = - 2, b= - 3 (d) a = - 3, b = - 2 sin x, x n n, n e / 13. If and 2 , otherwi se 4 5

N ;

JC

0

18

bx + ax + 4 ; x > - 1

2

'1

x sin

17. Let f{x) =

(d) con not be defined for JC < - 1 1 >0

22 . I f f i x )

_ J [cos 7t .r] , x < 1 [

I JC— 1 I ,

1 <.T <

2

,J C* 0


107

([.] denotes the greatest integer function) then f(x) is (a) Continuous and non-di fferentiable at x = — 1 and x = 1 (b) Continuous and differenti able at x = 0 (c) Discontinuous at x = 1/2 (d) Continuous but not differentiable at x = 2 23. Iffix) = [-42si nx] , wh ere [x] represe nts the greatest integer function < x, then (a)/(x) is periodic (b) Maximum value of/(x) is 1 in the interval [ - 271, 271] (c)/(x) is discontinuous at x =

+ ^ ,n e I

(d )/ (x ) is differentiable at x = n 71, n e I 24. The function defin ed b y / ( x ) = (-1) [J C 5 ([.] denotes greatest integer function) satisfies (a) Discontinuous forx = n

1/ 3

, where n is any

integer ( b) / (3 / 2) = 1 (d) None of these

( c ) / ' (x) = 0 for - 1 < x < 1

25. If fix) be a conti nuous function defin ed for 1
the set of values, of p for which / " (x) i s everywhere continuous is (a) [2, o°) (b )[ -3 , oo ) (c) [5, oo) 27. The value of

(d) None of the se p

for which the function 3 (41-1) 2 >,x*0

sin (x/p) In

1 +

T

= 12 iln 4) , x = 0 may be continuous at x = 0 is (a) 1 (b )2 (c )3 (d) 4 28. The value of / ( 0 ) so that the funct ion ,, . 1 - cos (1 - cos x) /(x) = is continuous X

everywhere is (a) 1/8 (c) 1/4

(b) 1/2 (d) Non e of thes e

29. The jump of the function at the point of the discontinuity of the function

1 -k

l/ x

1 +k

\/x

(a) 4

ik > 0) is (b )2

"

(c) 3 (d) Non e of thes e 30. Let f" (x) be continuous at x = 0 and f " ( 0 ) = 4. The value of Lim (a) 6 (c) H

(x) - 3/ ( 2 x ) + /(4x)

is

(b) 10 (d) 12

, x = 0

MULTIPLE CHOICE -II Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer (s). 1 31. Let / (x) = • ([.] denotes the greatest [sin x] integer function) then (a) Domain of fix) is (2n n + 7t, In n + 2n) U {2«7t + 7t/2}, where ne/ (b)/(x) is continuous when x e (2n 7t + 7t, 2/i 7t + 271) (c)/(x) is differentiable atx = 7t/2 (d) None of these

32. Let git) = [r (1 /0 ] for f > 0 ([.] deno tes the greatest integer function), then g (g) has (a) Discontinuities at finite number of points (b) Discontinuities at infinite number of points (c)s(l/2)=l (d) g (3/4) = 1 33. Let f(x) = \x} + -lx - [x], wh ere [x] d enotes the greatest integer function. Then (a)/(x) is continuous on R+ (b) fix) is continu ous on R

Obj ecti ve Mathematics

108

ic) fix) is continuou s on R ~ I (d) None of these 34. Let f{x) and (J) (X) be defined by fix) = [J C] and l

(d) k (a) = 1 + I x I, a e R

(a) Continuous at x = n/2 (b) Discontinuous at A = 7i/2 (c) Discontinuous at A = — 7T/2 (d) Discontinuous at an infinite number of points 40. If /( x) = tan"

(b) Lim /( JC) does not exist an d/ is not continu ous X

1

cot x, then in) fix) is per iod ic with period 71 (b) f{x) is disco ntin uous at x = n/2, 3n/2 (c)/(x) is not differentiable at x = n, 99n, 100n (d )/ (x ) = - 1, for 2nn < x < (2 n + 1) 7t

1

atx = 1 (c) <> | of is continuous for all

A)2", then/is

39. Let/(x)= Lim (sin

x

(d) fo (J) is continuous for all JC 41. Let/

35. The following functions are continuous on (0,71) (a) tan x (b)f

t sin

(c)

(d)

36. If

JC <

371 2 sin — JC, —- < X < 7 I 9 4 x sin JC , 0 < x < re/2 7C

.

f(x):

—
tan [x] 71

+

I 1

" "

5x + ,1

) A

' X > x<2

2

42. fix) = min (1, cos x, 1 - sin x } , - 7t < x < n then

, 7 1

— s i n ( r t + x),

1

{a) f{x) is not conti nuou s at x = 2 (b)/(x) is continuous but not differentiable at x=2 (c)/(x) is differentiable everywhere (d) The right derivative of fix) at x = 2 doe s not exist

3K <

=f 0 [

Then

> 0

W

where

[.]

[1 +l /n (sin x+1 )1] denotes the greatest integer function, then fix) is (a) continuous V x e R (b) discontinuous V x e I (c) non-differentiable V x e / (d) a periodic function with fundamental period not defined 37. If / ' (A) = g{x) (JC - a)1 where g {a) * 0 and g is continuous at x = a then (a)/is increasing near a if g (a) > 0 (b) / i s increasing near a if g {a) < 0 (c)/is decreasing near a if g {a) > 0

43. If /( x) =

log (1+x 2 ) ' " then 0 , x= 0 (a)/is continuous at x = 0 (b)/is continuous at x = 0 but not differentiable at x = 0 (c)/is differentiable at x = 0 (d)/is not continuous atx = 0

44. Let /(x) =

S1

frfr-K]) where [.] denotes l+[x 2 ] the greatest integer function. Then/(x) is

a if g (a) < 0 (d)/is decreasing near 38. A fun ctio n which is continuou s and not differentiable at the srcin is (a) fix) = x for A < 0 and fix) = x for A (b) g (JC) = x for x < 0 and g (A ) = 2x for (c) Ii(A ) = A i A I , A e R

(a) f{x) is not differe ntiable at '0' (b)/(x) is differentiable at TC/2 (c) fix) has local maxi ma at '0' (d) None of these

> X

0 > 0

"

(a) Continuous at integral points (b) Continuous everywhere but not differentiable (c) Differentiable once but/' do not exist (d) Differentiable for all x

( A ) , / " (A), .. .


45.

46.

109

I 2x - 3 I fx] ; x > 1 KX x<1 2 ([.] denotes the greatest integer function) (a) is continuous at x = 0 (b) is differentiable at x = 0 x =1 (c) is continuous but not differentiable at (d) is continuous but not differen tiable at x = 3 / 2 2 a !x - x - 2 I x <2 2+ x- x Let/(x) : x=2 b

The function

fix)-

x-[x] x> 2 x- 2 ([.] denotes the greatest integer function) If/(x) is continuous atx = 2 then (b)a=\,b=\ (a) a= 1, 6 = 2 {c)a = 0,b=\ (d) a = 2, b = 1 tan

47. The func tio n/( x) =

e

x

1

t an A" ,

,

+1

51. The function/(x) = I x - 3x + 2 I + cos I x I is not differe ntiable at x = (a) - 1 (c) 1

52. Let h (x) = min {x, x 2 } for every real number x. Then (a) h is continuous for all x (b) h is differentiable for all x (c) h' (x) = 1 for all x > 1 (d) h is not differentiable at two values of x 53. Let

= X, x = 71/4 A, is

(a) 1

(b) 1/2

(c) -1 / 2

(d) None of these

54. A funct ion / (x ) is def ine d in the i nterv al [1,4] as follows :

f(x)=

(b) nn + n/2 (d) nn + n/8 • 1, x < 0 48. Let/(x) = < 0 , x = 0 and [ 1, x>0 g(x) = sin x + cos x, then poin ts discontinuity o f f { g (x)} in (0, 27t) is 3n In 7t 37C1 f (a), ( b ) 2 ' 4 j j 4 ' 4 27t 5rc (c). 3 ' 3 <«{t-T

lo g, [x] , 1 <

X

<3

I log, x I , 3 < x < 4

the graph of the function/(x) (a) is broken at two points (b) is broken at exactly one point (c) does not have a definite tangent at two points of

(d) does not have a definite tangent at more than two points 55.

If

g (x) =

where [x] function and

49. The points of discon tinui ty of the funct ion In

(2 sin x) /(x)= Lim are given by s2 n » - > ~ 3" - (2 cos x)z (b)nn±n/6 (a) nn ± 7T/12 (c) nn ± n/3 (d) None of these

—

,. '

Jfix) v

[/Ml 3 denotes

U , x = 7t/ 2 the greatest

f.K integer

2 (s in x- sin" x) +, .I sin x - ..... sin" ~x , I = ^ 2/ (sin • x- si• n" x) \ -l = i nx • -s i,neR si n• nxl I

_

then

50. Let / ( x ) = [cos x + sin x] , 0 < x < 27t where [x] denotes the greatest integer less than or points

and

* 71/4

If/(x) is continuous in (0,7t/2] then

is discontinuous

of

1 - tan x •, X 4x — n

fix)

x e [0,71/2)

atx = (a) nn + n (c) nn + tc/4

equal to x. The number discontinuity of/(x) is (a) 6 (b) 5 (c )4 (d) 3

(b) 0 (d)2

of

(a) g (x) is continuous and differentiable at x = 7C/2 when 0 < x < 1 (b) g (x) is continuous and differentiable at x = 71/2 when n > 1 (c) g {x) is continuous but not differentiable at x = n/2 whe n 0 < n < 1 (d) g (x) is continuous but not differentiable, at x = 7I /2 when n > 1

110



Time : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct 1. f (x) = 1 + x (sin x) [cos x] , 0 < x < n/2 ([.] denotes the greatest integer function) it/2) (a) is continuous in (0, (b) is strictly decreasing in (0, ji/2) n/2) (c) is strictly increasing in (0, (d) has global maximum value 2 2. If f (x) = min (tan x, cot x) then (a) f (x ) is discontinuous at x = 0,Jt/4, 57t/4 ( b ) f ( x ) is continuous atx = 0,71/2, 37T/2 rn/2

f(x)dx = 2ln<2 0 (d) f(x) is periodic wi th period n If f (x) is a continuous function V x e R and the range of fix) is (2.V26) and (c)|

g(x) =

is continuous V

x e R, then

c the least positive integral value of c is, where [.] denotes the greatest integer function (a) 2 (c) 5

f(x) =

(b) 3 (d) 6 [1/2+*] - [1/2] - 1
has ([.] denotes function).

the

(a) Discontinuity at (b) Discontinuity at (c) Discontinuity at (d) Discontinuity at

x =0 x = 1/2 x=1 x = 3/2

a +5. If /( *) =

x 2 , h + sin x — x

greatest

integer

answer(s). [10 x 2 = 20] 6. Which of the following functi ons are different iable in (-1, 2) (a) J ^ (log xfdx

Z t

(b)J

^< £c

l -1+r dt (d) Non e of these o l+ t+ t 7. If f(,x) = [x] + [x + 1/3] + [x + 2/3] , then ([.] denotes the greatest integer function) (a ) fix) is disc onti nuous a t x = 1, 10, 15 (b) fix) is continuous at x = n/ 3, where n is any integer (c) J

f 2/3

(c )J

fix)dx=

1/3

(d) Lim fix) = 2 X-+2/V 8. Let (1 + I cos i I ) b/[ /•(*) = ]

I

e .e e

,

cot 2z/cot 8x ,

1

, n K
cos

x

"

If f (x) is continuous in {nn, (n + 1) 7t) then b2 (a) a = 1, = (b) a = 2, b = 2 a = 3, b = 4 (c) a = 2, b3= (d) 9. Iffix) = | ( s i n ~ 1 x)2q

c o s (1/x

t;

* * 0 ° then

(a) fix) is continu ous everywh ere in x e [ - 1, 1] (b) f (x) is contin uous no whe re in x e [- 1, 1] (c) f(x) is differe ntiabl e everywhe re in X £ ( - 1, 1)

x>0 x= 0

x<0

(where [.] denotes the greatest integer function). If f (x) is continuous at x = 0 then b is equal to (a) - 2a (b) a - 1 a +2 (c) + 1a (d)

(d) f (x) is differenti able no wher e in * e [ - 1, 1] Le t fix) = [a + p sin x] , x 6 (0, 7t), a e / , p is a prime number and [x] is the greatest integer less than or equal to x. The number of points at which fix) is not diffe renti able is (a) p (b) P - 1 (c) 2P + 1 (d) 2p - 1


111


must be 100%

Answe r Multiple

Choice

-I

1. (c) 7. (a)

2. (a) 8. (c)

3. (d) 9. (a)

13. (d) 19. (b)

14. (b) 20. (b)

25. (c)

26. (c)

27. (d)

32. (b), (c)

33. (b)

34. (a), (b), (c)

35. (b), (c)

36. (a), I

38. (a), (b), (d)

39. (b), (c), (d) 40. (a), (c) 45. (a), (b), (c), (d)

41. (a), (d) 46. (b)

42. (a), I

50. (c)

52. (a), (c), (d)

Multiple

5.(d)

6. (d)

15. (a)

11. (b) 17. (a)

12. (a) 18. (b)

21. (a)

22. (c)

23. (c)

24. (a)

28. (a)

29. (b)

30. (d)

Choice -II

31. (a), (b) 37. (a), (d) 43. (a), (c) 48. (b)

44. (a), (d) 49. (b)

54. (a), (c)

55. (b)

Practice

4. (c) 10. (d) 16. (a)

51. (c)

47. (b) 53. (c)

Test

l-(a) 7. (a), (c)

2. (c), (d) 8. (b)

3. (d) 9. (b), (c)

4. (a), (b), (d) 10. (d)

5. (c)

6. (c)

DIFFERENTIAL COEFFICIENT

^ (iii) ^

w h e r e k is an

~ ^dx

(x) • h (x)} = h (x)

c onsta

y

h (x) + h (x) ^

nt

h (x).

In particular Chain Rule ^

{f l (X) . h

( X ) . h (X)

dx

h (x) . ^

} =

fe(x)

fi (x) - /1(x ). ~

(iv)

(X)

( /1 (x) f 3(x)

( f2 (x) /(x) 3

)

)+

( h (X )) . h

(X )

) + .

h (x)

2

t fe (x)] (v) If y = fi (t/), u = f2 (v) and v = b (x) then

di dx

=

d

1

du

du

dv dv

dx

dt

If x = f (f) and y = g (f ), then

dx

g'(f) f'd)

If f(x, y) = 0, then on differentiating of f(x, y) w.r.t. x, we get d/dxf(x, dy/dx and solve.

dy _ dx ~

y)

f I dx df_ dy

In particular if f(xu X2, X3,

xn) = 0 and x2, X3 df

df

df

x n are the functions of xi then dx2df dx3 df dx

n

= 0, Collect the terms of

Differential

Coefficient

113

§13.4. y= [h(x)]

If

k w

o r y = h (x). h (x) . h (*)

h (x ). fe (x) . fe (X) or gi (x). 512 (x). 93 (x) then it is convenien t to t ake t he logarith m of the function fir st and the n diffe rentiate. This is called derivat ive of logarithmic function -.Write [f(x)]

Va 2 Va 2

./ ?

' a - x

- x

9W

= e 9 p ° ,n(f(x))

and differentiate easily

x = a sin 6 or a co s 0

2

x = a tan 0 or a cot 0

+ X2 ?

x = ase c 0 or a cos ec 0 x = a cos 0 or a cos 20

>a+x

V(2ax- x 2 )

x = a (1 - cos 0)

§ 13.6. The points on the curve y = points.

f (x) at which dy/dx = 0 or dy/dx

§ 13.7. If a function f(x) is def ine d on [a, b] satisfying (i) f is continuous on /(a) = f(b) then c e (a, b) such that f (c) = 0.

b] (ii) f \s differentiable

§ 13.8. If a function f(x ) is defin ed on [a, fa] sati sfying (i) f is continuous on [a,

b]

(ii) f is differentiable on (a,

b) then c e (a, b) su ch tha t f (c)

Hb) - f(a) b-a

§ 13.9. If at all points of a certain interval interval.

d r^^

f' (x) = 0, then the function f(x)

c/6

b)

114


MULTIPLE CHOICE - I

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. x

1. l f J = e

\ then

dx

is equal to

8.

If 5f(x) + 3/1 -

(a)(l+logJc)(b)(l

(c) log x (1 + log x) (d) None of these 2.

- 2

then k is equal to

A

dx

dy (а)0 (б)2

(b)l (d) none of these

3. If y = (1 + x) (1 + x2) (1 + / ) . . . (1 + x

2

) then

at x = 0 is

dx (a)0 (c)l

(b)-l (d) None of these

If x = e>

then ^ is dx (b)

(c)5.

(d) None of these

l+x

2x

The derivati ve of sin

2x

to tan 1

l+x2

with respect

is

( d )

(c)

= 16 then ^

(a) —1 (c) 1

(b) 7/8

(c) 1

(d) None of these

9. If 2X + 2y = 2x+y, then the value of ^

T~2 1 +x

x = y= 1 is (a) 0 (b)-l (c) 1 (d) 2 If f(x) = \x-2 \ and g(x)=fof(x), then for x > 2 0,g\x) = (a) 2 (b) 1 (c) 3 (d) Non e of these 11- I f f ( x ) = sin" 1 (sin x) + cos" 1 (sin x) and <)> (x) =f(f(f(x))), then 4.' (x) (a) 1 (b) sin x (c) 0 (d) Non e of these If f{x) = (log cot x tan x) (log tan x cot x)~ 1 + tan

-1

then f (0) is equa l to

(a) - 2 (c) 1/2

(b) 2 (d) 0

x2 + y2 = t - j a n d / + y

dx

4

(a) - 1 (c) 1 If

(b)0 (d) Non e of these

equation

equals

(a)P (x) + P" x (b) P " (x). P"' (x) (c) P (x). P"\x) (d) Non e of these

= f 2 + - 2r ,


variables

x and y

x -J

are

related

1 V1 + 9y 2

(c) 1 + 9y

by

the

* =rr*du, the n ^ dx

equal to (a)

then

equals

at (2 ,2 ) is

2

at

dx

xy

7. If y = P (x) is a polynomial of degree 3 then 3 (Py

(a) 14

13. If (b)l

1

Uxy.yx

=l

-x

(a) 0

6.

1 -x

then

is equal to

dx

+log *r

\ = x + 2 and y =xf(x)

(b) Vl +9y 1 1 + 9y

2

is

Differential Coefficient

15.

If yl/n = [x +

115

^l+x

2

then (1 +x )y 2 + xy1

(a)

is equal to

(b) ny2

(a) n y , , 2 2 (c) n y 16.

If /( x) = cot

then/'(l) is

(a)0 (c) - 1

set of f'(x) > g' (A) wh er e an d g(x) = 5 x + 4x log 5 is

2

and/(2) = 4 =/'(2) then/

, , a 2

(b) 32

(c) 64

(d) No ne of thes e the

( 1) = <{> (— 1) and au a 2 , a 3

are in A.P. then ((>' (a^, (a) AP (c) HP

(b) GP (d) No ne of these

(a) - cosec x. cos x

,.

(c)±2 23. If

dy dx

y =

then

is equ al to

2

(b )0 (d) Non e of these

V2TT

Pix) is

a

polyn omial

such

t hat

P (A2 + 1) = {P (A)}2+ 1 and P ( 0 ) = 0 then P'i0) is equa l to (a) - 1

(b )0

(c) 1

(d) No ne of thes e

(b) cos a

(d) None of these dy_

at 0 - 37 t/ 4 is

(b) 2 (d) None of these sin x + ^sin + Vsin x +.. . 00

(c) sin a

(d) Non e of these

29. The third deriva tive of a fu ncti on fi x) varishes for all A. I f / ( 0 ) = 1,/' (1 ) = 2 an d f" = - 1, the n fix) is equal to

x/x

cos 20 J ' dx

(a)-2

:~ 1 at x=

(a) 2

x) w.r.t x, where

21. The diff. coeffi. of /(log f ( x ) = log x is (a) A/log x (b) log

22. Ify =V f j i

+

= sin x + sin 4x . cos x

^ = , ^ then the valu e of A is dx 1+ x -2 x c o s a

n (c) - — - cosec x. co s x 1 ou (d) None of these

co s 2 0 ^

(d) not exist

(a) - 1 (c) - 2

a > 0 then y" (0)

28. If sin y = x sin (a + y) and

cosec x°. co s x 1 o(J

A)

2x'

,

n/2

f\x)

27. If

dy If y = sin A" and u = cos x then ^ is equa l to

(c) (A lo g

f

(y/x)

(b) ae-

a 26. If

a po lyn omi al fun cti on of

second degree. If

-k/2 S

for all x

(19) + g1 (19) is

(a) 16 19. If (j) (x) be

(b)l (d) Non e of these

25. If V(x 2 + y2) = ae™

(b) (0, 1) (d) (0, oo)

18. If / ' (x) = g (x) an d g'(x) =-f(x)

) , «'(2 ) = 4, v'(2)

= 2, u (2) = 2, v (2) = 1, then f'(2) is equal to

2l +1

(a )( l, oo ) (C) [0, oo)

fix) = log {^

24. Let

-x

(b) 1 (b) -log 2

17. Th e solution f{x) = (1 /2 ) 5

(b)

(b)

(d) None of these

x

(a)-l (c) log 2

20 .

co s X 2y- 1 cosy (d) 2jc— 1

2y -i cos X 2x- 1 co s y

the n

(a) ( - 3/2 )

A2

(b) (-1/2)

A

(c) (- 1/2)

A2

+

3A

+

(d) (- 3/2 )

A2

-

7A

+ 2

2

+

3A

- 3 A

+ 9 +1 1

30. If y = lo g/ ( A - 2 ) 2 fo r A * 0 , 2 the n y' (3) is equal to (a) 1/3 (b) 2/3 (c) 4/3 (d) Non e of these

116

Ob j ective Mathemati cs

MULTIPLE CHOICE-II Each question, in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). Let f{x)-x" , n being a non-negative integer, the value of n for which the equality /' (a + b) =/' (a) +/' (b) is valid for all a, b > 0 is (a )0 (c) 2 1

\+x 1

-1

+x

1

\

(c)

x + 3x + 3

/

, + up to n terms,

(d) None of these

1 +nz

33. If f{x) = x J + x 2 f \ 1) + xf" (2) + / " ' (3) fo r all x e R then (a)/(0)+/(2)=/(l) (b)/(0) +/(3) = 0 (c)/(l)+/(3)=/(2) (d) None of these L e t / b e a func tion such th at /( x + y) =f{x) +fiy) for all x a n d y and f{x) = {2x + 3x) g{x) for all x where g{x) is continuous and £(0) = 3. Then/'(x) is equal to (a)9 (c) 6

(b)3 (d) none of these

(d) None of these

V( 1 + 1 2 ) - V(1 - 1 2 ) —47 J If y = i ' 2— I ' and x = \(1 - t ) , 2 V(l + r )+V(l-r )

Tf

then

( b) - —^ 1+n

1+ n n

1 log, (2x)x

dx

1

+ tan

x + 5x + 7 then y' (0) is equal to (a)-

38.


32. If y = tan + tan

(c)

( a

=

\ 2 {l+Vl-

(b) (c)

{V(i-f

f 4

<}

)-i}

1 t 2 {l + V(i - r 4 )!

(d) 39. Let x cos -v + y c o s * = 5. The n (a) at x = 0, y = 0, y' = 0 (b) at x = 0, y= 1, y' = 0 (c) at x = y = 1, y ' = - 1 (d) at x = 1, y = 0, y ' = 1 40. If

/ ( x ) = ( 1+ x) "

then

36.

2/a

(a) 2/a

(b) -

(c) 2/a

(d) None of these

dy If sin (x + y) = log (x + y), then ^ = (a) 2 (c)l

37.

equal s dxL

(b) - 2 (d)-l

If y = log 2 {log 2 x} , then i ^ (a)

e lo g 2 x lo g e X

„.

(b)

ic equa l to is

1 x log, x log, 2

value

2n

(a )n

(b)

( c ) 2" ~ 1

(d) None of these

If

fix)

sin x sin {nn/2)

a 35. If V(x + y) + V(y - x) = a, then

the

2

cos x cosin n/2)

a

3

then the value of

(fix)) at x = 0 for n = 2m + 1 is dx" (a) -1 (b) 0 (c)l (d) independent of 2x I f f { x ) = sin" 1 +x , then (a)/is derivable for all x, with (b)/is not derivable atx = 1 (c)/is not derivable atx = - 1 (d)/is derivable for all x, with

IxI<1

Ix I >1

a

of

Differential Coefficient

117

If P{x) be a polyno mial of degre e 4, with P(2) = (2)-\,P' = 0,

P" (2) = 2, P"'(2) = - 12 then P"(l)is equal to (a) 22 26(c)

and

P<-(2) = 24 45>

(b) 24 (d) 28

m=x1

Le t

+

g'(2) = 8 + #'( 1) g"{2) +/"(3) = 4

(b) (c)

(d) No ne of the se {f

I f / W = si n

j

[x]

fo r

2

and

I 3 [x] den ote s the grea test inte ger less than or

xg'(\)+g'\2)

and

e q u ^ x , t h e n / ' ( V ^ i s ^ a l to

2

gix) = + x / ' ( 2) +/ "( 3) then

[")-V

(d) N o w of these

TT

(a)/'( l) = 4+/ '( 2)

Practice Test M.M. 20

Ti me : 30 Min.

(AJThere are 10 parts in this question. Each part has one or more than one correct

1. I f y = J X f(t)

si n [k ix - <)] dt, then

+

k 2y

If5. f

{4

answer(s). [10 x 2 = 20]

12 - 2 F" (t)} dt, then

F (4)

dx

2

-

equals (a) 0

(b) y

(c)k.fix)

(d)

If

V(1 - x 6 ) +V(1 -y

dy _ fix dx then (a

6

) = a (JC3 -y 3)

and

1-xfc

)f(x,y)=y/x 2

3.

k2f(x)

(b

)f(x,y)=y2/x2 2 , 2

2

ic) fix, y) = 2y /x (d) /•(*, y) = * 2 / y 2, \ If - i f 3 + 2 log e x - 1 log c ( e/x y = tan + tan 1 - 6 log e x log e (ex 2) d2 then —^ is dx (a) 2 (c) 0

(b) 1 (d) - 1

d3 If x = a cos 0, y = b sin 0, then —^ is equal dx to -36 4 4 (a) cosec 0 cot 0 3 (b) (c)

a 3b

3

equals (a) 32/9 (b) 64/3 (c) 64/9 (d) No ne of th es e 6. lffix-y),fi x).f(y) a n dfix +y) are in A.P., for all x, y and /"(0) * 0, then (a)/"(2) ==/"(- 2) (b)/"(3) +/"(- 3) = 0 (c)/" (2)+/"(-2) = 0 ( d ) f (3) = f (- 3) 7. If y ='\jx + ^y +

+ Vy + .. .

then

is equal to (b)

/ - * 2yf - 2ry - 1 (c) (2y - 1 ) (d) None of these x -x 8. The derivat ive o f cos 1 at x = - 1 X 1 +X is (b) —1 (a) -2 (c) 0 (d)l If

J

V(3 - 2 si n

then

dl)

2

t) dt + f" cosfd* = 0, 0

k/2

is

cosec 0 cot 0

-3b \

4

cosec 0 cot 0

(d) None of these

dz dx

(a) - 3 (c )%/3~



118

10. If fn (x) = / » " 1 (x) for all n e N an d f0 (x) = x

(b

)fn(x).fn_X{x)

(c) fn (x). f „ . l W. . . f

then -j- [fn (x)} is equal to

2

(x). fx (x)

(d) n fi(x) t=i

Answers Multiple Choice -I 1. (a) 7. (c) 13. (c) 19. (a) 25. (c)

2. 8. 14. 20. 26.

(a) (b) (b) (c) (c)

3. 9. 15. 21. 27.

(c) (b) (a) (c) (c)

5. (b) 11. (c)

4. (b) 10. (b) 16. (a) 22. (b) 28. (c)

17. (d) 23. (b) 29. (c)

34. (a) 40. (b) 44. (a), (b), (c)

35. (c) 41. (b), (d) 45. (b)

6. (a) 12. (c) 18. (b) 24. (a) 30. (b)

Multiple Choice -II 31. (a), (c) 32. (b) 37. (a), (b) 38. (a), (b) 42. (a), (b), (c), (d)

Practice 1. (c) 7. (b)

33. (a), (b), (c) 39. (c) 43. (c)

36. (d)

Test 2. (d) 8. (b)

3. (c) 9. (c)

4. (c) 10. (a), (c), (d)

5. (a)

6. (a), (c)

TANGENT AND NORMAL The equation of the tangent at a point (xo, yd) to the curve y = f(x) is ' dy\ y-yo dx (*- X0) S Ax>, M>) equation of the normal at a point (xo, yo) to the curve y = f (x) is - 1

y - y o = / ...x

(x -xo )

dx yo) : (i) If tangent is parallel to the x-axis or normal

is perpendicular to x-axis then

dy dx

Fig.14.1. = 0. (ib, yb)

(ii) If tangent is perpendicular to the x-axis or normal is parallel to the x-axis then

Length of the

Tangent:

yo^f

KPT) =

dy dx

(*>. >b) dx V > Jfc. K>)

Length of the

Normal:

Length of the Subtangent:

l(PN)

-sf

Subnormal:

(*> ,yo)

I (TM) = —— yo dx

Length of the

dy dx

= y>

l(MN)

=

:*>. yo)

ya X»o, yo)

If

x = f(f), and y =

g(t)

then

dx f'(t) In this case, the equations of the tangent and the normal are given by

= f r j j (* -' (« and

yy-

g(t

))s f(f

) + (x -f (t )) f' (t ) = 0.

= o.

120


H

§ 14.3. The angle between two Curves

p z

Let y = f(x) and y = g(x) be two curves and P(K), yo) be a common point of intersection. Then the angle between the two curves at P (xo< >D) is defined as the angle between the tangents to the curve at P and it is given by ,-ne''(x o)-g '(xo )

8V /A01 0

If the curves touch each other at P, then 9 = 0'; so that f (xt>) = GR (AD) If the angle 9 is a right angle, the curves are said to cut orthogonally then f'(xo).cf(x0) = -1

Y' Fig.14.2

MULTIPLE CHOICE-I Each question is this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. The eq uation

of the tangent to the curve y = 1 - ex/ at the point of intersection with the y-axis is (a )x + 2y = 0 (b )2 x + y = 0

(c)x-y 2. The

line

=2

(d) Non e of these

a

touches

b

the

curve

( a )U ?

(d) None of these

(1, 1) is inclined at an angle tan

(c)a = - l , b = 2

+ p2 2

=a -

(c)a2-b2

p2

= a2+p2

(d) a2 + b2 = a2 - p 2 slope of the tangen t to the

curve

x

y= [ ^ at the point where x = 1 is J o 1 +xi (a) 1/4 (b) 1/2 (c) 1 (d) Non e of these —

4. If the tangent to the curv e xy + ax + by = 0 at

jt- axis, th en (a) a = 1, b = 2

(b) a -b

2

7. The equation of the tangent to the curve (d) None of these

3. The cho rd joi nin g the point s where x = p an d x = q on the curve y = ax 2 + bx + c is parallel to the tangent at the point on the curve whose abscissa is (a) 1 / 2 ( p + q) (b)1/2 (p-q)

(c)'

2

6.The

y = be~ x/a at the poin t

(c)Uf

(a ; a2 + b =a

-1

2 with

(b)a=l,b = -2 (d)a = - l,b = -2

2 2 x2 x 5. If the cur ves ~ + \ and - ^ = 1 cut 2 2 2 a b a (3 each other orthogonally, then

IXI

at the point wher e the curv e cuts the y=e line x- 1 is (a) * + )> =e (b) e(x + y ) = l (c) y + ex= 1 (d) Non e of these 8. If the tangent to the curve V JC" + = Va~ at any po int on it cuts the a xes OX and OY at P and Q respectively then OP + OQ is (a) a/2 (b) a (c) 2a (d) 4a 2

9. If the tangent at (1, 1) on y

=x(2-x) meets the curve again at P, then P is (a) (4 ,4 ) (b) (-1 , 2) (c) (9/4, 3/8) (d) No ne of the se The 10. tangen t to the curv e x = a V(cos 20) cos 0, y = a Vcos 20 sin 0 at the point corresponding to 0 = rc/6 at (a) parallel to the x-axis

2

Tangent and Normal

121

(b) parallel to the y-axis (c) parallel to line y = x (d) None of these

(a) 1/2 (c) 3/2 14.The

11. The area of the triangle f orme d by the pos itive x -axis, and the nor mal and tange nt to the circl e x + y 2 = 4 at (1, V3) is (a) V3~ (c) 4 -If

(b) 2

^

(b) 1 (d) 2

acute

a ngles

2

betwe en

the

curves

2

y = I x - 11 and y = I x - 3 I at thei r poin ts of intersection is (a) Jl /4

(b) tan

1

(4 -12/7)

1

(c) tan" (4 -IT) (d) none of these g (x) = x/ ta n x where 15. If /( x) =x /s in x and 0 < x < 1 then in this interval

(d) None of these

12. The dist ance b etwee n the srcin and the Ix 2 normal to the curve y = e + x at x = 0 is (a) 2/V3" (b) 2/-I5

(a) both/(x) and g (x) are increasing functions (b) both/(x) and g (x) are decreasing functions (c)/(x) is an increasing function (d) g (x) is an increasing function

(c) 2/ -I T (d) none of these 13. The value of m for which the area of the triangle included between the axes and any tangent to the curve x my = bm is constant, is MULTIPLE CHOICE -II

Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). 16. If the line ax + by + c = 0 is a norm al to t he curve xy = 1. Then

(a) a > 0, > b0 b0 (c) a < 0, >

(b) (d)

a > 0, b < 0 a < 0, b < 0

20.Let the parabola s y = x + ax + b and y = x(c —x) touch each other at the point (1,

0) then (a) a = - 3

17.The normal to the c urve represente d par ametrically by x = a (cos 0 + 0 sin 0) and y = a (sin 0 - 0 cos 0) at any point 0, is such that it

(b)f> =l (d) b + c = 3

(c) c = 2

(c) touches a fixed circle

21. The point of intersection of the tangen ts drawn to the curve x y = 1- y at the points where it is meet by the c urve xy = 1 — y, is given by (a) ( 0, -1 ) (b) (1, 1) (c) (0, 1) (d) none of these

(d) passes through the srcin

22. If y = 4x - 5 is a tan gent

(a) makes a constant angle with the x-axis (b) is at a constant distance from the srcin

18. If the tangent at any point on the curve

x + y 4 = a cuts off intercept s p and q on the co-ordinate axes, the value of (b ) a

(a )a

p~

+ q~~

-1/2

1 /2 (c) a (d) None of these 19. If y= / ( x ) be the eq uation of a parabola which is touched by the line y = x at the point where x = 1. Then (a)/'(0) =/'(l) (b) /' (l ) = 1 (c)/(0)+/'(0)+/"(0)=l (d) 2/(0) = 1 - / ' ( 0)

to the

cur ve

y 2 = px + q at (2, 3) then is

(a) p = 2, q = - 7 (b)p =-2,q = l (c)/ ? = -2, <7 = - 7 (d)p = 2,<7 = 7 23. The slope of the normal at the point with abscissa x = - 2 of the graph of the function /(x) = I x - x I is (a )- 1/ 6

(b) —1/3

(c) 1/6

(d) 1/3

24. The subtangent, ordinate and subnormal to 2

the parabola y = 4 a x at a point (different from the srcin) are in

122


(a) A.P. (c) H.P.

(b)G. P. (d) None of these

( 9

4

>

_9 VTT' Vl T

w

25. A point on the ellipse 4x 2 + 9v = 236 whe re the tangent is equally inclined to the axes is

(

c

)fJL _ — V3~' v u

(d) None of these


Ti me : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct 1- The curv e y -exy + x =0 h as a verti cal tangent at the point (a) (1, 1) (b) at no po in t (c) (0, 1) rx

2 | t | dt, the tangent lines 0 which are parallel to the bisector of the first co-ordinate angle is (b)y =x +-

(a)y = x - (c )y=x-~

form an angle of and TI/6atand the pointx *=3= 2 an angle of 71/3 theatpoint an angle of 7t/4. The value of +\

8. If

f" (X) dx

1 2 if" (x) is suppose to be continuous) is 3 V3-1 ( a ) 4^ 3- 1 (b)' (a) 3>/3 4 - V3~ (c) (d) None of these

-1.2,., cos t dt atx

y-r

(a) (c)

Si 2

3l 4

th e

subnor mal

1 - n

slope of th e t an gen t to the 1 . = -. — is 4 V2" ( 4V8" 1 (b) 3

4

the

at

any

point

on

n

y= a" x is of con sta nt leng th, th en th e value of n is (a)-2 (b) 1/2

J

4.The

2

(c)f = <3

3

f'{x)f"(x)dx

32 = 80 i 243 ~ 81

7.For the curve x =t -1 ,y = t -t, tangent line is perpendicular to x-axis when t = oo (a) 0 t= (b)

3. The tangent to the graph of the function y=f(x) at the point with abscissa x =1

j

(d)y

8 ~~3

X

2

(d )y =x + |

3

4098 _ 2048 27 81

3 6. If the pa ram etr ic equati on of a curve given by x = e cot t,y -e si n t, then the tangent to the curve at the point t = ti/4 makes with axis of x the angle (a) 0 (b) tc /4 (c) ti /3 (d) 7t/2

(d) (1, 0)

2. For F(x)=1

(c) y

answer(s). [10 x 2 = 20]

(d)2

(c) 1 9.The tang ent

and norma l at th e

2

curve

(a) tan 71

4 3 5. The equations of the tangents to the curve

y = x4 from the point (2, 0) not on the curve, are given by (a )y = 0 (b)y-l = 5(x-l)

1

(c) tan

,

(d) None of these

point

2

P {at , 2at) to the parabola y = 4ax meet the x-axis in T an d G respectively, then the P to the angle at which the tangent at pa ra bo la is incl in ed to the tangent at P to the circle through T,P,G is

5J[_1

12

(b) cot

t

(d) cot" 1 1

V

10. The value of parameter a so that the line 2 (3 - a) x + ay + (x - 1) = 0 is nor ma l to th e curve xy = 1, may lie in the interval (a) (- oo, 0) u (3, ~) (b) (1 ,3 )

(c) (- 3, 3)

(d) Non e of th es e

Tangent and Normal

123 Record Your Score

Max. Marks 1. First attempt 2. Second attempt 3. Third atte mpt

must be 100%

Answers Multiple

Choice

-I

1. (a) 7. (d)

2. (d) 8. (b)

3. (a) 9. (c)

13. (b)

14. (b)

15. (c)

Multiple

Choice

4. (b) 10. (a)

6. (b) 12. (b)

-II

16. (b), (c)

17. (b), (c)

18. (a)

19. (a), (b), (c) (d)

21. (c)

22. (d)

23. (a)

24. (b)

Practice

5. (c) 11. (b)

20. (b), (d) 25. (a), (b), (c)

Test

1. (d)

2. (a), (b)

3. (d)

4. (b)

7. (a)

8. (b)

9. (c)

10. (a)

5. (a), (c)

6. (d)

MONOTONOCITY §15.1. Monotonocity

A function defined on an interval said to be (i) Monotonically increasing function : If X2>X1 => > f(x-i), V x i, x2 e[ a , (ii) Str ictly incr easin g function : If X2> x-] => f(x2) > f(x 1) , V xi, X2 e [a, b] (iii) Monotonically decreasing function : If x 2 > X1 f (x 2) < f(xi ), V X1, x 2 e [a, b] (iv) Strictly decreasing function : If

x 2 > x i => /(x2 )
XI,X

2

G

§ 15.2. Test for Monotonocity

(i) (ii) (iii) (iv)

Th e Th e Th e The

functi on functi on fun cti on functi on

(x) is is (x) is (x) is

in th e interval [a, if (x) > 0 in [a, in th e interval [a, if (x) > 0 in [a, in th e interval [a, if (x) < 0 in [a, in th e interval [a, if (x) < 0 in [a,

MULTIPLE CHOICE-I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whicheven is appropriate. 1. Let y = x2e \ the n the interval in whic h y increases with respect to x is (a) (--. .oo) (c) (2,00)

(a) (- 00, 00) (b) - 4 , 3 - —

u [ l ,o o)

(b )( -2 ,0 ) (d) (0, 2) (d) (1, 00) 4. On which of the following intervals is the

the interval

function x 1 0 0 + sin x - 1 dec rea sin g ? (a) (0,7 t/2) (b) (0, 1)

(a) (2/; + 1, 2/;), n e N

(c) ( ^ , 7t ] 5. The (d) None of these 3. The set of all values of function

decreases for all real x is

a for which the

funct ion

(d) No ne of th es e / ( x ) = log (1 + x ) -

increasing on (a) (0, oof (c) (- oo, oo)

(b) (-co, 0) (d) None of these

is

Monotonocity

, 6.

125

a sin x +cos b x . , It / (A) = — : : is decre asing for all c sin A + a cos A A then (a) ad - be 0> (b) ad - be < 0 (c) ab -cd> 0 (d) ab - cd < 0

7. If f ( x ) = (ab-b 2-2)x

+ J (cos 4 6 + sin 4 0) o c/8 is decreasing function of A for all A e and /;> E R, /J being inde pende nt of A, then (a) a e(0, VfT) (c) a e ( - V(T, 0)

(b) a e ( - V6, V6) (d) non e of the se

8. The valu e of a in order that/( A ) = V3~sin A - cos A - 2ax + b decreases for all real values of A, is given by (a) a < 1 __ (b) a > 1 (c) a >4.2 (d) a <42 + X

9. The function /( A ) =

} is

(c) increasing for all (— .- 1) .

11. If a <0, the funct ion /( A ) = e"x + e "x is a monotonically decreasing function for values of A given by (a)x>0 (b) A < 0 (c) A > 1 (d) A < 1 2 12. y= {A ( A - 3) } increases for all values of A lying in the interval (a) 0 < A< 3 / 2 (b) 0 < A < °° (c) - °o < A< 0 (d) 1 < A < 3 13. The

(a) (b) (c) (d)

B I ( E + A)

(a) increasing on [0, (b) decreasing on [0, (c) increasing on [0,7i/e) and decreasing on [n/e, •») (d) decreasing on [0, n/e) and incr eas ing on [n/e, A ) = (x + 2) e

10. The function/defined by/( (a) decreasing for all A

A

(d) decre asi ng in ( - 1, «>) and inc rea sin g in

fun cti on /(A) = tan A - A

always increases always decreases never decreases some ti mes increa ses decreases

and some

time s

14. If the funct ion / ( A ) = cos I A I - 2 ax + b increases along the entire number scale, the range of values of a is given by (a) a - 3/ 2

15. The func tion / (A) = A V(AA - A2) , a > 0 is

(b) decreasing on (—<*>,— 1) and increasing in

(a) (b) (c) (d)

increases on the interval (0, 3a/4) decreases on the interval ( 3a/4 , decreases on the interval (0, 3a/4) increases on the interval ( 3a/4 ,

a ) a

MULTIPLE CHOICE -II Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answerfs). 16 . L e t / I(A)=/(A)-(/(A)} ) } for all +{/(A (c) increasing in j — — , 0 real values of x. Then (a) h is increasing whenever/( A) is increasing (d) decreasing in I 0, (b) h is increasing whencver/'(x) < 0 (c) h is decreasing whenever/is decreasing 18. Whic h of the fo llow ing func tions are (d) nothing can be said in general decreasing on (0,71/2) ? 2 (a) cos A (b) cos 2A 17. If <)>(A) = 3 / 3 A x +/(3 - A ) V AG (-3,4) (c) cos (d) tan 19. If ((> (A)=f(x) +f(2a - x) and /" ( A ) > 0, where/"( A ) > 0 V A e ( - 3, 4), then 0, 0 < A < 2a then (a) increasing in [ , 4 (a) <()(A ) increases in (a, 2a) (b) <|)(x) Increases in (0, a) (b) decreasing in — 3, — (c) (])(A ) decreases in (a, 2a) 2

3

126


(d) <)>(x) decreases in (0, a) - If f{x) = 2x + cot ' x + log (V1 + x 2 - JC), then/(x) (a) increases in (0, °o) (b) decreases in [0, (c) neither increases nor decreases in (0, (d) inc rea ses in (— «) 3 2 21. If f (x) — ax - 9x' + 9x + 3 is increasing on R, then a>3 (a)< a 3 (b) (c) a 3< (d) Non e of thes e 22. Functio n / ( x ) = I x I - xI - 1 I is monotonically increasing when (a) x < 0 (b) x > 1 (c) x < 1 (d) 0 < x < 1 23. Every invertibl e funct ion is (a) monotonic function (b) constant function

(c) indentity function (d) not necessarily monotonic function

20

24. L et /( x) = 2x 2 - log I x I, x * 0. Th en /( x) is >

(a)m.i.in|--,0 (b) m.d. in (c) m.i. in (d) m.d. in

1

)uf^,«

.0

1 2' U 0, 1

u

2

J

~

2

1

| U

4

25. Let / ' (x) > 0 and g'(x) < 0 Then (a)/{g(x)}>/{g(x+l)} (b)/U(x)}>/{g(x-l)} (c)g{/(x)}>g(/(x+l)} (d)g{/(x)}>g(/(x-l)}

for all x e R.


M.M. 20

(A) There are 10parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] 3 2 2 4. If f\x) = | x | - {x}wh er e {x} den ot es th e 1. Let f(x)=x + ax + bx + 5 sin x be an increasing function numbers R. Then condition

in the set of real a an d b satisfy the

(a) a 2 - 3b - 15 > 0 (b) a 2 - 36 + 15 > 0 (c) a 2 - 36 - 15 < 0 (d) a > 0 and 6 > 0 2. Le t Q (x) =/"( x) + / ( 1 - x ) and f"(x) < 0, 0 < x < 1, th en (a) Q increases in [1/2, 1] (b) Q decreases in [1/2, 1] (c) Q decreases in [0, 1/ 2] (d) Q increases in [0, 1/2] 3. I f f: R —> R is the function defined by

fix)-

2 2 , then — X e +e (a)f(x) is an incre asing functi on (b)f(x) is a decreasing function (c)f(x) is onto (surje ctive ) (d) None of these

fractional part of x, then in

(a) |- f ,0

f(x) is decreasing

(b) | - f ,2

1 < 2 ' 5. The interval to which 6 may belong so that the function

(0| - f , 2

/(x) =

(d)

—

V21 - 4 6 - 6 2 6+1

x 3 + 5x + V6

is increasing at every point of its domain is (a) [ - 7, - 1] (b) [ - 6, - 2] (c) [2, 2 5] (d) [2, 3] 6. For x > 1 , y = log c x satisfies the inequality (a )x - 1 >y

(b )x - 1 >y

(c) y > x - 1

,,, x - 1
7. The function interval (a) (0, e) (c) (0, 1/e )

f(x)=xx decrease s on th e (b) (0, 1) (d) None of th es e

Monotonocity

132

8. Let the funct ion f(x) - si n x + cos x, defined in [0, 2n], then f (x) (a) increases in (n/4, n/2) (b) decreases in (7r/4, 5n/4) (c) increases in

0,J

(d) decreases in

o.j

571

lu

be

j ex (x - 1) (x - 2) dx.

9. Let f(x) =

Then

decreases in the interval (a) (- oo,- 2)

(b) ( - 2, - 1)

(c) (1, 2)

,2n

(d) (2, oo)

10. For all x e (0, 1) (a ) e * > l + x

M f . a *

(c) sin x > x

(b) log e (1 +*) (d) log

e

x >x

Record Your Score

Max. Marks 1. First attempt 2. Second attempt

3. Third attempt

must be 100%

Answers Multiple

Choice

-I

1. (d)

2. (d)

3. (b)

4. (d)

5. (a)

6. (b)

7. (b)

8. (b)

9. (b)

10. (d)

11. (a)

12. (a)

13. (a)

14. (c)

15. (a)

20. (a), (d)

Multiple

Choice

-II

16. (a), (c)

17. (a), (b), (c), (d)

18. (a), (b)

19. (a), (d)

21. (b)

22. (d)

24. (a), (d)

25. (a, c)

Practice

23. (a)

Test

1. (c)

2. (b), (d)

3. (d)

4. (a)

6. (a), (b), (d)

7. (c)

8. (b), (c)

9. (c)

5. (a), (b), (c), (d) 10. (b )

MAXIMA AND MINIMA §16.1. Maxima and Minima

From the figure for the function y =

f{x), we find that the function ge ts local maximum a nd local

minimum. The tangent to the curve at these points are parallel to x-axis,

i.e.,^

dv

= 0.

Greatest

Fig. 16.1. § 16.2. Working Rule for Finding Maxima and Minima (a) Firs t Derivative T e st : To che ck the maxi ma or minima at x = a (i) If f' (x) > 0 at x< a and f (x) < 0 at x> a i.e. the sign of f (x) ch an ge s from + ve to - ve, then ha s a local maxim um at x = a.

f (x)

Maxima and Minima

129

Fig. 16.3. (ii)

If (x) < 0 at and (x) > 0 at has a local minimum at x =

the sign of

(x) changes from - ve to + ve, then

(iii) If the sign of (x) does not change, then (x) has neither local maximum nor local minimum at x = a, then point 'a' is called a point of (b)

Se co nd deriv ative Te st :

(i)

If

(a) < 0 and

(a) = 0, then 'a' is a point of local maximum.

(ii)

If

(a) > 0 and

0, then 'a' is a point of local minimum.

(iii)

I f f " (a) = 0 and f (a) = 0 then further differentiate and obtain f" (a).

(iv )

I f f (a) =

=

(a) = ....=

= 0 and

* 0.

If is odd then (x) has neither local maximum nor local minimum at x = a, then point 'a' is called a point of If

is even, then if

then (x) has a local maximum at x = a and if

" (a) > 0 then f( x) has a

local minimum at x = a. Note : Maximum or minimum values are also called local extremum values. For the points of local extremum either o or (x) does not exist. § 16.3. Greatest and Least Values of a Function

Given a function

the value (c) is said to be

in an interval [a,

(i) The greatest value of (ii) The least value of Two important tips : (i)

(x) in [a,

(x) in [a,

if (c) > (x) for all xin [a, b], if (c) < (x) for all x in [a, b].

If

Let

=

{|f degree of

degree of f(x)}

if z is minimum => y is maximum, if

is maximum => y is minimum.

(ii) S tat ion ary val ue : A function is said to be stationary for x = c, then 0.

(c) is called stationary value if

130


MULTIPLE CHOICE -1 Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. The

great est

value

of

function

the

f i x ) = 2 sin x + sin 2x on the interval

(a) 1 (c) 3

is (a)

3 VI

2. The poin ts of extr emu m of the functio n

e '2/2 (1 - t2)dt are

(a) ± 1

(b) 0

(c) ± 1 / 2

(d) ± 2

(a) (x — 2 ) 4

(b) 3 - (x - 2)

4

(c) - 17 - (x - 2) 4

(d) Non e of these

4. The difference between the greatest and the

least value of the function

(a) a = — 3/4, b — - 1/8 (b) a = 3/4, b = - 1/8 (c) a = - 3/4, b = 1/8 (d) None of these 2 + 2 2 / + ... + 2h°x20. f{x) has (a) more than one minimum (b) exactly one minimum (c) at least one maximum (d) None of these

11. Let/(x)

= j s' n ~ ' a 2x

(a) 1 (b) 47/6 (d) 59/6 8/9

3 ) , then maxima o f f ( x )

x. If in [1, 6],

6. Le t/ (j c) be a different ial function for all

/(l) = — 2 and /' (x) > 2 for all then minimum value of/(6) = (a)2 (c)6 7.The

JC

[ 0, 2n ]

where

/(JC) = ex sin x has maximum slope is

,, (a)-

n.

(b) ^

„.

(c)7t

n

(d) None of these

8. Let

^

IX " +JC

2 +

3J C + si n

JC I

]

(d) No ne of thes e

function 2

in the interval

* < x>1

/ ( x ) has a relative minimum at x = 0 then the function y =f ix) + ax + b has a relative minimum at x = 0 for (a) all a and all b (b) all b if a = 0 (c) all i > 0 (d) all a > 0

13. The

(b)4 (d)8 poin t

0 <

12. A differentiable function

(b) a —3 (d) Non e of these

(c) a

' ,

(b) - 1

(c ; 0

is (a) 3

+

Then

f ( x ) can have a minimum at x = 1 is the value of a is

f ( x ) = j* (t 2 + t+ 1) dt on [2, 3] is

5. L e t f ( x ) = a-(x—

I JC I + bx2 + x has extremum

10. Let / ( x ) = 1 + 2x

f'{2) =/ "( 2) = 0 an d/ 3. A function/such that has a local max imu m o f - 1 7 at 2 is

(a) 37/6 (c) 57/6

f

at x - 1 and x - 3 then

(d) None of these

F( x) =f

(b) 2 (d) inf init e man y

9. If /(JC) = a log

(b)3

2 (c) 3/2

then number of points (where/( JC) attains its minimum value) is

3 +s in - , x *0 * J , X = 0

/(x) = £

{

2 ( , - 1) (r-2)>

2

+ 3 {t- 1) (r - 2) } dt attains its maximum at x = (a) 1 (b) 2 (c) 3 (d) 4 14. Assu ming that the petrol burnt i n a motor boa t varies as th e cu be of its velo ci ty , th e most economical speed when going against a current of c km/hr is (a) (3 c/ 2) km/hr (b) (3 c/ 4) km/hr (c) (5 c/ 2) km/hr (d) ( c /2 ) km/hr

Maxima and Minima

131

IS . N Characters of information are held on magnetic tape, in batches of A characters each, the ba tch proc ess ing time is a + (3x 2 seconds, a and p are constants. The optical value of .v for fast processing is,

/(a)p \ a

(b)

wVf

(d)

a

JF 11

a

MULTIPLE CHOICE-II Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, (1corresponding to the correct answer(s). 16. L et/ (.v ) = cos x sin 2x then (a) min f(x) > - 7/ 9 v G ( - n, 71)

per ho ur at 16 mil es per hou r. Th e most i.e., economical speed if the fixed charges salaries etc. amount to Rs. 300 per hour.

(b) min f(x) >9/ 7 .v e (- 7i, JI)

f(x)

(c) min

>-1/9

(d) 40

be ins cribed in a c ir cl e of radius r is

>-2/9

v e [ - 7t, JT]

(a)

17. The minimum value of the function defined b y/ ( A ) = maximum {A. A + 1. 2 - A) is (a) 0 (c) 1

(b) 20

(c) 30

22. The maximum area of the rectangle that can

.v e [ - 7t. 7t]

(d) min f(x)

(a) 10

(c)

71 R 71/"

(b) r (d) 2r 2

(b) 1/2 (d) 3/2 A

+

-sin

18. Let/(x): -

1 -

3X

,

A < 0 0
-

1 <

A, COS A ,

—

<

X <

7T

Then gl obal ma xim a of /(. v) equals and global minima of/( A ) are (a) - 1 (b) 0 (c )- 3 (d) - 2

19. On [l,e], the least and greatest values of In

A

IS

(b)l,e

(a) e, I 2

;(c) 0, eA

(d) none of these

20. The minimum value of 1 +sin

1

a

(a) 1

(a) three equal real roots (b) three distinct real roots (c) one positive root if/(a) < 0 and/(P) > 0 (d) one negati ve root i f / ( a ) > 0 an d/ (P ) < 0 24. Let/( A ) be a function such that /'(a) Then at x = a,/(A)

* 0.

(a) can not have a maximum (b) can not have a minimum

2, /'( .V ) = A

23. Let / (A) = OA3 + bx~ + cx + I have extrema at A = a , p suc h that a p < 0 and f(a).f(p) < 0 then the equation/( A) = 0 has

1 1 +cos a

is /

(b)2

(c) (1 + 2" /2 ) 2

(d) None of these 21. The fuel charges for running a train are pr opor ti onal to the sq uare of the spe ed generated in miles per hour and costs Rs. 48

(c) must have nei ther a ma xi mum minimum

nor a

(d) none of these 25. A cylindrical gas container is closed at the top and open at the bottom; if the iron plate of the top is 5/4 times as thick as the plate forming the cylindrical sides. The ratio of the radius to the height of the cylinder using minimum material for the same capacity is (a) 2/3

(b) 1/2

(c) 4/5

(d) 1/3

132


Practice Test M. M: 20

Tim e : 30 Min

(A) There are 10 parts in this question. Each part has one or more than one correct 1. Let f(x) = (x 2 - 1)" (x2 +x + 1) then f(x) x = 1 when local extremum at (a ) n = 2 (b) n = 3 n = 6 (c) = 4n (d) 2. The critica l poi nts of th e func tion , . I x-2 I . where f (=x ) ^—is x (a ) 0 (b) 2 (c) 4 (d) 6 3. Let/(x) =

x - x

+lOx - 5,

has

decrease function exist

f\x)

x
- 2r + log 2 (b - 2) , x > 1

- 1)

(d) None of these

dt (x>0); then/ 4. Let f (x) = J* o t (a ) maxima when re = - 2 , - 4 , - 6 ,

-

(x) has

(b) maxima when re = - 1 , - 3 , - 5 (e) mi ni ma wh en re = 0, 2, 4, (d) minima when n = 1, 3, 5, 5. Le t/" : [a,6] -» R be a function such that for c e (a, b),f' (c)=f"(c) =f"\c) =f w{c) =f"(c) =0 th en (a) /"h as local ex tr em um a t x = c (b)/has ne it he r local ma xi mu m nor local mi ni mu m at x = c (c) fienece ssa ril y a co nst an t func tion

(b) Function has interval of increase and (c) Greatest and the least values of the

the set of values of 6 for which /(x) have x = 1 is given by greatest value at (a) 1 < 6 < 2 (b) 6 = ( 1 , 2 | (c) be { -

answer(s). [10 x 2 = 20]

(d) Function is periodic 7. Let fix) = (x - l ) m (x - 2)", xe R, Then each critical point of (x) f is ei th er local maximum or local minimum where (a) m = 2, n 3= (c) m = 3, n= 4

(b)

m = 2, n = 4

(d) m = 4,

R

= 2

8. Th e nu mbe r of soluti ons o f th e e qua tio n

0 , g ( x ) * 0 and has minimum value 1/2 is (a) on e

(b) tw o

(c) inf ini te ma ny

(d) zero 0
9, Let / > ) =

then

x> 1 (a) fix) ha s local max ima a t x = 1 (b)/"(x) has local minima atx = 1

(c)f(x) does not hav e an y local ext rem a at

x=1

(d) f (x) has global minima at x = 1 10. Two town s A and 5 ar e 60 km ap ar t. A school is to be built to serve 150 st ud en ts in town A and 50 st ud en ts in town B. If the tota l dist ance to be trave lled by all 200 st ud en ts is to be as smal l as possible, th en the schoo l should be buil t at

(d) it is diffi cult to say wh et he r (a) or (b)

(a )t o wn fl

(b) 45 km fro m town A

Fr om th e gr ap h we can conclude th at the

(c) town A

(d) 45 km from town

(a) Function has some roots Record Your Score Max, Marks 1. First attempt

i

2. Second attempt 3. Third attempt

must be 100%

B

Maxima and Minima

133

Answ er s Multiple

Choice

-1

1. (a)

2. (a)

3. (c)

4. (d)

5. (c)

6. (b)

7. (b)

8. (a)

9. (a)

10. (b)

11. (d)

12. (b)

13. (a)

14. (a)

15. (c)

20. (c)

21. (c)

5. (d)

6. (b)

Multiple

Choice

-II

16. (a), b)

17. (d)

18. (b), (d)

19. (c)

22. (d)

23. (b), (c), (d)

24. (d)

25. (d)

1. (a), (c), (d)

2. (a), (b), (d)

3. (d)

7. (b), (d)

8. (d)

9. (a)

Practice

Test 4. (a), (d) 10. (c)

INDEFINITE INTEGRATION 17.1. Methods of Integration (i) Integration by Substitution (or change of independent variable):

If the independent variable x in J f{x) dx be changed to t, then we substitute x = (f) i.e., dx = 0 ' (f) dt j f(x) dx = | /(<{> (t)) '(0 dt which is either a standard form or is easier to integrate. (ii) Integration by parts : If u and vare the differentiable functions of xt he n J u.vdx = uj

udx-j

"

\-jj-uIf/

dx'

vdx

dx.

17.2. How to choose 1st and llnd functions : (i) If the two functions are of different types take that function as 1st which comes first in the word ILATE where I stands for inverse circular function, L stands for logarithmic function A stands for Algebraic function, T stands for trigonometric function and E stands for exponential function. (ii) If both functions are algebraic take that function as 1st whose differential coefficient is simple. (iii) If both functions are trigonometrical take that function as llnd whose integral is simpler. Successive integration by parts : Use the following formula

J UVdX = UV-\ - 1/ V2 + U" V3 - u"' V4 + .... + + +(-1)"" 1 u " " 1 Vn + (- 1)" J if vn dx. where if stands for nth differential coefficient of u w.r.t. xa nd vn stands for nth integral of vw.r.t. x. Note - J

— 1 ( 2 a x - x2 )

(••• j

-

-iM. (âj

1 - cos 6 = vers 0 1 - sin 0 = covers 0

Cancellation of Integrals:

i-e.

{ e"{ f(x) + f (x)} dx = exf(x) + c.

17.3. Evaluation of Integrals of Various Type Type I: Integrals of the type.

(0

•

ar + bx+c

C' )f

12 ^(ar

d

*

+ bx+ c)

(iii) J ^(aS + bx+c) dx

Rule : Express ax 2 + bx+ c in the form of perfect square and then apply the standard results. Type II: Integrals of the type

(i) j

2

PX+q

ar + bx+c

Rule for (i): V '

^ (ii) / j;

. f

3

yaxr + bx+c

(iii)dx f

(P*+cPd* = - M ( 2 f + b> d x 2 bx+c 2a) J (ax 2+ .bx+c) , K „ , „ ( (2a\ i ax+

+

+

fax 2 + bx + c) dx J q- & I f \ ^ 2 a J J ax2 + bx+c

The other integral of R.H.S. can be evaluated with the help of type I.

dx

Inde finit e Integration

135

Rule for (ii):

(px + q) dx

f

jd_ t

=

Vfax 2 + bx+ c)

2a

J

( 2 ax+b)

( dx+

ob\r dx 2 a j J Va ^ + bx +c {

V( aJ + bx+c)

dX q-g-1/ • 2 2 a J sla^ + bx+c \ j The other integral of R.H.S. can be evaluated with the help of type I.

= ^

a

W

+

to+O+f

+ ^ — and then (a>c + bx+ c)

f(x) + —

Rule fo r (iii): In this case by actual division reduce the fraction to the form integrate. Type III : Integrals of the form

dx ... f (OJ

....

— — a + bs in x

dx f (»)J

a + bcos

2

.....

dx ("0—1 g asinx+bcosx

x

(iv) | — I (v) 1 ,2 „ * (t an x) dx 3 — (asm x+ bc o s x) a sin x+ bs in xcos x + cc o s x + d where 0 (tan x) is a polynomial in tan x. Rule : We shall always in such cases divide above and below by cos s e c 2 x dx = dt the n the que stio n shall re du ce to th e for ms J —r—^— (ar + bt+c) Type IV : Integrals of the form J

v> / <• (ii) J Ja-t-nciny a + bs in x . r (p co s x + qsin x) , iv JI f H — r dx (acos x+ bs in x) Rule for (i), (ii) and (iii):

fv '

the numerator shall become sec

2

put tan x = t i.e.

bt+c)

1' a sin x+ b co s x + c

x/2

2 tan

co s x = 1 + tanx/ =2 2 ' ,

x ; then

or f — ^ — (ar + dx

(iii )J a -i- n rnc v a+bcosx . , r p co s x + q sin x + r , (v JJ dx a cos x+ bs in x + c JJ

1 - x/ tan2 2 write

2

sin x 1 + tan

2

x/2

x/2 and the denominator will be a quadratic in tan x/2. Putting tan x/2 =

2

i.e. s e c x / 2 dx = 2 dtthe que stio n shall re du ce to th e form I —^

ar +

— bt+c

Rule for (iv): Express the numerator as /(£/) +m(d.c. of Cf) find I and m by comparing the coefficients of sin x and cos xand split the integral into sum of two-integrals as

i j dx+ m |

d.c. of

rf

Cf = lx+ m In I £/1 + c Rule for (v) : Express the numerator as

I (D r) + m (d.c. of D r) + n, find /, m, and n b y comparing the coefficients of sin x, cos x and constant term and split the integral into sum of three integrals as ,r . r d.c . of Cf . / dx+ m J — dx + n J Cf

= Ix + m In I Dr I + n J ~ and to evalua te j Type V : Integrals of the form 2

0)J (x(x+ +k ax ) +dxa ) 2

4

(") j

2

ix 2

-

a 2)

4

dx

tx 4 + A x 2 + a 4 )

f dx J — Dr

proceed b y the method to evaluate rule (i), (ii) and (iii).

t

136

Obj ective Mathem atics

where k is a constant, + ve, - ve or zero. 2

2

Rule for ( i) and (ii ) : Divide above and below by x 2 then putting (i) t = x- — and (ii) t = x+ —

i.e.,

,2

«

dt =

a

dx

7 then the questions shall reduce to the form

and dt =

dt

or

t 2 + t?

dx

r

dt

3

t 2-c 2

Remember:

... r
=1 r x2 dx (x 2 + a 2) dx x4 + /cx 2 + a 4 " 2 J ( x 4 + / r x 2 + x 4 )

.... r (

"

,J

nm f 1

n

dx (x4

+

fcx2

d x

(x2 + /r)"

+ =

a4)

2

=J

_ r

(x

~

2a 2

(x 4

1, r (x2 - a2 ) dx 2 J tf + k f + a 4 )

+ a2 ) dx

+ /rx

2

x

k(2n-2)

+

(x 2 + A) " - 1

+

_ J_

a4)

r

2a 2 J

+ J2£Ln3Lf

*(2"-2)

J

(x 2 - a2 ) dx (x4 + kx 2 + a4) dx +

Type (VI) : Integrals of the form

... f

dx (Ax+ B) V(ax + b) .... dx

(«)

(/4X2 + Bx+ C) V(ax+b)

(iii) I (iv) J ^

dx (Ax+ B) ^(ax 2 + bx+ c) dx

1 j (/Ax2 + 0x + C) V(ax^ + bx + c)

Rule for (i) and (ii) :

Putax+b

Rule for (iii):

Put Ax + B = ~

Rule for (i v) :

Put —

„

= P

. ax2+ b x + c

2.

= r

5

A^+Bx+C Note : The integral J

f

, dx „ = where (Ax+ B) ry(a)C + bx+ c)

r is positive integer, may also be evaluated by

substitution Ax+ B = -y Type VII : Integral of the type J x m(a + bxnf

dx, where m, n, p e Q

Case (i): (i) If p e /+ then expand by the formula of Newton binomial,

(ii) If p < 0, then we put x = t k, where k is the common denominator of the fractions m and n.

m 1 Case (ii ) : If ——— -

Integer then we put a + bx" = P

where a is the denominator of the fraction p. Case (iii)

: If

m

— 1 +p = Integer then we put a + bx" = t ax n where a is the denominator of the

fraction p. Type VIII : Integrals of the type :

j f{x, (ax+ b) Pl/£ " , (ax + £>)P2/Cfe ) dx where f is irrational and pi , pz

<71, <72

e I then put ax + b = f ) , where m is the L .C .M. of q-i, qz ....

Indefinite Integration

137

Type IX : Integral of the form J s i n m xcos"xdx t. Ca se (i ): If m is odd and n is even then put cos x = Ca se (i i): If m is even and n is odd th en pu t sin x = t Ca se (iii ): If m and n both are odd then if m > n, put sin x = t m< n, put co s x = f m = n, put sin x = t or co s x = t. Case (iv) : If m + n = -ve and even. Then convert the given integral in terms of tan xand sec xthen put tan x = t. Case (v) : If m an d n are even integer, then convert them in terms of multiple angles by using the formulae 1 + co s 2x .2 1 - cos 2x sin 2x 2

co s x = and

; sin x = 2 cos x c os y = cos (x+

; sin x c o s x = — - —

y) + cos (x- y)

17.4. Standard Substitution

4 V

Expression _ a \ or V( x- ct )( p- x) (P X)

f

'(- a

V

Substitution x = a cos 2 9 + p sin 2 9

or V( x- a) ( x - P)

x = a se c 9 - p tan 9

X - p

1 V(x-a) (x-P)

x - a = t2 or x- P =

t2

MULTIPLE CHOICE-I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. x 2 + 2x (x ' + x ^ + l ) 10

1. If a part icle is mov ing with veloci ty v (?) = cos ntalong a straight line such that at t = 0, s =its4 posit ion fun cti on is given by 1 1 (a) —co s 7ir + 2 (b) - sin— + 4 n n (c) 1/71 sin nt + 4(d) Non e of these 2. Let

f{x) be

a

functi on

/( 0) =/ '( 0) = 0,/"(x) = function is (a) log (sin x) + ^ tan

3

sec

4

( that,

x + 4, then

2

2

x + 2x 2

x + J

(d) None of these 12 + SJC9) f ^x —^5 ^ ^ dx is equal to ( x + x 3 + l) 3

°

(b) r^ -i 1 + c 2( x + x + l ) 5 3 I—7 T c ) ln(x +x + \ + S2x + 5x ) + c (d) Non e of these

the

4. If _[/(*)

C os xdx

= ~ f [xyr c thcn/(x) is 2" (b) sin x (d) I

(a) x

x + xb

(b) | log (sec x) + i tan (c) log (cosx) + £cos

such

2

(c) cos x 5

j (sin 2x - cos 2x) dx = ^

U)a = ~,bGR (c) «= —, /? e

4 K

sin (2x — a) + 1

( b = '

R

4

(d) Non e of these

138


6. The

primi tive

,, , f(x)=

f unct ion

of

the

(d) None of these

"(c) 0 10.

T , 2 2^3/2 \a -x (a ~x) (a)c + (b)cT 2 2 , 2 3 2a x 3a x , 2 _ 2..3/2 (c) c _ ,t 2 % 3 — (d ) N on e of th es e 3a x 7. The

antiderivative 1 whose 3 + 5 sin x + 3 cos x pass es th ro ug h th e po in t ( 0 , 0) is

/«=

log

1 - ^ tan x / 2

'log V

1 + 1 ta n x/ 2

log

1 + | cot x / 2

(a,i

(b) 5c + 5d + x

(a) constant

funct ion

4{cf-x) . 4 IS

1

2/ 4 ..3/4 dx is equal to X (X + 1) 1 \V4 + c (b) (xH + l ) 1 / 4 + c (a) I I+-7 1/4 W

1—I

f

+c

of graph

j I+-4I X

Nl/4

+C

11. Let the equation of a curve passing through f 2 x the point (0, 1) be given by y = J x . e dx. If

the equation of the curve is written in the form x =f(y) the n/( y) is (

a

/

(c,I

( d) -

(d) None of these 12.

2

^ 3

(d) None of these

(c)

- dx is equal to

J

8. J x (1 + log x) dx is equal to

In

^ (b)

(1+*)

X

(a) x log, x + c

(c) x+c

(a)

(b) e + c

x+ 1

+ c

(b) ex(x+l)

+c

(d) Non e of these ^ - 7+ c

(c)-—

9. J (x - a) (x - b) (x - c)... to

(x - z) dx is equal

( d ) —

(*+l)

T + C

1

MULTIPLE CHOiCE -II Each question, in this part, has one or more than one correct answers). a, b, c, d corresponding to the correct answer(s). 13.

CLF

(a) A - sin (b) B = co s (c ) A = cos (d) B = sin 14.

(b) 3 (1 + x~ 1/ 2 )~ 2 / 3 + c / \ 1 r\ , l/2\— 2/3 , (c)3 (1 +X ) +C (d) None of these

sin X dx = Ax + B log sin (x - a ) sin (x - a ) + C then Ifj-»

a a a a

16. if

2/3

(1 +x

(a) 3(1 + x~

1/2 1/2

f4 e+ 6e - x dx = Ax + B log,(9e -4) 9ex -4e~"

J

1 • 2 2-sin x If the derivative of fix) w.r.t. x is ———— fix) then/(x) is a periodic function with period (a) n / 2 (b) 71 (c ) 2n (d) not def ine d

15. jx'


) " 5 / 3 dx is equal to

)~l/3 + c

17.

then (a) A = 3/2

(b) B = 35/36

(c) C is indef inite

(d) A + B = -

19 — 36

2^ f, x 1/2 Let j r — T - dx = — gof (x) + c then 3 vl - x .3/2 (a)/ (* ) = V7 (b) /( x) =x' .2/3 (c)/(x)=x (d) g (x) = sin x

18. If J cosec 2 x d x = f ( g (x)) + C, then

+C

Indefinite Integration

139

(a) range g (x) = ( (b) do m/ (j c) = ( - oo, oo) - {0}

+ e +1

(c) g'{x) = sec 2 x ( d ) / ' (x) = 1 /x fo r all x e (0, 19. If

Jg

|

=

(c)sW

VT 77 .

(d)/(.v)«2(A--2)

x

• v dx =f (x) V(1 +e ) y(l + e) - 2 log g (x) + C, then

f co s 4.i - 1 dx is equal to 20. J cot x - tan x i (a) - - cos 4x + c (b) - ± co s 4.t + c (c) - ^ sin 2x + c

(d) Non e of these

(a) / ( x )= x - 1 V1 + e + 1


Time : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct

1.

n -n If f(x) = Li m —n—, x > 1 then

I

2

x f ( x ) In (x + V(l + S )) dx is V(1 + x 2 )

(a) In (x + V(l+x 2))

-x + c

(b) | (x 2 In (x + V(l+a; z

2.

2

) ) - x2) + <

(c) x In (x + V(l+X 2 )) - Zn (x + V ( l + x 2 )) + c (d) None of these Th e va lue of the integral dx , n.l/n , n e N is J X- (1.. +X ) 1

(a)

f

X

1

(a) sin x - 6 tan

(1 + n)

x

(b) sin x - 2 (sin x)

1

+c

(c) sin x - 2 (sin x)

1

- 6 tan

1

(sin x) + c

(d) sin x - 2 (si n x) If

1

+ 5 tan

1

(sin x) + c

\f{x) si n x cos x dx = 2 (6

1 (O-

(d)-

1 (1-D

(l + re)|

1 -

1 +

1

— — log f(x) + c, -a)

then f (x) = 1 (a) 2 . 2 ,2 2 a si n x + b cos x (b)

2 . 2 ,2 2 a si n x - o cos x

(c)

+c

+c

n

2 = 20]

(sin x) + c

2 2 , 2 . 2 a cos x + b si n x (d) None of these 5. If l' means log log log

(b)

answer(s). [10 x

+c

+c

repeated

r

{[ x I (x) I2 (x)l3 (x) ...t (x)f , , ,r + 1 . , (a ) l (x) + c ir + l , , ( (b) ^ f c r +1 +

x

3 , 5 X + cos x 3. The valu e of the int egra l J —^ dx 2 . 4 sin x + sin x

x, the log being times

(c) (xj + c (d) None of these

1

then

dx is equal to

140

Ob j ective Mathemati cs Record Your Score

Max. Marks 1. First attempt 2. Second attempt must be 100%

3. Third attempt

Answers Multiple

Choice

-I

1. (c)

2. (b)

3. (b)

4. (b)

5. (b)

6. (c)

7. (b)

8. (c)

9. (a)

10. (d)

11. (c)

12. (a)

13. (c), (d)

14. (b)

15. (b)

16. (b), (c), (d)

17. (b), (d)

18. (a), (b), (c)

19. (b), (d)

20. (d)

Multiple

Practice 1. (d)

Choice

-II

Test 2. (a)

3. (c)

4. (a)

5. (a)

DEFINITE INTEGRATION § 18.1 Definite Integrals

Let f be a function of x defined in the interval [a, F' (x) = f (x), for all x in the domain of f, then

J

b], and let F be another function, such that

f (X ) dX = [F (X)]a

= F(b)-F(a) {Newton-Le ibnitz formula} is called the definite integral of the function f{x) over the interval [a, b] a and b are called the limits of integration, a being the lower limit and b the upper limit. Note : In definite integrals constant of integration is never present. 1. Properties of Definite Integrals :

•b Prop. I: J

a

f{x) dx =

Prop. II:

j

rb

f(t)dt.

a

f(x) d x = - |

J

f(x) dx

a

b

,b

J f(x) dx

Prop. Ill:

a

c

, t>

a

c

f

= J f(x) dx+j

f(x)dx b].

where c, is a point inside or outside the interval [a, f f(x) dx =

Prop. IV:

J

Prop. V:

J

0

-a

f(x)dx

f(a-x)dx

f

0

f(x)dx

= 2 J

0

or

0

According as f(x) is ev en or odd func tion of x. •a

J ' ' f(x) dx = j 2 J0

Prop . VI:

0

b

f

Prop . VII:

J

I

rb

rb+nT

a

(• b

f(x) dx = J f(x) dx, a+ nT a If f (a + x) = f (x), then

J

Prop. IX :

f"

I

J

C na

Prop. X : J f{x) dx=

( n - m) J

o

a

where f (x) is periodic with period Tan d n e I.

ra

f(x) dx = n I

o

f(x) dx.

/• a

f{x) dxlf f (x) is a periodic function with period a, i.e.,

f(a + x) = 1 (x)

0

ma

Prop. XI :

=

if f( 2a - x) = - f(x )

f (x) dx = J f(a + b - x) dx

a

Pr op . VIII:

if

'<*>

0,

If f is continuous on [a, fa], then the integral function

r*

g defined by

g(x) = J f{t) dt, xe [a, b] is diff ere ntia ble in [a, b) an d cf (x) = f(x) for all x e [a, b].

142


If f(x) <

Prop. XII :

,i>

,b

(x) for x e [a, b], then J f (x) dx < J

0 (x) dx

a

Prop. XIII:

J

f{x)dx

I dx

I f (x)

Prop. XIV : If m is the least value and (estimation of an integral) then

M is the greatest value of the function fb

m(b-a)<

)

a

f(x) dx <

f (x) on the interval [a, b].

M(b-a)

Prop. XV : (i) If the function f (x) increases and has a concave graph in the interval

(b-a)f(a)

< \

a

(ii) If the function f (x) incr ea se s a nd h as a conv ex graph in the interval (b-a)

f(a) + f(b)

[a, b], then

f(x)dx<(b-a) f{a)+0 f{b);

b

'

b

< J f(x) dx < (b-a)

[a, b] then

f(b).

Prop. XVI : If f(x) and g(x) ar e integrable on the interval (a, b), th e Schwarz-Bunyakovsky takes place : rb f (x) dx J f(x) g(x) dx j S 2 (x) dx

inequality

a

1

[a, b] and ther e exists a poi nt c e

Prop. XVII : If a function fis integrable and non-negative on continuity of ffor which f(c) > 0,

[a, b] of

b

r J f(x) dx > 0

then

(a
a

Prop. XVIII: Leibniz's Rule :

If f is continuous on

[a, b] an d u (x) and v(x) ar e diffe rentiab le func tions of x whos e v al ues lie in [a,

b]

then

d_ • M*)

dx Prop. XIX : If /is continuous on

f(c) = —

1

(b-a)

I f(x) dx

dv

du

d x f(t)dt = f{v(x)} — - f{u(x)} dx [a, b], then there exists a number c in [a, b] at which

u

(JO

is called the mean value of the function

f(x) on the interval [a, b],

a

Prop. XX : Given an integral b jr f (x) dx

a

where the function f(x) is continuous on the interval [a, b]. Introduce a new variable x = (> (t) If (i) 0 (a) = a, <> | (P) = b, (ii) (f) and <|>'(t) are continuous on [a, (3], (iii) /[()) (0] is defined and is continuous rb on [a, p], then rp J f (x) dx = J (f)] <|)' (f) dt.

t using the formula

r

Prop. XXI : If f(x) > 0 on the interval [a, b), then J f(x) > 0 Prop. XXII : Suppose that

f(x, a) and fa' (x, a) ar e continuo us functions when

c < a < d and a < x <

then /'(a)

= J

a

f'(x,a)dx,

where/'(a)

is the derivative of I (a) w.r.t. a an d f' (x, a) is the derivative of f(x, a) w.r.t. a, keeping x constant.

b,

Definite Integration

143

r*

Prop. XXIII : If f is continuous on [a, b], then J f(t) dt

for x e [a, b] is differentiable on [a,b].

Prop. XXIV : Improper integrals :

(i) f

f{x)dx

a

fc>->

= Lim

f(x)dx.

j

+°°

a

(ii) f f(x) dx -oo

= Lim J f(x)dx a —> - oo a »+C O -C .+ 00 (iii) J f (x) dx = j f(x) dx + J — oo

— oo

C

f(x) dx

18.2. Definite Integral as the Limit of a Sum

Let f(x) be a continuous function defined on the closed interval [a, b]. Then eb J f(x) dx

(n 1) " = Lim h ]T h-> 0

n oo nh= b- a

f(a+rh)

r= 0

18.3. Some Important Results to Remember

0) (ii) w

(iii)

gt2

=

r= 1

"(" +1) g"+ 1> 6 2

£/ = ^ ( n + l ) r= 1 4

(iv) In G.P., sum of n terms, Sn = —^—— =

, r> 1

a (1- f ) , r< 1 (1-r)

(v) sin a + sin (a + (5) + sin (a + 2p) + .... + sin (a + n- 1 P) sin n p/2 s j n i" 1st angle + last angle sin p/2 (vi) cos a + cos (a + P) + cos (a + 2P) + .. .. + cos (a + n - 1 P) sin n p/2 f 1st angle + last angle C0S ~ sin p/ 2 1

1

22

1

32

42

1

1

+

32

1

(viii) 1

1

+

22

1

+

52 " 62

7t_ 12

1

1

j?

62

6

5

2

(ix)

1

1 32

+

1 W

j

(xi) cos 0

42

1 + .... 52 1

+

62

Tt

- + ...

e / e + e~

/8

JI2 24

e'° - e" * 2/


144

18.4. Summation of Series by Integration

Let f (x) be a continuous function defined on the closed interval [a, b], then n-1 1 ( r \ r1 Lim Z f- (x) dx = f Rule : Here we proceed as follows :

1 ( r (i) Ex pre ss the given ser ies in the form Z — f — (ii) Th en th e limit is its su m whe n n — > <=o lim I - f oo n

n

n

1

Rep lac e - by x and - by dx, an d lim Z by th e sign of /?->oo n n ("0

Th e lower and up pe r limits of integrati on will be the va lu es of - for th e first an d last term (or th e limit of these values) respectively.

18.5. Gamma Function

e~ x x " - 1 dx is called the second Eulerian integral

The definite integral f

and is denoted by the

symbol rn [read as Gamma n]. Properties of Gamma function

r(n+ 1) =

nVn;

n

= 1 ; T( |) = rrc

= n I Provided n is a positive integer.

F(n+1)

.n/2

si n m x cos" x dx

we have J

' m+M + 2 "i 1[VJ 2 r 2nri'j

, ,

where m>- 1 and n>- 1 (This formula is applicable only when the limit is 0 to

n/2).

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate.

1. If f

I ' J -5 - 4 =

b then fJ

o r + 1

/,_i

~Ij - V " - is equal to H /-/; - I

, , -b (a) ae

. -b (b ) — ae

(c)-be'°

(d) ae

The 2.

value sin I n + C (a) n (c) 3 K

I ,

of

the

f 3. J

o

[VTt an *]

=

([.] denot es the greatest integer funct ion) (a) 5 n / 6 int egr al

,

n

-if 2 (c) — - tan

(b) y

- tan"

1

(2/V3)

V.,,, (d) Non e of thes e

x

T

sin x / 2

^

(ne/V)iS


4. The value of J "

(a)0 (c) - n / 2

7

'-^-

(b) 1

(d) n/2

is equal to

tion

145

Indefinite

(•1 5. The value of I [x [1 + sin tu ] + 1] dx i -1 ([.] denotes the greatest integer) (a) 2 (b) 0 (c) 1 (d) None of these C«JC - rt/4 6. J I sin x + cos x I dx (n e N) is -It/4 (a) 0 (b) 2n (c) 2 V2 n (d) None of these frlt/2 „;„ „ dx then I lies in the interval Let / = J 'o x (a) (0 ,1 ) (b)[ 0, 1] (c) [0, 3/2] (d) [1, 2) 8. For any t & R a n d/ b e a continu ous function, pl + cost xf(x(2-x))dx Let/, = J 2 sin i 2 , 1 + cos1 I f{x (2 - JC))dx then — is and/ 2 = J 2 sin / '2 (a )0 (b) 1 (c)2 (d)3 9. The value of the integral » nit +1 (I cos x I + I sin x I) dx is o (b) In + sin t + cos t (a) n (d) 4 n + sin t - cos t + 1 (c) cos t log sin x dx = k, then the value of o .7C/4 J,tc/4 log (1 + tan x) dx is 0

10. If |

(b )f

(a)-4

, ,

k ~~ 8 K/4

x

(d)|

(b)2

(a) 0 (c)e 2,1

12. The value of J 'o (a) 7t 2 TI

sin

'-r'1

(d) None of these of

value (x2 - 2) x J V ( x 2 - 1)

the

integral

-dx is

(a) 0 (c) 4/3

(b) 2/3 (d) Non e of these

15. If /, = J 3 / ( cos 2 x) dx and I 2 = | /( cos

2

x) dx

then (b)/,=2/

(a ) I x = h

2

(d) None of these (c)A = 51 2 p50n 16. If / = I V(1 - c o s 2 x ) dx, then the value of o / is (a) 50 V2~ (b) 100 <2 (c) 25 ^2 (d) Non e of these 17. Let fix) be an odd functi on in the interval T T T. Then with a period 2 ' 2

Fix) = \X fit)

dt is

a

(a) periodic with period T (b) non periodic (c) periodic with period 2T (d) periodic with period aT 10 f i f io 18. If I fix) dx —5, then 2 I / ( i t - l +x ) < i c o *= 1 o is (a) 50 (b) 10 (c)5 (d) None of these ,2k 19. The value of the integral J [2 sin x] dx

e

(d) None of these dx is +\ (b) 0 (d) JC/2

is ([.] denotes the greatest integer function) (a) 7t (b) 2ti (c) 37t (d) 47t 20. Let / (x) =

s,nx

4 1 3 .J

(c)l

14. The

2

x dx . f<- e .. sec xdx 11. J j, is equal to e 1— 1 - n/4

(c)

(a) 1/2

x d (x - [x] ) is eq ual to

mini mum

lxl ,l- lxl ,j 4

V x e R, then the value of j fix) dx is equal -l to ">V2

146


c)

(d) none of these

32 2

L

21.

1

cosec

101

x — — 1 dx = k

value of k is 8)1

a)

dx, a > 0 is

~ and J 2 -o

-fit

(b)f 2a

, 0 -In c) 2 — a 23.

r' 2 f j ex ( x - a)
equal to (a) a k

(b) bk

(c) nak

(d) nbk

24.

7^

30. Value of f

(b) a < 0

" _2" dx is equal to e

(b) e 2 (d)

c) 0

(b) greater than 2 (c) lies between 3 and 4 (d) None of these n, integer 31. Suppose for every .4 ,n + 1 I f(x)dx = n the value of J / (x) dx is

X

2

a) log e 2

? is +x )

(a) less than 1

c) 0 < a < 1 (d) a = 0 f fx] denotes the greatest integer less than or Jf" o

n

(a) 16


(c) 19 25.

f J / (t) dt = x + J tf (t) dt then the value of 0

x

/(I) is a) 1/2

311/4 x 26. The value of Jr 77T77™ • k/4 1 + sin x

(c) 71

(b)

x

sin

1

-ITdt + j 0

0

(d) -1 /2

(a) (-12 — 1) 7i

32. The value of 2 • sin

(b) 0

c) 1

is

f1 x dx —5 — J o (x + 16) lies in the interval [a, b]. Then smallest such interval is 1 (b) [0, 1] (a) 17 J_ (d) None of these (c) 27

2 V(1

a) 1< a < 2

f11 then J f(x) cos kxdx - it

29. The value of the defin ite integral

2

fj

the

(d) 1/101

poo

22.

then

(b) 1/2

c) 0

bjsin (/.r)

+ 2 l j=

'

s ec ua t0

l

(-12 + 1) 7t

'

cos

]

-17 dt is


(a) 7C/2 (c) 71/4 33. If / = J 1 cos

2 cot

1

dx then

(d) None of these

27. L e t / b e a positive function. If

tk / , = J1 \-k

xf{x(\-x)}dx,I2

= J* \—k

f{x(\-x)}dx

where 2k - 1 > 0, then I x : / 2 is equal to (a) 2 : 1 (b) k : 1 (c) :12 28. Let/(x) =

(d) 1 : 1 1 " — a 0 + X a, cos (ix) 2 j=l

(a ) / > 2 (c) 0 < / <

(b)/ = - 2 1

(d) None of these

f2 2 34. Th e valu e of J [x - 1 ] dx, where [x] denotes the greatest integer function, is given by (a)3 -V3T -V2~ (b) 2 (c) 1 (d) Non e of these

„_ _ 35. Let

{" ( bt cos At -a sin At I 5 t 0[ then a and b are given by

dt =

a sin 4x

Integration

Indefinite

147

(a) a = 1/ 4, /? = 1 (c) a = — 1, 6 = 4 36. If

(b) a = 2,b = 2 (d) a = 2, 6 = 4

/(x) = c o s x - [

f"(x) +f{x) (a) - cos A

(a)-/,=/

dt,

( x- 0 / ( 0 o

then

A

dt (d)-J

0

(A

- 0/ (0

(d)/, < /

r" —

dx = 6 then

(1

2

>/3

< /3

6 - A )"

b= 12, n e /?

(a)«=4,

(b)0

(c) J" ( A - 0/ (0 0

2

2

< /3

rh 43 . I f j « A " +

equals

(b) /,>/

2>/3

(c) /, = /

(b) a = 2, b — 14, n e /?

dt

(c) a = - 4, 6 = 20, ne

37. Let /( A ) = max { x + I A I , A - [A] }, wher e [ A] denotes the greatest integer
R

(d) a = 2, 6 = 8, n e R 44. If/( A ) and g{x) are continuous functions, then

/(A) dx is equal to

J

(b) 2 (d) No ne of these .2 c i•n 2 v J dx, where 1

(a)3 (c) 1

38. The

value

of

+

ffa 1A/[

[ A]

2

41.

(c) ^ ( e - 3)

(d) None of these

o


< 4 46. The value )

f

J

«(

0

x sin • 2n A of J 2tc o sin .v+ co s v(b)-*

4

3

I a,, co s' 1=0

'

A

sin x dx

(hi a

]

and a 2

(c) a and a (d) a, and a } 42. Let f':R->R such that f{x + 2y)=f(x) +f{2y) + 4xy V A, y e R and/'(0) = 0. If /, = J f{x) dx, 12 = J fix)

1/2

dx

then

dx and

.v

dx is

2

' 2

(d) 27c2 (-1

I 2=]

e

1

2

co s

A

1

dx,

e- x'l J dx o o let / be the greatest integral among /,, 1 2, / 3 , / 4 then (a)/ = Z, (b )/ = / 2 (c ) / = /

e

3

}

fix)

1 + 2 co s A , dx is (2 +c os A)

™ - 2

h =

depends on

/3= f

(d) none of thes e

45. The value of J

(a)

( ~ A ) ]

(b) a non-ze ro consta nt

47. Consi der the integrals fi _ 2 /, = j e x cos xdx,

(b) -U' -2)

0

(a) depe nd on \

+ G

(c) k~

(a) \ ( e - l )

(aV) 0 and a 2

[G(A)

K/1

dx is

Tlie value of

dx is

n

A

(c) zero

-1 (a) 4 (b) 9/2 (c) 2 (d) Non e of these 40. Let/( A ) be a function satisfying/'( A) =/( .V wi th /( 0) = 1 and g be the function satisfying /(A) 4- g (A) = A" the va lue of the in tegral (A)

I

5

max {2 - A, 2. 1 + x } dx is

The value of I

J ' f( x) g

p*)-/(-*)]

/

A

denotes the greatest integer < x, r is (a) T (b) 0 (c) 4 -f sin 4 (d) Non e of these 39.

7

J, ,

48 . If /( x) = f •2

1

dx and /, = j

(d )/ = / (sin"'VT) V7

4

2

rfr then the value of

(1 - A ') If " (x)} - 2 / ' (x) at x = ^

is

(a) 2 - 7t

(b) 3 + Ji

t,c)4-n

(d) None of these

148

Obj ective Mathematics :

49. If fo r x * 0 af(x) + bf

— - 5 whe re a x

b

(c) 1001 (e- 1)

then f xf(x) dx = l

. . (a) (c)

b - 9a

(b)

9 (a2 - b2)

b-9a

b (a2 — b2)

(d) None of these

51. L et /( x ) be a continuou s function such that / ( a - x) + / ( x ) = 0 for all x e [0,

dx

J(

o

l + /

w

fca/12

(b )

(c) -

foa/24

(d) None of these

ka/24

fn Th e value of J [ Vx ] dx, ([.] denot es the

(a)

(b) 1 (d)0

Then [

(a ) -

+ ~ f " (x) } = 3 !"

greatest integer function), n e /Vis

[g(x)-g(-x)}dx

+/( -* )}

( a )n (c)-l

Y\f'(x)

2

56.

50. La eK/2 t / and g be two continuous functions. Then

-kn. is equal to

) j j ( x dx-\xf(x)~

b-9a

6 (a 2 - fc2)

J

(d) No ne of the se

55. I f / " ' (x) = k in [0, a] then

n(n+

1)

(b)7« (n-l)(4 n+l) (c) £ n « ( « + 1) (n + 2) (d)

a],

is equal to

6

. l/n

57.

(a) a

(b)

(c )f(a)

(d)\f(a)

52. The equation ,n/4 -iz/4

a I sin x I + b sin x + c d t = 0, 1 + cos X

where a, b, c are constants, gives a relation bet we en (a ) a, and b c (b) a an d c (c ) a an d b (d) b and c I6n

Lim n —>

'' ft

o

I sin x

58. If [x] stands for the greatest integ er funct ion, .10 [x 2] dx is the value of J 2 4 [x - 28 x + 1 96] + [x 2]

54.

fiooo The value of J ex

w

59.

cos (cos

(b) 1000

'

x)

dx is equal to

x)

(d) No ne of thes e

60. If na = 1 always and II {l+(ra)

1 1

( b )0

(c) p - a

dx, is

- 1 ) (e

sin (sin~

a

(a) 1

2

n —» °o then the value of

}1/ris 2

(b) e

(a) 1

([.] denotes the greatest integer function) (a) 1000 e

(b) 1 (d) none of these

(a) 0 (c)3

I dx is


(a) 17/2 (c) 21/2

(b)
(a )« (c) 1

cos cos

53. The value of J

e quals

(c) e

n2/24

(d)e

K /8 -n /\ 2

MULTIPLE CHOICE -II Each question in this part has one or more than one correct answer (s). For each quesion, write the letters a, b, c, d corresponding

to the correct answer (s).

r100 -i 61. Th e val ue of I [tan x] dx is o (when [.] denotes the greatest integer function)

(a) 100 (c) 100-tan 1

1 (b) 100-tan 1 (d) None of these

Integration

Indefinite

149

ra + n/2 70. The value of J (sin x + cos x) dx is

62. If x satisfi es the equation xif

r'

dt

—x

'o tL + 2t cos a + 1

t3 sin 2t , ) dt -3 r + 1

nY (a) independent of a (b) a | —

3

- 2 = 0

(c) 371/8

(0 < a < 7i), then the value of x is (a) sin a a

(a) - 71/2 (c) - n / 3

(d) None of these

f [xl IXI 63. The value of the integral J — dx,a
is

X

{a) b-a (c) b + a

J

(b) 1 (d)-l

(a) less than e (c) less than 1

o

ex dx is

,1

2X dx, I2 = J

(2

3

2

2X dx, h = J 2A dx

(b) I 2 > /,

(b) more than 10"

-7

(d) Non e of these f4 dx 68. If / = JJ 3, , , then 3 V(log x) (a) / < 1 (b) / > 1 (c) / > 0-92 (d) / < 0-92 A dt r1 f' dt 69. If /, = — r2 and I 2 - J\ r f o r x > 0 , then •v l+t i 1 +1 (a) /, =1 2

(b) /, > I 2 (d) /, = cot"

2 >/,

[x2 - x + 1 ] dx, (where [x] 0 denotes the greatest integer function) is given by , ,5-VS . 6 V5~ (a) — =2 — (b) • 2 The value of I

7- VJ

1

(d)

8-V3T

74. If / = f V(1 +x 3 ) dx then 0

(a)

(c) less than 10'

(c)/

(d) Non e of these

(b) / V 5 / 2 (d) Non e of these

^ l* + 2*+ ... +n* 75. The value of Lim * +1 n —> =0 n

(d)/ 4 >/3 f 19 sin x 67. The absol ute value of j ^ dx is 10 1 + x 7

(b) 1

(c) / < V7"/2

(c) h > U

(a) less than 10

x]) dx

(a )0

(a) / < 1

t2 1 and / 4 = J 2" dx then (a) /, > h

min (x-[x],-x-[-

(c) 2

(c)

(b) greate r than e (d) greater than 1

. 1 2

66. If / s = J

J

2

72. The value of J

is ([.] denotes the greatest integer function)

64. If/( JC ) satisfies the requirements of Rolle's theorem in [I, 2] and/'(x) is continuous in 7 [ 1 , 2 ] , then f {x) dx is equal; to

65. The value of the integra l

(b) — 7t (d) 0

-2

(b) a - b (d) Non e of these

(a)0 (c) 3

2

71. The value a in the interval [ - n, 0] staisf ying ,2a sin a + J cos 2xdx = 0 is

(b) — 2

(c)4

(d) 17ia

x - 7C/4

(c)

1 k+2 2

k+3

( b )

1 *+l

(d) 0 I In x I dx is

76. The valu e of the integral I \/e

(a) 1 - \/e -1

(b) 3 ( 1 -

We)

(d) None of these

(c)e 1 77. f(x) is continuous periodic

func tion with ,a + r period T, then the integral /= J /( x) dx is a (a) equal to 2a (b) equal to 3a (c) independent of a (d) None of these

150

78.


12/1 ( r Lim Z lo g J 1 + - Iequals n —» oo n r = n+ 1 n 1 27 4e

(a) log

(b) log

(c) log

27

(d) None of these

84. The point s of extrem um of f 0

are (b) JC = 1 (a) x = —2 (c )x = 0 (d) JC = - 1 85. The values of a which satisf y

79. The maximum and minimum value of the

f™ I sin x dx = sin 2 a ( a e [0, n/2

dx

f integral I o

• are (1 + sin x)

80. Given that n is odd and m is even integer.

(a) (c)

2

2

(b)

2

2

(d) None of these

n —m 2 2 m +, n

2

n —m

2

87.

l / 4

dx is equal to 1

15

(a) 10

(b) 11

(c) 9

(d) 11/2

\

(a) one

(b) infini te

(c) two

(d) zero

I f f ' { x ) — f (x) + J* /(JC) dx and given

then/(x) =

+c

(a)

X

,,,

e 2—e 2e

+

1 -e

x

1 +e

^ X

y(ai, - x + Vx /— dx is equal to

(c)

e 2—e

(d)

2ex 3—e

(b) a (c) — a

dx is

(d) 4tc

x

\5/4

(d) None of these 83.

^ KX ^

89. The number of positive continuous functions f(x) def ine d in [0, 1] for which

90.

/ 5/4

+c

•W 4

cos

(b) 7t/ 2

(c) 3/4 7i

/

J a/4-

(d) None of these

88. The value of J (a) 2n

x)2

(b) i (log 2

1A x V

+C

1 -

,

15

dx , the n/( x) + /

If/(x) = J y ^

(c)-(xlogA")

>5/4

n

(c)

[jc]) dx equals [Here [.] denotes

(a) (log x)2

\ h fix) dx has the

b- a a (a) maximum val ue/ (6) (b) minimum value/(a) (c) maximum value bf(b) (d) minimum value f(a) b - a

(a)

(d ) 1 I T I / 6

f

n —m

82. f(/-x)

(b) 37t /2

(c) 7 T I /6

greatest integer function]

81. Le t/ ( a) > 0, and le t/ (x ) be a non decrea sing continuous function in [a, b], Then 1

(a) 71/2 ,10 86. J Sgn (x -

The value of J cos mx sin nx dx is

2m

2TI ])

are equal to

(b) k (d) 3)1/4

(a) 7t/4 (c) 71/2

-—^LLi 2 + e'

(d) None of these

/(0)

= 1

151

Integration

Indefinite


M.M. : 20

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] J.199 JC/2

a

1. The value of the int egra l J

2. If /„ = I

dx , is

ta n" x dx , then

o i (a)J„+J„_ 2 n-1 _ 1 1 is a , < 2 IUn < (c) 71+1 71-1 natural number (d)(/„ + 2 + / „) (n + l ) = 1 • 2n

3. Th e va lu e of J

[2 sin x] dx , (when [ .

n represents the greatest integer function) is 5k (b) - 71 (a)r571

(d) - 2k

(c) 3 4.

Lim n->°°k

n =1

r

l 1 a- — ,a -— n a+k a a+1 n

(a) 1 (b) 2 (c) 3 (d) Non e of th es e 5. Le t f(x) = min ({.*}, (- .r l) Vx e R, where {.r} denotes t he frac tional pa rt of x, then p.100 J.UU

f (x) dx is equal to

100

(a) 50 (c) 200

6. The valu e of J

0 loga (b) 2 log (a + 1) (d) None of these

(a)log a (c) 3 log a P 7t/4

fr l x — 11


(a) 50-12

(b) 100^/2"

(c) 150V2"

(d) 200V2"

7. The val ue of ,71/4

r

J

0

(tan" x + tan"

[x]),

1 n+1

(d) None of these

= 4

-1

x)d(x-

(b)-

4

8. If | f(x)dx

-2

denotes the greatest integer

(where [ . function) is 1 (a) n-1 2 (c) 71-1

and ff 4 (3 -f(x))dx '22

f(x)dx

the value of j (a) 2

= 7

is

(b) - 3 (d) None of these

(c) - 5 9. If 7, = J\X e" e' z o then (a) / j = e l2 x

(b ) / j = e

e

(b) 7j = 2

I2

I2

(d) Non e of th es e .4 r

ar g 2 L greatest J

where [.] denotes the function, is (a) 3 \2~ (b) 6 V3" (c) V6

Max. Marks 1. First attempt 2. Second attempt must be 100%

z -z2 /4 e~ '*dz

dz an d U = \*J o

10. If z - x + 37th en value of

Record Your Score

3. Third atte mpt

2x) dx is

V(T + cos -it/2

(d) no ne

_ • "

7 dx, J integer

2 + 1

152

Ob j ective Mathemat ics

Answers

Multiple

Choice-!

1. (b)

2. (a)

3.(c)

7. (d) 13. (b)

8. (b)

9. (d) 15. (d)

14. (a) 20. (b)

19. (a) 25. (a)

26. (a)

31. (c)

32. (c)

37. (d) 43. (b)

38. (b) 44. (c)

21. (c) 27. (c) 33. (b)

49. (d)

50. (d)

39. (b) 45. (c) 51. (b)

55. (c)

56. (b)

57. (b)

Multiple

40. (d) 46. (c) 52. (b) 58. (c)

5. (a)

6. (c)

11. (a)

12. (a)

17. (a)

18. (c)

23. (c)

24. (b)

29. (a)

30. (a)

35. (a)

36. (a)

41. (d) 47. (d)

42. (c) 48. (d)

53. (c)

54. (b)

59. (c)

60. (c)

Choice-ll

61. (c)

62. (a), (b)

67. (a), (c)

68. (a), (c)

73. (c)

74. (a), (c)

79. (a), (c)

80. (b)

84. (a,) (b), (c), (d) 88. (d)

Practice

4. (d) 10. (c) 16. (b) 22. (d) 28. (c) 34. (a)

89. (d)

63. (a), (b), (c) 64. (a) 69. (a), (d) 70. (a), 75. (b) 76. (b) 82. (a) 81. (a), (b) 85. (a), (b), (c),(d) 90. (b)

65. (a), (d)

66. (a), (d)

71. (b), (c), (d) 77. (c)

72. (b)

78. (a)

83. (a) 86. (b)

87. (b)

4. (a)

5. (a)

Test

l.(d)

2. (a), (b), (c), (d)

6.(d)

7. (a)

3. (a) 8. (c)

10. (d) 9. (d)

AREAS § 19.1 The ar ea boun ded by the continuou s curve

y = f{x), Fig. 1 th e axis of

x and the ordinates

x = a a nd x = b (where b > a) is given by ,b

A = J f(x) dx a

§19.2

= J y dx a

The area boun ded by the continuous curve y = c and y = d (where d> c) is given by f tf cd A = J g(y) dy = J xdy

x = g(y), (Fig. 2.) the axis of y and the ab sc is sa e

Y'

Fig. 1. § 19.3

Fig. 2.

The area boun ded by the straight lines x = a, x = b ( a < b ) and the curves y = f (Fig. 3) provided f(x)
A =J

[g(x)-f(x)]

(x) and y = g(x),

dx kY

ZuU^fa S '

ra II

x' •

x=b

0

Fig. 3.

Note : If some part of curves lies below the x-axis, then its area is negative but area cannot be negative. Therefore we take its modulus. (i) If the cur ve cr os se s the x-axis in two points (i.e., c, d), Fig. 4 then the area between the curve y = f (x) on the x-axis and the ordinates x = a and x = b is given by

154

Ob j ective Mathema tics rb

c

A =

r J f(x) dx a

J f(x) dx c

+

+

d

f(x) dx

Fig. 4.

(ii) If the curve c ro ss the x -axis at c, then the area bounded by the curve y = f(x) and the ordinates x = a and x= b{b>a) is given by (Fig. 5)

A =

j f(x)dx

+

j

f(x)dx

MULTIPLE CHOICE-I Fig. 5

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate : 1. The area between the curve y = 2x — x , the x-axis and the ordinates of two minima of the curve is 7 (a) 120 13 / * 11 (d) 120 720 The area bounded by the x-axis, the curve y = / ( x ) and the line s x = 1 and x = b is equal to pl(b2+ 1) - \' 2) for all b > 1, then/(x) is (a)V(x-l) (b) V(x+1) (C) Vcx 2+\)

(d)

* V(1 +x 2 )

area of the region bounded 2 x I + I y I = 1 is 1 —y =1 x1 and (b) 2/3 (a) 1/3 (d) 1 (c) 4/3

3. The

by

4. Area bounded by the curve y = (x - 1) (x - 2) (x - 3) and x-axis lying between the ordinates

x = 90 and x = 3 is equal to1 1(in square units) ^ ™ (a)(b) — 13

(c)" T

(A\ ( d

15 )

T

5. The area o f the figure bounded by the cur ves y = I x - 1 I and y - 3 - I x l is (a )2 (c)4

(b) 3 (d)l

6. Area bounded by the curve _y = x sin x and x-axis between x = 0 and x = 2k is (a) 2k (b) 371 (c) 4TC (d) 5TT 7. Let f(x) = min [x + 1, V(T -x)], Then area bounded b y / ( x ) and x -axis is

Areas

155

15. The area of the fig ure bound ed

by two branches of the curve (y - x)" = x and the straight line x = 1 is

">6

8. The area boun ded by the graph y = I [x — 3] I,

the x-axis and the lines x = — 2 and x = 3 is ([.] denotes the greatest integer function) (a) 7 sq. unit s (c) 21 sq. units

(b) 15 sq. unit s (d) 28. sq. units

9. The value of c for which the area of the figure bounded by the curve y = 8x" - x , the straight lines x = 1 and x = c and the x-axis is equal to 16/3 is (a) 2

(b)^8

(c)3

(d) -1

10. Area

x 64 x-axis is v=

boun de d +

by

the

curves

, y = x — 1 and x = 0

above

The slope of the tangent to a curve y =/(x) at (x,/(x)) is 2x + 1. If the curve passes through

the point (1, 2), then the area of the region bo unde d by the cur ve, the x-axis and the lin e x = 1 is (a) 5/6 (b) 6/5 (c) 1/6 (d) 6 12. The area of the region boun ded by the curve 4

2

5

a y - (2a - x) X is whose radius is a, is (a) 4 : 5 (c) 2 : 3 13.

to that of the given by the ratio (b) 5 : 8 (d) 3 : 2

(b) 4/5 sq. unit s (d) 3 sq. units

16. If the area bounded by the x-axis, the curve

y=/(x) and the lines independent of d,Vd>c then/is (a) (b) (c) (d)

x = c and x = d is ( c is a constant),

the identity function the zero function a non zero constant function None of these

17. The area bounded by the curve and y = - I x I + 1 is :

•vry

([.] denotes the greatest integer function) (a )2 (b) 3 (c) 4 (d) Non e of these 11

(a) 1/3 sq. unit s (c) 5/4 sq. units

circl e

I X I

The area bounded by y = x e and lines 1x 1= 1, v = 0 is (a )4 (b) 6 (c) 1 (d)2 14.The area of the figur e bounde d by fix) = sin x, g (x) = cos x in the first quadrant is (a)2(V2-l) (b)<3 + [ (c) 2 (VT— 1) (d) No ne of thes e

y — Ix I— 1

(a) l" (c) 2 ^ 2

(b) 2 (d) 4 a/2" 2 18. The area bounded by y = x , y = [x + 1], x < 1 and the y-axis is ([.] denotes the greatest integer function) (a) 1/3 (b) 2/3 (c) 1 (d) 7/3 19. Let f(x) be a cont inuou s fu ncti on such that the area bounded by th e curve y = / ( x ) , the x-axis and the two ordinates x = 0 and x = a is 2 a a . K , ,, . — + — sin a + — cos a, then/(7r/2) is n~

(a)(c) 20The .

( b )

K+ 1

y

TC +

4

(d) None of these area

boun ded

by

the

curve

= x — 2x + x 2 + 3, the axis of abscissas and two ordinates corresponding to the points of minimum of the function y (x) is (a) 10/3 (b) 27/ 10 (c) 21/1 0 (d) Non e of these 21. The are bounded by the curv es and y = In x, y - In I x I, y = I Injc I y = I In i x I I is (a) 5 sq. units (b) 2 sq. unit s (c) 4 sq. units (d) Non e of these 22. The area bound ed by the curve •y = x + 2x + 1 and tan gent at (1, 4) and y-axis is


156

(a) — sq. units

... 1 (b) — sq. units

(c) 2 sq. unit s

(d) No ne of these

(b)

23, The triangle formed by the tangent to the curve /(x)=x + bx - b at the point (1, 1) and the co-ordinate axes, lies in the first quadrant. If its area is 2, then the value of is (a) - 3

(b) - 2

(c) - 1

(d) 0

24 . Th e area bounde d by y = 2 -

'b'

(c)

2-In

3

2

4-3

In 3

(d) None of these 25. If /(•*) = max \ sin x, co s x, — \ then the area of the region bounded by the curves y = f(x)> *-ax is, y-axis and x = 2n is (a)

3 I 2 - x I, y = -—r I xI

5n

12

(b)f ^

+, 3

sq. units

+ V2"j sq. units

IS

(a)

5 - 4 In 2

(c) (d)

sq. units

12

57t • -12+ V3~ sq. units 12'


M.M : 10

(A) There are 10 parts in this question. Each part 1. Area of the region bounded by the curve

y =ex,y -e given by (a) e - e~+12 (c) e + e - 2

%

and th e stra ight line (b )e -e

_ 1

x = 1 is

-2

one or more than one correct

Area bounded by the curve y =/(*), y = 0 and x = ± 3a is 9a/2. Then a= (a) 1/2 (c) 0 4. The area bounded

(b) - 1 / 2 (d) No ne of th es e by th e curves


2. The area bounded by the curve y =f(x), the x-axis and the ordinate x = 1 and x = 6 is (•b - 1) cos (36 + 4). Th e n f(x) is given by

(a) (x - 1) sin (3x + 4) (b) 3 (x - 1) sin (3x + 4) + cos (3x + 4) (c) cos (3x + 4) - 3 (x - 1) si n (3x + 4) (d) None of these

2

y = (1 - x 2 )" n e N, then

5. If An is the area bounded by and coordinates axes, (a)A

n=An_1

(b)A„
n

1 n_1

= 2An_1

Record Your Score


2

| x | + | y | > 1 an d x +y < 1 is (a) 2 sq. un it s (b) 7t sq. un it s (c) (n - 2)sq. un it s (d) (71 + 2) sq. units

(e)A„>A

: 3. L e t f ( x ) = \* [x : x > 0

answer(s). [10 x 2 = 20]

must be 100%

Areas

157

Answers Multiple

Choice

-I

1. (a)

2. (d)

3. (b)

4. (b)

5. (c)

6. (c)

7 .( c)

8. (b)

9. (d)

10. (c)

11. (a)

12. (b)

13. (d)

14. (a)

15. (b)

16. (b)

17. (b)

18. (b)

19. (a) 25. (d)

20. (b)

21. (c)

22. (b)

23. (a)

24. (c)

2. (c)

3. (a)

4. (c)

5. (b)

Practice 1. (c)

Test

DIFFERENTIAL EQUATIONS § 20.1. Variable Sepa rabl e :

If the differential Equation of the form

h (x) dx = f 2 (y) dy where U and f2 being functions of x and y only, then we say that the variables are separable in the differential equation. thus, integrating both sides of (1), we get its solution as J U {X) dx = | f

2

...(1)

(y) dy+ c

where c is an arbitrary constant. Method of Substitution : If the differential equation is not in the form of variable separable but after proper substitution the equation reduces in variable separable form in the new variable. § 20.2. Homogeneous Differential Equations :

A differential equation of the form _y _

dx

are homogeneous functions of Working Rule : To pr ocedure:

h K(x, yy) and f2 (x, y) ' y /J

x and y of the same degree, is called a

homogeneous equation.

get the solution o f a hom og en eo us differential equa tion, we follo w the follo wing

(i) Put 7y = vx so that

dx

1 y2 where fztx, y)

v+ x^f dx

= ^

•

(ii) The equation thus obtained will be of the form in which variables are separable. (iii) After integration replace vby y/xand get the gen era l solution. Equations Reducible to Homogenous Form : A differential equation of the form

dy _ dx

ax+ by+ c aix+biy+ci

a b If — = — = m ( sa y) ai bi dy m (ai x+ biy ) + c . then -f - = :— whe re m is any number. dx (ai x+ bi y+ ci) In such case the substitution Case(i)

aix+ biy = v,so that a i + b A ^ r = ^ r dx dx

transform the differential equation to the form

dv 1 srl 5 -

) J

dv = ai + bi dx

mv+ c V+ Ci fmv+ c s C1

which is a differential equation in variables separable form and it can easily be solved.

Different ial Equation s

If

Gase(ii):

then put

159

— * ~ ai bi x = X+ h, y = V+ k, so that

dY _ dy dX dx transform the differential equation to the form

dY _ aX+bY+(ah + bk+c) dX~ aiX+ £hV+ (aib+ b\k+ ci) Now, cho ose h and /csuch that

ah+bk+c then for these values of

=0

a ih + bi k+ ci = 0 h and k, the equ ation bec omes

dY dX

aX+bY

aiX+ Y This is a homogeneous equation which can be solved by putting Y = vX. ( x - h) an d / by ( y- k) we shall get the solution in srcinal variables xand y.

finally by replacing

X by

§ 20.3. Linear Differential Equation :

Py = Q, where Pand

A differential equation of the form

Q are functions of x alone or constants is

known as linear differential equation and its solution is given by y e l p d=*

\Q(e$

Pdx)dx+c

dx Note : If the differential equatio n c an be re pre sen ted in the form — + Px =

Q. wher e P an d Q are

functions of y alone or constants and its solution is given by

Q (e> Pdy) dy+ c

x e W = j

Equations Reducible to the Linear form : (i) Bernoullis' Equation : An equ at ion of th e form ^

alone or constants and

+ Py = Qy°, where

n is constant, other than 0 and 1, is called a

Dividing by y", we get

ynfx+pyn+1 = ° y - " + 1= y, sot hat

Now pu t

(- n + 1) y

= ^ , d x d x

we get

dv+ (1-n)Pv —

=

(1

-n)Q.

which is linear differential equation. (ii) If the equation of the form

^+P0(y)

Dividing by

(y), we get V (y) dx

lM >(y)

N o w p u t

or

= Q y (y),

Q are functions of xalone or constants.

where P and

=

^ go that '

dx[\\i (y) j ^ == k/,-. — dx

y (y)

dv dx

=

»where , y (y) dx ' 1

d

P and Q are functions of x

Bernoulli's equation.

k is constant,

160


dv + A-Pv = M3 dx

we get which is linear differential equation.

(i) d(xy)

=

xdy+ydx xdy - ydx

N

(iii) d ' y

f o \

(V)

d

x

n \

d X

(ix)

/dx

(vi)

2 2 _ 2x y dy - 2xy dx

2

x

d tan-1 y | _ xdy-

ydx

2

(xi) d\ In

=

j , , d

(viii)

4

x +y

(xiii)

(x)

2

ydx-xdy xy xdy- ydx

(xii) (xiv)

xy x ye* dx - e dy

(XV)

(xvi)

2

(xvii)

d

y

y2

I

2xydx-

y 2

2xydy x2

X

/

(vii)

(iv)

x2

X

\

ydx - xdy

(ii)

d I / c

)?dy

/

2

2 x y dx-2x

.tan - 1 *-

d[ln(xy

)] =

ydx - xdy =

J + f

^ ±

^ xdx + ydy

d ^ /n (x 2 + y 2 ) d

ydy

2

~

x +y

2

J_ 1 _ *dy + ydx ~* yj ~

x

2

_ xe^dy-

d

/

e^dx

y d (x^y") = x m " 1 y" ~ 1 (mydx + nxdy).

MULTIPLE CHOICE -I

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. If y - f ( x ) passing through ( 1, 2) s atisfies the differential equation y (1 + xy) dx - x dy = 0, then 2x

(a)/ (* ) = • 2~--x 2 (c)/(x) =

x- 1 4 -x2

(b)/(x) =

x+ 1

x2+l 4x (d)/(x) = 1 -2x

z

y.^lc

(x + Vc) whe re

is a positive parameter, is of

c

(b) <> | iy/x) = kx (d) 4> (y/x) = ky

4. Let a and b be respectively the degree and order of the differential equation of the family of circles touching the lines 2

2. The differential equation representing the

family of the curves

(a) x <> \ (y/x) = k (c)y^(y/x) =k

2

y —x = 0 and lying in the first and se cond quadrant then (a) a = 1 , 6 = 2 (b) a =1,6=1 (c) a = 2, b = 1 (d) a = 2, b = 2 5. The i ntegrating fact or of the differ ent ial

(a) order 1, degree 3

equation ^

(b) order 2, degree 2 (c) order 3, degree 3 (d) order 4, degree 4

by

(x log e x) + y = 2 log e x is given

(a) x

(b) e

(c) log,, x

(d) log e (log e x)

3. The solution of the different ial equation =

dx

l x

+

ii
s

Differe ntial Equations

161

6. A differential equation associated with the 5x lr primiti ve y = a + be + ce is (a)y

3

+ 2 y 2 -yi =0

(b)y

3

+ 2 y 2 -35y,=0

(c)4y

+ 5 y 2 - 20y, = 0

3

(d) None of these

dx — dt o 1 - cos tcosx

7. The functio n f( t) = -7"J J

• -ix/2 (a) y = 1 — 11 e • -ix/1 (b) 3y = 1 + 11 e - -ix/2 (c )3 y= 1 - 11 e~ (d) None of these 12. The particular solution of I n & j = 3 x + 4y, y(0 ) = 0, is 4

3x

>=7

satisfies the differential equation

(a) 3e 4y + 4e~ 3x (b) 3e -4e~

(a) ^+2 /( r)c otr = 0

(c) 4ex + 3e (d) None of these

(b) f

- 2 / « cot r = 0

t

2/(0 = 0 (d)

2/(0 = 0

8. A continuously differentiable function y = / ( x ) , * e (0, Jl) satis fying y' = 1 + y 2 , y(0) = 0 = y(7t) is (a) tan x (b) x (x - 7t) (c)(

x-*)

\

(l-/

)

(b) n

K

<

(a)x = y f f =0 2

/y dv xy —^2=>'; ' dx (d) None of these 11. The particular solution of the differential equation y' + 3xy = x which passes through (0,4) is

(a )y + 2x + ^ = 0 (b)y=

1+ J - J + J

14. A spherical raindrop evaporates at a rate

10. The form of the differential equation of all central conics is

(b)* +y

13. If the slope of tangent to the cur ve is maximum at x = 1 and curve has a minimum value 1 at x = 0, then the cu rve whi ch also satisfies the equation y'" = 4x - 3 is

(c)y= 1 +x + x 2 + x 3 (d) None of these

(d) not possible 9. The largest value of c such that there exists a differential function h(x) for -c
=7

propo rti ona l to its su rfac; e are a at any instant t. The differential equat on giving the rate of change of the radius of the raindrop is ( a ) ^ + 2r = 0 dr ,2 (b)~-3r =0 dt

(d) None of these 15. Through any point (x, y) of a curve which pas ses through the origin, lines are drawn parallel to the co-ordinate axes. Th e curve, given that it divides the rectangle formed by the two lines and the axes into two areas, one of which is twice the other, represents a family of

(a) circle (b) parabola (c) ellipse (d) hyperbola

162


MULTIPLE CHOICE -II Each question, in this part, has one or more than one correct answer (s). For each question write the letters a, b, c, d corresponding to the correct answer (s). 16. The

degree

of

-yVl

the diffe rential

equation

+x2 + Vl +y2) = A (x Vl +y2

satisfying (Vl

+x2)is

(a) 2

(b )3

(c) 4

(d) Non e of these

17. Solu tion

2y sin x

of

21. Solution of the different ial equation = 2 sin x cos x dx

2 n . . - y co s x, x = — , y = 1 is given by (a) y 2 = sin x (b) y = sin 2 x

of all

'Simpl e 2 71

—, >) 271 x = a cos (cof + <( -

dt A (c)^4-n2x dt

=0

(c)y = x tan

1

c-1

c+1

= (tan

-

y — 1) e

+c tan

J

+ c

(c) log y = 2jc - log c (d) None of these (*)r2 <±t and< t>( l) = 0

<> t W = (a){2(*-l)} (t>) {5

then

1/4

(x-2)}

1/5

1/3

(c){3(x-l)} (d) None of these 25. Solution o f different ial equation 2

2

(2x co sy + y cos JC) dx + (2y sin x - x siny) dy = 0 is

In

2

1 In

(d) None of these 20. The differ entia l

y

r \ » tan —' y -1 1i + c- etan y ' (c) x= (d) None of these

24. If

In

(b) y = x tan

(b) xe

is twice the square of the ordinate is given by (a) log y = 2x + log c (b) y = ce2x

.. i d X 2 r, (b) — j + n x = 0 dt d2x 1 ( d ) ^ + -Vx = 0 dt n

y - cos" — --cos |2 | ) then the equation of the curve is X [l I

-l

equation

23. Equat ion of the curve in whic h the su bnorm al

19. The slope of the tangent at (x, y) to a curve passin g th ro ug h ( 1 , j i / 4 ) is given by

(a)y = tan

(a) xy = sin x + c cos x (b) xy sec x - tan x + c (c) xy + sin x + c cos x = 0 (d) None of these

-1

Harmon ic Motions ' of given period — is

0

j + y (x sin x + cos x) = 1 is

/ \ tan )' =_ (1 - tan y) e (a) xe'

18. The diff erenti al equation

( A I - - R + / UC =

x cos x ^ ^

22, Solution of the differen tial 2 _] (1 }dx= (tan y-x)dy is

(c) y = 1 + cos x (d) None of these

,2 / ^d X

(a) (y'-\)(y + xy') = 2y' (b) (y' + 1) (y-xy')=y' = 2y' (c)(y'+l)(y-xy') (d) None of these

2

(a)jc cos. cos y + y sinj c = c (b) x cos y - y sin JC = c equation

= 1 is given by

of

the

curve

(c) JC2 cos 2 y + y 2 sin 2JC = c (d) None of these

163

Differential Equations

Practice Test M.M, : 10

Ti me : 15 Min

(A) There are 10 parts in this question. Each nart has one or more than one correct 1. Solutio n sin

of

th e

= co s

(1- *

diff ere ntia l cos

equa tion

y) is

x

(a) sec y = x - 1 - cex (b) sec y = x + 1 + ce (c) y = x + ex + c (d) None of these 2. The differential equation whose solution is 2

2

(d) None of these 4. Solution of th e diff ere ntia l equa tion dy x + y + 7 . dx ~ 2x + 2y + 3 1S (a) 6 (x +y) + 11 log (3x + 3y + 10) = 9x + c (b) 6 (x + y ) - 11 log (3x + 3y + 10) = 9x + c (c) 6 (x +y) - 11 log (x + 10/ 3) = 9x + c (d) None of these 5. The solution o f th e eq uat ion

2

(x - h) + (y - k) = a is (a is a con sta nt) 2 3

(a) {1 + (y') l = ay" (b) (1 + (y'ff

4 ^ - 3 y = sin 2xis dx

= a2 (y"f

(c) (1 + (y') 3] = a Cy")2 (d) None of these 3. Solutio n of th e equ ati on + — t a n y=

answer(s). [10 x 2 = 20]

ta n

(a )ye

3l

=- A e

^ (2 cos 2x + 3 sin 2 x) + c

i d

1 Ir (b) y = - — (2 cos 2x + 3 si n 2x) + ce

y sin y is

Id

(a) 2x = si ny (1 + 2cx 2) (b) 2x = si ny (1 + cx2) 2 (c) 2x + sin y (1 + cx )

(e) y = {- 1 /VI3 } cos (2x - t a n "

1

(3/2)) + ce

31

(d) y = [- 1/VlJ) sin (2x + tan"

1

(2/3)) + ce

31

Record Your Score


must be 100%

Answers Multiple

Choice

-I

1. (a)

2. (a)

3. (b)

4. (c)

5. (c)

Mb)

7. (a)

8. (a)

9. (c)

10. (c)

11. (b)

12. (c)

13. (b)

14. (c)

15. (b)

20. (c)

21. (a), (b)

Multiple

Choice-ll

16. (d)

17. (a)

18. (b)

19. (c)

22. (b), (c)

23. (a), (b), (c)

24. (c)

25. (a)

Practice 1. (b)

Test 2. (b)

3. (a, b)

4. (b, c)

5. (a), (b), (c), (d)

CO-ORDINATE GEOMETRY

21 STRAIGHT LINE § 21.1. Dis tan ce Formula

The distance between two points P (xi , yi ) and Q (x2, y2) is given by

PQ = I

(X2-X1)

2

+ (y2-yi)

2

Distance of (xi , yi) from srcin = Note : If distance between two points is given then use ± sign. §21.2. Section Formula

If R (x , y) divides the join of P (xi , yi ) and Q (X2 , yz) in the ratio m i : m2 (mi , rr/2 > 0) then + x = mi x^ x i . y = (djv . des jntema ||y) m1 + 77 72 m i y 2 - ft^yi (divides externally) and x = mi - /7T2 ; y' /771 - r772 § 21.3. Area of a Tria ngle

The area of the triangle 4 BC with vertices /A (xi , y i ) ; 8 (X2 , ys) and C (X3 , ys). The area of triangle ABC is denoted by A and is given as : xi yi 1 1 A = X2 yz 1 xs y3 1 1 X1 = o t 0* - yz) + x-2 (ya - yi ) + xs (yi - >2)] Note : (a) If one vertex (xa , ys) is at (0, 0), then

A = g

1

(*iy2-* 2yi) I

(b) If area of triangle is given then use ± sign (ii) If a rx+bry+cr = 0, (r= 1, 2, 3) are the sides of a triangle, then the area of the triangle is given by A =

1 2C1C2C3

az

bz cz

33 t>3 C3 where Ci , Cz , C3 are the co-factors of c i , C2 , C3 in the determinant. § 21.4.

Area of Pol ygo n

The area o f the poly gon whose verti ces are (xi , y i ) , (* 2, y 2), (X3, y3) , • • • (x n , = 2 Kxi>2 - *2 yi ) +

(X2 y3 - X3/2) + ... +

- xiy n )]

is

Straight Line

165

§2 1. 5. Nine point Centre

The centre of nine point circle (which passes through the feet of the perpendiculars, the middle points of the sides, and the middle points of the lines joining the angular points to the orthocentre) lies on OP and bisects it. When O and P are the orthocentre and circumcentre of a triangle. Also the radius of the nine-point circle of a triangle is half the radius of the circumcircle. § 21.6.

Relation i n Orthocentre, Nine point

centre, Centroid and Circumcen tre

The orthocentre, the nine point centre, the centroid and the circumcentre therefore all lie on a straight line and centroid divides the orthocentre and circumcentre in the ratio 2 : 1 (Internally). § 21.7.

Collinearity of thr ee Give n Poin ts

The three given points are collinear lie on the same straight line if (i) Area of triangle ABC is zero. (ii) Slope of AB = slope of BC = slope of AC (iii) Distance between A & 6 + distance between B & C= distance between A & C (iv) Find the equation of the line passing through any two points, if third point satisfied the equation of the line then three points are collinear. §21.8. Locus

The locus of a moving point is the path traced out by that point under one or more given conditions. a point : Let (xi , How to find the locus of be the co-ordinates of the moving points say P. Now apply the geometrical conditions on xi , yi . This gives a relation between x\ and yi . Now replace xi by x and yi by y in the eleminant and resulting equation would be the equation of the locus. § 21.9. Pos it ion s of Poin ts (*i,

ax2

and (xa, /2) Relative to a give n Li ne

If the points (xi, yi) and (X2, are on the same side of the line ax+ 0, then axi + both are of the same sign and hence ^ + by 2 + °c> 0> a n d if t h e P oints (*i > y i ) & (X2,

the opposite sides of the line ax+ to each other and hence

2 +toys+ c both are of signs opposite

c = 0, then axi + byi + c and axi + byi + c

and are on

<0

§ 21.10. Angl e bet wee n two Lines

Angle between two lines whose slopes are

and

Corollary 1 : If two lines whose slopes are

and

is 0 = tan

1

. I + mi mz are parallel iff 6 = 0 (or n ) « t a n 0 = 0 <=>

Corollary 2 : If two lines whose slopes are mi and mz are perpendicular iff 0 =

cot 0 = 0 <=> mi mz = - 1 Note : Two lines given by the equations aix+

(i) Parallel if ^

=^

ci = 0 and a2X+

(slopes are equal)

(ii) Perpendicular if aia2 +

0 (Product of their slopes is -1)

(iii) Identical if — = ^ = — (compare with the conditions in (i)) ctz bz cz

0 are

j or -

= j


166

§2 1. 11 . Equation of a Line Parallel to a giv en Line

The equation of a line parallel to the line ax + by+ c = 0 is of the form ax+ by+ k = 0, where k is any number. § 21.12. Equation of a line Perpen dicul ar to a giv en Line

The equation of a line perpendicular to the lineax+by+ any number.

c = 0 is of the form bx - ay+ k= 0 where k is

§ 21.13. Lengt h of Perp endi cula r from a Point on a Line

The length of perpendicular from (xi , yi ) on ax+ by + c = 0 is ax-i + byi + c

§ 21.14.

Dist ance bet we en two P arall el Lines

Let the two parallel lines be ax + by + c = 0 and ax+ by+ ci

= 0

First Method : The perpendicular distance between the lines is

ci •

V + b2 ) Second Method : Find the co-ordinates of any point on one of the given lines, preferably putting x = 0 or y = 0. Then the perpendicular distance of this point from the other line is the required distance between the lines. § 21.15. A line Equally Inclined with two Line s

Let the two lines with slopes mi and m 2 be equally inclined to a line with slope m, then

§ 21.16.

mi - m _

rr)2 - m

1 + mi m ~

1 + m2m

Equat ions o f Straight lin es through ( xi , yi) m aking Z a with y= mx+c

and where

are

y - y i = tan (9 - a) ( x - x i ) y - y i = ta n( 9 + a) (x -x i) tan 6 = m.

§2 1. 17 . Family of Lines

Any line passin g thr oug h the poin t of intersection of the lines a i x + biy+ ci = 0 and azx + t>2 y + C2 = 0 can be repres ented by the equation (ai x + biy + ci) + X (a2X+1>2 y+ cz) = 0. § 21.18. Concurrent Lines

The three given lines are concurrent if they meet in a point. Hence to prove that three given lines are concurrent, we proceed as follows : Method I : Find the point of intersection of any two lines by solving them simultaneously. If this point satisfies the third equation also, then the given lines are concurrent. Method II : The three lines aix+ bty+ a = 0, /'= 1, 2, 3 are concurrent if

ai a2

bi b2 ciC2

33

t>3

= 0

C3

Method III : The condition for the lines P = 0, Q = 0 , and R = 0 to be concurrent is that three constants I, m, n (not all zeros at the same time) can be obtained such that IP + mQ + nR = 0.

Straight Line

§ 21.19.

The

167

image of 0 be

(xi , yi ) with respect is given by *2

to

*i a

21.20.

A(X1 ,yi)

The I mage of a poin t with Res pe ct t o the Line M irror :

the

line

_ - 2

~

b

~

\

mirror byi + c)

+ t?)

tf

Equations o f the Bis ect ors o f the Ang les betwe en two Lines

Equations of the bisectors of the lines Li : a i x + b i y + c i = OandZ .2 : (aib2 * a2bi) where ci > 0 & C2 > 0 are (a ix + friy+ ci ) V

^(az2

+ bi 2 )

Conditions

Acute angle

+

B (X2 .y2) Fig.21.1

= 0

bz2) X

Obtuse angle

bisector

bisector

axa2 + b\b 2 > 0

-

+

axa2 + b]b2 < 0

+

-

obtuse bisector Fig.21.2

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. If the distance of any point srcin is defined as

(x,y) fro m the

(x, y) = max {I x I, I y I}, (x, y) = a non zero constant, then the locus is

3. One of the bisector of the angle between the lines

a (x - 1 ) 2 + 2h (x - 1) 0 ' - 1) + b (y - i f

(a) a cir cle

(b) a stra ight line

(a) 2x - y = 0

(b)2x + y = 0

(c) a squa re

(d) a triangle

(c) 2x + y - 4 = 0

(d) x — 2y + 3 = 0

2. The poi nt (4, 1) under goes t he follo wing three transformations successively (I) Reflect ion about the l ine y = x . (II) Transformation through a distance 2 units along the positive direction of x-axis (III) Rotation through an angle n/-4 about the srcin in the anticlockwise direction. The final position of the point is given by the co-ordinates (b) ( - 2. 7V 2) (d)

12., 7 V2)

= 0

is x + 2y - 5 = 0. The othe r bis ect or is

4. Line L has intercepts a and b on the co-ordinate axes, when the axes are rotated

through a given angle; keeping the srcin fixed, th e sa me line has inter cep ts p and q, then

\ 1 1 1 1 (b)—+ — = — + — a' b~ p~ q~ .. 2 2 , 2 ">„sl 1 1 1 ( c ) a +p =b +q (b) — + — = — + — a' p b q~ 5. The poi nt A (2, 1) is tra nsl ate d para lle l to the line x - y = 3 by a distance 4 units. If the newposi ti on A' is in third quadrant, then the co-ordinates of A' are , \ 2 r2 (a)a +b=p

2 2 +q

168


(a) (2 + 2 ^ 2 , 1 + 2 V2) (b) ( - 2 + V2",- 1 - 2 V2) (c) (2 - 2 <2,1 - 2 V2) (d) None of these 6. A ray of light comming from the point (1,2) is reflected at a point A on the x-axis and then pas ses thr ough the poi nt (5, 3). The co-ordinates of the point A is (a )( f, 0

(b)|

T3

(c) (- 7, 0) (d) Non e of these 7. The orthocentre of the triangle formed by the lines x + y = 1, 2x + 3y = 6 and Ax — y + A = 0 lies in, (a) I quad rant (c) III quad rant

(b) II quadra nt (d) IV quadr ant

8. The reflecti on of the point (4, - 13) in the the line 5x + y + 6 = 0 is (a) ( - 1 , - 1 4 ) (b) (3, 4) (c )( l, 2) (d) (- 4 , 13) 9. All the points lying inside the triangle formed by the points (0, 4), (2, 5) and (6, 2) s ati sfy (a) 3x + 2y + 8 > 0 (b) 2x + y - 10 > 0 (c )2 x- 3y - 11 >0 (d) - 2x + y - 3 > 0 10. A line passing through

P (4, 2) meet s the x and y-ax is at A and B respectively. If O is the srcin, then locus of the centre of the circumcircle of A OA B is 1 (b) 2x 1 +>•"' = 1 (a)x"'+y=2 1 1 (c) x + 2y~ = 1 (d) 2x~1 + 2y~ 1 = 1

11. Let n be the number of points having rational co-ordinates equidistant from the point (0, VT) then (a) < n 1 (b) n=1

n(c)< 2

(d)n>2

12. The co-ordinates of the middle points of the sides of a triangle are (4, 2), (3, 3) and (2, 2), then the co-ordinates of its centroid are

(a) (3, 7/3 ) (c) (4, 3)

(b) (3, 3) (d) Non e of these

13. The line segment joinin g the points (1, 2) and (- 2, 1) is div ided by th e lin e 3x + 4y = 7 in the ratio (a) 3 : 4 (b) 4 : 3

(c) 9 : 4

(d) 4 : 9

y{) and

14. If a straight line passes through (xj,

its segment between the axes is bisected at this point, then its equation is given by (a )- +

X

= 2

(b)2(xy

1 +yx,)=x,y,

(c) xy x + yx| = X]y! (d) Non e of these 15. If Pi,P2,Pi be the perpendiculars from the 2

point s ([m ,2m), {mm', m + m ) and (m respectively on the line

, 2m')

- 2

sin a x cos a + y sin a + cos a thenp\,pi, (a) A.P.

2

\

0

p-} are in

(c) H.P.

(b) G.P. (d) Non e of these

16.The

acute angle 9 throug h whi ch the co-ordinate axes should be rotated for the point A (2, 4) to attain the new abscissa 4 is given by (a) tan 9 = 3 / 4 (b) tan 9 = 5 / 6 (c) tan 9 = 7 / 8 (d) Non e of these

2" 2 2 1 then - + ~ . „ . +• 1 !9 ! 3 ! 7 ! 5 ! 5 ! n! orthocentre of the triangle having sides x - y + 1= 0, x+y+ 3=0 and 2x + 5y - 2 = 0 is (a) (2m — 2n, m —ri)(b) (2m — 2 n, n — m) (c) (2m — n,m + n) (d) (2m — n,m — n)

17, If

18. The equation of straight line equally inclined

to the axes and equidistant from the point (1, -2) and (3, 4) is

(a) x + y = 1 (c) y - x = 2

(b) y - x 1 = 0 (d) y - x + 1 = 0

19. If one of the diagonal of a square is along the line x = 2y and on e of its verti ces is (3, 0)

then its sides through this vertex are given by the equations (a) y - 3x + 9 = 0, 3y + x - 3 = 0 (b) y + 3x + 9 = 0, 3y + x - 3 = 0 (c) y — 3x + 9 = 0, 3y — x + 3 = 0 (d) y - 3x + 3 = 0, 3y + x + 9 = 0 20. The graph of the funct ion 2

y = cos x cos (x + 2) - cos (x + 1) is

169

Straight Line •

(a) A straight line passing through (0, - sin with slope 2 (b) A straight line passing through (0, 0)

2

1)

2

(c) A para bola with vertex (1 , - sin 1) (d) A straight line passing through the point • 2 . ' 7t , sin 1 are paralle to the x-axis 2 V. 21. P (m, n) (where m, n are natural numbers) is any point in the interior of the quadrilateral formed by the pair of lines xy- 0 and the two line s 2x + y — 2 = 0 and 4x + 5y = 20. The pos sible number of pos itions of the poi nt P is (a) Six (b) Fiv e (c) Four (d) Eleven 22. The image of the point A (1, 2) by the line mirror y = x is the point B and the image of B by the line mirro r y = 0 is the point (a, P) then (a) a = 1, (5= - 2 (b)a = 0,p = 0 (c) a = 2 , P = - 1

(d) Non e of thes e

23. ABCD is a square whose vertices A, B, C and D are (0, 0), (2, 0), (2, 2) and (0, 2) respectively. This square is rotated in the X - Y plane with an angle of 30° in anticlockwise direction about an axis passing through the vertex A the equation of the

diagonal BDof this rota ted squa re is If E is the centre of the square, the equation of the circumcircle of the triangle ABE is (a) V3~x + (1 - V 3 ) y = V3~,x 2 + y 2 = 4 (b)(l +V3)x-(1

— V2) y = 2, x 2 + y 2 = 9

(c) ( 2 - V3 )x + y = 2 (V3~- 1), x 2 + y 2 -xV3~-y = 0 (d) None of these 24. The straight line y = x - 2 rotates about a poi nt wher e it cuts x-axis and becomes perpendicu la r on the straight line ax + by + c = 0 then its equation is (a) ax + by + 2a = 0 (b) ay - bx + 2b = 0 (c) ax + by + 2b = 0 (d) None of these 25. The line x + y = a meets the axis of x and y at A and B respectively a triangle AMN is inscribed in the triangle OAB, O being the N, M and N lie srcin, with right angle at respectively on OB and AB. If the area of the

triangle AMN is 3/8 of the area of the triangle OAB, then AN/BN = (a) 1 (b) 2 (c )3 (d) 4 26. If two vertices o f an equilate ral triangl e have integral co-ordinates then the third vertex will have (a) integral co-ordinates

(b) co-ordinates which are rational (c) at least one co-ordinate irrational (d) co-ordinates which are irrational 2 , a 3 , P 1; P2 , p 3 are the values'of

27. If a,, a

n - 1

which L x r=0

r

n for

n — 1

is divisible by Z

r=0

x , then the

triangle having vertices (ci] P]), (a 2 , P2) and (a 3 , p 3) can not be

(a) an isosceles triangle (b) a right angled isosceles triangle (c) a right angled triangle (d) an equilateral triangle 28. The points (a , p), (y, 5), (a , 8) and (y, P) where a, p, y, 8 are different real numbers are (a) collinear (b) vertices of a square (c) vertices of a rhombus (d) concyclic

29. The equations of the three sides o f a triangle are x = 2 , y + l = 0 and x +2 y = 4. Th e co-ordinates of the circumcentre of the triangle are (a) (4 ,0 ) (c) (0, 4)

(b) (2, - 1) (d) Non e of the se

2 + a/V2) be any point on a 30. IfP (1 + a/ line then the range of values of t for which the point P lies between the parallel lines x + 2y = 1 and 2x + 4y = 15 is . , 4^2 5 V2 (a) < a < —r— (b) 0 < a < (c) - — <

5/V2

a <0

(d) None of these 31. If the point (a, a) fall between the lines I x + y I = 2 then (a) I a= I2 (b) Ia I = 1

170

Objective Mathematics (d) I a I <

(c ) I a I < 1

1

32. If a ray travell ing th e line x = 1 gets reflected the line x + y = 1 then the equation of the line along which the reflected ray trevels is (a) y = 0 (b)x-y=l (c) x = 0 (d) Non e of these 33. I f / ( x + y) =/ (x )/ (y ) Vx, y e R and/(l) = 2 then area enclosed by 3 I x I + 2 I y I < 8 is (a) /(4) sq. units (b) 1/2/(6) sq. units 6

(c) j / ( )

s

q-

un its

(d) ~/(5) sq. units 34. The diagonals of the parallelogram whose sides are Ix + my + n = 0, lx + my + n = 0, mx + ly + n = 0, mx + ly + n' =0 include an angle (a) 7t /3 (c) tan

-1

(b) 71/2 2

l£ • m fl , 2 I +m

(d)tan

2 Im ,2 ,

/ +m

2

35. The area of the triangle having vertices (-2, 1), (2, 1) and f Lim

Lim cos "' (n ! 7tx); x is

m —> oo n —> oo

38. The position of a movin g point in the xy-plane at time is given by I 2 (u cos a.t, u sin at — — gt ), where u, a, g are constants. The locus of the moving point is (a) a circle (b) a parab ola (c) an ellipse (d) Non e of these 39. If the lines x + ay + a = 0,bx + y + b = 0 and cx + cy + 1 = 0 (a, b, c being distinct * 1) are concurrent, then the value of

a-I (a) - 1 (c) 1

b-1

c-t

is (b) 0 (d) Non e of thes e

40. The medians AD and BE of the triangle with vertices A (0, b), B (0, 0 ) and C (a, 0) are mutually perpendicular if (a)fc = V2"a (b )a = V 2 b (c)b = --l2a (d) a = 5-l2b 41. P is a point on either of two lines y - V ^ l x l = 2 at a dista nce o f 5 unit s from their point of intersection. The co-ordinates of the foot of perpendicular from P on the bis ector of the ang le betwee n them are (a) ^0, ^(4 + 5^3) jo rj ^0 ,| (4 -5 V3 ) j,

rational, Lim Lim cos 2m(n ! kx); where x is m —> 0 0 n —» oo

(depending on which line the point

irrational) is

( b ) ^ ( 4 + 5V 3)

(a) 2 (c) 4


36. Consider the straight line ax + by = c where a,b,ce RH this line meets the co-ordinate axes at A and B respectively. If the area of the A OAB, O being srcin, does not depend upon a, b and c then (a) a, b, c are in A.P. (b) a, b, c are in G.P. (c) a, b, c are in H.P. (d) None of these 37. If A and B are two points having co-ordinates (3, 4) and (5, -2) respectively and P is a point such that PA = PB and area of triangle PAB= 10 square units, t hen the co-ordin ates of P are (a) (7, 4) or (13, 2) (b) (7, 2) or (1, 0) (c) (2, 7) or (4, 13) (d) Non e of these

(c)j

V|(4

-5V3

P is taken)

)

5V3" 42. Let AB be a line segment of length 4 with the poi nt A on the line y = 2x and B on the line y = x. Then locus of middle point of all such line segment is a (a) parabo la (b) ellipse (c) hyper bola (c) circle 43. Let O be the srcin, and let A (1, 0), B (0, 1) be two poi nts . If P (x, y) is a point such that xy > 0 and x + y < 1 then (a) P lies either inside A OAB or in third quadrant (b) P can not be inside A OAB (c) P lies inside the AOAB (d) None of these

171

Straight Line

If the point P (a, a) lies in the region corresponding to the acute angle between the lines 2y = x and 4y = x, then (a) a e (2, 4) (b) a e (2, 6) (c) a e (4, 6) (d) a e (4, 8) A ray of light coming along the line 45. 3x + 4y - 5 = 0 gets reflect ed from the line ax + by —1=0 and goes alon g the line 5x- 1 2 y - 10 = 0 then 64 , 112 (a) a = -115 ' 64 (b) a = 115 , , Tl5

2 '

115 8

115 64

8

46 . If line 2x + 7y - 1= 0 inte rsec t the lines L, s 3x + 4y + 1 = 0 and L2 = 6x + 8y - 3 = 0 in A and B respectively, then equation of a line parallel to L, and L2 and passes through P such that AP:PB = 2:1 a point (P (internally) is is on the line 2x + 7y - 1 = 0) (a) 9x + 12)1 + 3 = 0 (b)9x + 1 2 y - 3 = 0 (c) 9x + 12y - 2 = 0 (d) Non e of these 47 . If the point (1 + cos 9, sin 0) lies betwee n the region corresponding to the acute angle betwe en the lines x - 3y = 0 and x-6y = 0 then (a) 0 e R (b) 0 e R ~ tin, ne I (c) 0 e R ~ I nn + ^ ], n e /

(d) None of these In a A AB C, side AB has the equation 2x + 3y = 29 and the side AC has the equation x + 2y- 16. If the mid point of BC is (5, 6) then the equation of BC is (a) 2x+ y = 7 (b) x + y = 1 (c)2x-y=17 (d) None of these 49. The locus of a point which moves such that the square of its distance from the base of an isosceles triangle is equal to the rectangle under its distances from the other two sides is (a) a circle (b) a parabo la 48

(d) a hyper bola

371

15

64

(c )a =

(c) an ellipse

50. The equation of a line through the point (1, 2) whose distance from the point (3, 1) has the greatest possible value is (a) y = x (b) y = 2x (c) y = - 2x (d) y = - x 51. If the poi nt (cos 0, sin 0) doe s not fall in that angle between the lines y = I x - 1 I in whic h the srcin lies then 0 belongs to

(b)

-

2

n

n

2 ' 2

(c) (0, n) (d) None of these 52. ABC is an equilateral triangle such that the vertices B and C lie on two parallel lines at a distance 6. If A lies between the parallel lines at a distance 4 from one of them then the length of a side of the equilateral triangle is (a) 8

(b)

, ,4V7

(d) None of these

53. The four sides o f a quadrilater al are given by the equation xy (x - 2) (y - 3)= 0. The equation of the line parallel to x - 4y = 0 that divides the quadrilateral in two equal areas is (a) x - 4y - 5 = 0 (b) x - 4y + 5 = 0 (c) x - 4y - 1 =0 (d) x - 4y + 1 = 0 54. Two points A and B move on the x-axis and the y-axis respectively such that the distance betwe en the two poi nts is always the same. The locus of the middle point of AB is (a) a stra ight line (b) a circl e (c) a parabo la (d) an ellipse 55. If

th t2 and / 3

are

distinct,

the

points

(t\, 2atx + at]), (t2, 2at2 + at'2)

and

(f 3, 2af 3 + af 3 ) are collinear if (a)?,/

2 ?=3 -

l

(c) tx +12 + f 3= 0

(b)f,+f (d)

2

+ / 3 = f, t2t3

+ 12 + 13 = - 1

56. If sum of the distances of a point from two perpe ndicular lines in a plane is 1, the n its locus is (a) a square (b) a circle (c) a straight line (d) two intersecti ng lines

172


57. If the lines 2 (sin a + sin b) x - 2 sin (a - b) y= 3 and 2 (cos a + cos b) x + 2 cos (a — b) y = 5 are perpendicular, the n sin 2a + sin 2b is equal to (a) sin (a-b)-2 sin ( a + b) (b) sin (2a -2b)-2 sin {a + b) (c) 2 sin ( a - b ) - sin (a + b) (d) sin (2a - 2b) - sin (a + b) x+y- 1 meets 58. The line x-axis at A and y-axis at B, P is the mid point of AB

Px is the foot of the fig. No. 23.3 pe rpendicula r fr om P to OA\ Mx is that of Px from OP\P2 is that of M, fr om OA; M 2 is that of P2 from OP; P 3 is that of M2 from OA and so on. If Pn denotes the nth foot of the pe rpendicula r on OA from M„ _ j, then OPn = (a) l / 2 n

(C) 2"1-

(d) 2"

+ 3

59. If the area of the triangle who se vertices are (b, c), (c, a) and (a, b) is A , then the area of triangle whose vertices are 2

2

(ba - c , be - a ) a nd(cb-a (a) A (c) aA + bA

(ac - b2, ab - c 2 ), 2

,ca-b)

2

is

(b) (a + b +c) A (d) Non e of these

60. On the portion of the straight line x + y = 2 which is intercepted between the axes, a square is constructed away from the srcin, with this portion as one of its side. If p denote the perpendicular distance of a side of this square from the srcin, then the maximum value of p is (a) V2 (b) 2 <2 (c) 3 V2 (d) 4 <2

(b) 1/2 "

MULTIPLE CHOICE -II Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). 61. Th e poi nts (2 , 3), (0, 2), (4, 5) and (0, t) are concyclic if the value of t is (a) 1

(b) 1

(c) 17 62. The point

of

(d )3 intersec tion

of

the

lines

, ,

3

3

N

,

3

, 3

(b )( -l ,0 ) (d) (7, 10)

(c) x 2 - y 2 = 2 (ax + by)

63. The equations (b - c) x + (c - a) y + a - b - 0 3.

(a) (0,-1) (c) (- 11 7- 8)

66. If the point P (x, y) be equidis tant fr om the point s A(a + b,a-b) and B (a-b, a + b) then (a) ax = by (b) bx = ay

• + f = 1 and y + ^ = 1 lies on a b b a (a) x - y = 0 (b) (x + y) (a + b) = 2ab (c) (Ix + my) (a + b) = (l + m) ab (d) (Ix - my) (a + b) = (l - m) ab

,,3

65. If (-6 , - 4) , (3, 5), (- 2, 1) are the vertices of a par all elo gram then re maining ver tex can not be

(d) P can be (a, b) 67. If th e li nes x - 2 y - 6 = 0, 3x + y - 4 = 0 and n

(b - c )x+(c -a )y +a - b = 0 will represent the same line if a (a) b = c (b) c = (c) a-b (d)a + b + c = 0 64. The area of a triangle is 5. Two of its vertices are (2, 1) and (3, -2). The third vertex lies on y = x + 3. The co-ordinates of the third vertex can be (a) ( - 3/ 2, 3/ 2) (b) (3/ 4, - 3/2 ) (c) (7/ 2, 13/2 ) (d) ( - 1/4 , 11 /4 )

Ax + 4y + X2 = 0 are concurrent, then (a)= A, 2 (b) A= - 3 (c)= A 4 (d) A = - 4. 68. Equation of a straight line passing through the point of intersection o f x - y + l = 0 a nd 3x + y - 5 - 0 are perpend icular to one of them is (a) x + y + 3 = 0 (b )x + y - 3 = 0 (c) x - 3y - 5 = 0 (d) x — 3y + 5 = 0 69. If one vertex of an equilateral triangle of side a lies at the srcin and the other lies on the

Straight Line

line x vertex are

173

= 0, the co-ord inat es of the third

(a) (0, a)

(b)

(c) (0, - a)

(d) ( -

a/2, - a/2) 43 a/2, a/2)

70. If the lines ax + by + c = 0, bx + cy + a = 0 and cx + ay + b= 0 are concu rrent {a + b + c * 0) then (a) a3 + b 3 + C3 - 3abc = 0 (b) a = b (c) a = b = c (d) a2 + b2 + c2 - be - ca- ab = 0 71. If the co-ord inat es of the vert ices of a triangle are rational numbers then which of the following points of the triangle will always have rational co-ordinates (a) centr oid (b) incen tre (c) circumc entre

n > 1, the lengt h of a side of Sn equals the length of a diagonal of Sn + ,. If the length of a side of S x is 10 cm, then for which of the following values of n is the area of Sn less than 1 sq. cm ? (b) 8 (d) 10

73. A line passing through the point (2, 2) and the axes enclose an area X. The intercepts on the axes made by the line are given by the two roots of (a) x - 2 I XI x +1 X, I = 0 (b)x

/

\

... 7n , lire (b )- -a nd - —

7 1

(a)-and-

(d) None of these

^ and 78. A(c)line of fixed length (a its ends are always perpe ndi cular lines. Th e which divides this line lengths a and b is (a) a circl e (c) a hype rbol a

+ b) moves so that on two fixed locus of th e point into portions of

(b) an elli pse (d) Non e of thes e

79. If each of the point s (x, , 4), ( - 2, y x ) lines on the line j oin ing the points (2, - 1), (5, - 3) then the point P (x, , y,) lies on the line (b)2x + 6y+

+ IX,lx + 2IXl = 0

1 =0

(c) 2x + 3y - 6 = 0

(c) JC — I A. IJC -t- 2 1 X. I = 0 (d) None of these 2

(d) 6 (x + y) - 23 = 0

74. If bx + cy = a, where a, b, c are the same sign, be a line such that the area enclosed by 1 2 the line and the axes of reference is — unit

8 then (a) b, a, c are in G.P. (b) b, 2a, c are in G.P. (c)

76. Two roads are represented by the equations y —x = 6 and x + y = 8. An ins pec tio n bunglo w has to be so constr ucted tha t it is at a distance of 100 from each of the roads. Possible location of the bunglow is given by (a)(10(W2 + l , 7 ) (b) (1 - 100^2", 7) (c) (1, 7 + IOOV2) (d) ( 1 , 7 - 100V2) 77, Angles made with the x-axis by two lines drawn through the point (1,2) cutting the line x + y = 4 at a distance V 6 / 3 from the point (1,2) are

(a) 6 (x + y) - 25 = 0

2

2

(c) bisector of obtuse angle (d) None of these

(d) orthocentre

72. Let 5], S2, ... be squares such that four each

(a )7 (c) 9

(a) bisector of the angle including srcin (b) bisector of acute angle

are in A.P.

(d) b, - 2a, c are in G.P. 75. Consider the straight lines x + 2y + 4 = 0 and 4x + 2y - 1 = 0. Th e lin e 6x + 6y + 7 = 0 is

80. Two vertices of a triangle are ( 3 , - 2 ) and ( - 2 , 3) and its orthocent re is (- 6 , 1). Then its third vetex is (a) (1,6) (b) ( - 1 , 6 ) (c) ( 1 , - 6 ) (d) None of these

B on AC where A (- 1, 3), 5 (1, - 1), C(5, 1) is (a) VTsT (b) VTo (c) 2^3" (d) 4

81. Length of the median from

82. The point P ( l , 1) is translated parallel t o 2x = y in the first q uadra nt throu gh a unit distance. The co-ordinates of the new position of P are

174

Ob j ective Mathema tics

(a)

1 ±;

2

1 ±

(b )( l± ^, l±

J_ V5

(b) rotation through a distance 2 units along the positive x-axis.

_2_

Then the final co-ordinates of the point are

V5

(a) (4, 3) (c )l ,4 )

87. The incentre of the triangle formed by the lines x = 0, y = 0 and 3x + 4y = 12 is at

83. The mid points of the sides of a triangle are (5, 0), (5, 12) and (0, 12). The orthocentre of this triangle is (a) (0 ,0 )

(b)(l,l)

(b) (10, 0) J1 g (d) 3 '8

(c) (0, 24) 84. The a lgebra ic sum

of the

(c)

perpend icular

A (a\ , b{)\ B (a 2\ b2)

distances

from

C (a3,

to a var iab le line is zero, then the

and

line passes through AABC (a) the orthocentre of AABC (b) the centroid of (c) the circumcentre of A AB C (d) None of these

1

1,

'1

88 . If ( a , b) be an end of a diagonal of a square and the other diagonal has the equation x — y = a then another vertex of the square can be

(a)(a-b,a)

(b) (a, 0)

(c) (0, — a)

(d) (a +

b,b)

89. The points (p+ 1 , 1), (2p + 1, 3) (2p + 2, 2p) are colliner if

and

(b) p = 1/2

(a) p = - l

85. One vertex of the equilateral triangle with centroid at the srcin and one side as x + y - 2 = 0 is (a) ( - 1 , - 1 ) (b)(2 ,2) (c) ( - 2, - 2) (d) non e of thes e

(c)p = 2

86. The point (4, 1 ) under goes the followi ng two

successive transformations : (a) reflection about the line

(b) (3, 4) (d) (7 /2 , 7/2)

y-x

90. If 3a + 2b +6c = 0, the fa mily of str aigh t lines ax + by + c = 0 passes through a fixed poin t wh os e co -o rd in at es are given by (a) (1 /2 , 1/ 3) (b) (2, 3) (c) (3, 2) (d) (1/3 , 1/2)

Practice Test Tim e : 30 mi n.

M.M.: 20

(A) There are 10 parts in this question. Each part has one or more than one correct 1. Consider the equation

y —y\ = m (x -x^). If

m and xx are fixed and diffe ren t lines a re drawn for different values ofyj, then

( C )

(a) The lines will pass through a fixed point (b) Th er e will be a set of para lle l lines

x = xj (c) All the lines intersect the line (d) All the lines will be parallel to the line

(a) ^(3V

I

+

3-l)

answer(s). [10 x 2 = 20]

(b) 3V3+1 (d) None of these

7

Le t Lx = ax + by + a3-Ib = 0 and

y = xj

2. If th e line y V3x cut th e cur ve 3 3 2 2 x +y + 3xy +5x + 3y + 4x + 5y - 1 = 0 a t the points A ,B ,C then OA.OB.OC is

L 2 = bx -ay + b 3-la = 0 be two st ra ig ht lines. The equations of the bisectors of the angle formed by the loci. Whose equations ar e

A JLJ -

X 2 L 2= 0

and

A

J L+J

= 0,

Ki and \ 2 being non zero real numbers, are given by

Straight Line

175

(a) Li = 0

(b) L 2 = 0

(c) XiLj + X 2L2 =0

(d) K 2 L l - XxL2 = 0

4. If P (1, 0), Q ( - 1, 0) an d R (2, 0) ar e th re e S given points then the locus of point satisfying the rela tion' (SQ) 2 + (SR)

2

= 2 (SP )

2

7. A fam ily of line s is give n by th e eq ua ti on (3x + 4y + 6) + X(x + y + 2) = 0. Th e li ne situated at the greatest distance from the po in t (2, 3) be lo ng in g to this family ha s t he equation (a) 15* + 8y + 30 = 0

is

(b) 4x + 3y + 8 = 0 (c) 5x + 3y + 6 = 0

(a) a straight line parallel to x-axis (b) circle through srcin (c) circle with centre at the srcin

(d) 5x + 3y + 10 = 0

(d) a straight line parallel to y-axis 5. The base BC of a triangle ABC is bisected at the point (p, q) and the equations to the AB and AC sides ar e px + qy = 1 an d qx+py =1. The equ ati on of the med ian through A is (a) (p - 2q) x + {q - 2p) y + 1 = 0

P on the line 8. The co-ordinates of the point 2x +3y + 1 = 0, suc h t h a t \PA-PB \ is maximum where A is (2, 0) and B is (0, 2) is (a) (5, - 3) (b) (7, - 5) (c) (9, - 7)

(b) (p + g) (x + y) - 2 = 0 (c) (2pq - 1) (px +qy- 1)

x- 3 y+5 .x- 3 y+5 and are —I cos 6Z = si n G cos ()> = si n <> | x-3 y+5 , i - 3 y + z 5, , = an d ,, = then cos a s in a p y

= (p 2 + q2 - 1) (.qx +py - 1) (d) None of these 'b' for which the srcin 6. The set of values of and the point (1, 1) lie on the same side of the straight line 2

e 4) (a) b (2, (c) be [0, 2]

(a) a = ^6

(b) P = - si n a

(c) y=c osa 10

a x + aby + 1 = 0 V a e R , b > 0 are

(d) (11, - 9)

9. The equation of the bisectors of the angles between t he tw o intersecting li ne s. :

b e (0, 2) (b) (d) No ne of th es e

(d)P = si na

- If 4a 2 + 9b 2 -c2+

12ab =0, th en th e f ami ly ax +or byat + c =0 is

of concstraight urr ent lines at (a) (2, 3) (c) ( - 3, - 4)

(b) ( - 2, - 3) (d) (3, - 4)

Record Your Score


mus t be 100%

|

Answers Multiple Mb) 7. (a) 13. (d) 19. 25. 31. 37.

(a) (c) (c) (b)

Choice-I 2. (c) 8. 14. 20. 26. 32. 38.

(a) (a) (c) (c) (a) (b)

3. (a) 9. (a) 15. (b) 21. 27. 33. 39.

(c) (d) (c) (c)

4. (b) 10. 16. 22. 28. 34. 40.

(b) (a) (c) (b) (b) (a)

5. (c) 11. 17. 23. 29. 35.

(c) (a) (c) (a) (a)

41. (b)

6. (a) 12. 18. 24. 30. 36. 42.

(a) (d) (b) (a) (b) (b)

ei the r

176


43 . (a)

44. (a)

45. (c)

46 . (c)

47. (d)

48. (b)

49. (d) 55. (c)

50. (b) 56. (c)

51. (b) 57. (b)

52. (c) 58. (b)

53. (b) 59. (b)

54. (b) 60.

Multiple

Choice-ll

61. (a), (c) 65. (a)

62. (a), (d) (b), (c), 66. (b), (d)

70. (a), (d) (b), (c), 75. (a), (b)

76.

80. (b) 86. (b)

Practice

67.

71. (b), (c), (a), (d) 81. (b) 87. (b)

63. (a), (d) (b), (c), (a), (d) 68. (b), (d) 69. (a), (c), (d) 72. (b), (c),(d) 73. 77. (b) 78. (a), 82. (b) 83. (a) 84. (b) 88. (b), (d) 88. (c), (d) 90.

64. (a), (c) (a), (b), (c), (d) (c) 74. (b), (d) 79. (b) (a) 85. (c) (a).

Test

1. (b), (c)

2. (a)

3.(b) (a),

7

8. (b)

9. (a), (b), (c)

- (c)

(c)

4. (d) 10. (a), (b).

5.

(c)

6. (b)

PAIR OF STRAIGHT LINES § 22.1. Homogeneous Equation of Second Degree An equation of the form ax2 + 2hxy+ fay2 = 0 is called a homogeneous equation of second degree. It represent two straight lines through the srcin. (i) The lines are real and distinct if (ii) The lines are coincident if

h2

h 2 - ab > 0. - ab = 0

(iii) The lines are imaginary if fa2 - ab < 0 (iv) If the lines represented by ax 2 + 2 hxy+ fay2 = 0 be y then

(y-

m-\x= 0 and y - nmx = 0

mix) ( y - mix) = y 2 + ^xy+^x h 2 . mi + mz = - -r - and b

= 0

2

a

m 1 mz = ~r • fa

§ 22.2. Angle Between two Lines If a +fa* 0 and 9 is the acute angle between the lines whose joint equation is ax 2 + 2hxy+ fay2 = 0, then

tan 0 = § 22.3.

2 ^(fa2 - ab)

a+b

Equation o f the Bis ec tor s of the Ang le s Betw een the Lines ax

The equation of bisectors is

2

+

~ ^h

Corollary 1 : If a = fa, then the bisectors are = 0, i.e., x-y= Corollary 2 : If h = 0, the bisectors are x y = 0, i.e., x= 0 , y = 0.

x2

y2

= 0 ••(i)

0, x + y = 0.

Corollary 3 : If in (i), coefficient of x 2+ coefficient of y 2 = 0 , then the two bisectors are always perpendicular to each other. § 22.4. General equa tion o f Se co nd Degr ee

The equation ax 2 + 2hxy + fay2 + 2gx + 2fy + c = 0 ...(i) is the general second degree equation and represents a conics (Pair of lines, circle, parabola, ellipse, hyperbola). => represents a pair of straight lines if A = afac + 2fgh - af

i.e., if

2

a h g h fa f g f c

- fag2 - ch 2 = 0 = 0

178


otherwise it represents a conic (i.e., if abc + 2fgh - af 2 - fag 2 - ch 2 * 0). Corollary 1 : Angle between the lines : If the general equation ax 2 + 2hxy+ fay2 + 2gx+ 2fy+ c = 0 represents two straight lines, the angle 9 between the lines is given by

^(tf-ab)

2

9 = tan

( a + b)

these lines are parallel iff hi 2 = ab and perpendicular iff a + b = 0 Coro llar y 2 : Condition for co inc ide nc e of li nes :

The lines will be coincident if h 2 - ab = 0, g 2 - ac =0 and f 2 - be = 0. : The point of intersection of ax 2 + 2hxy+ fay2 +

Corollary 3 : Point of intersection of the lines

2gx+2fy+c

= 0 is

( bg-hf rf-ab V

af-gh ' l?-ab I

or 22.5. Equation of the lines joining the srcin to the points of intersection of a given line and a given curve :

Curve PAQbe (Fig. 22.1) ax 2 +2hxy+ bf + 2gx+2fy+ c 0= ...(i) and the equation of the line PQ be Ix +0 my + n = ...(ii) From the equation of the line (ii) find the value of (1) in terms of xand y, i.e. ^

^

= 1

- n

Fig. 22.1

...(iii)

Now the equation (i) can be written as ax 2 + 2hxy+ fay^ + (2gx+ 2fy) (1) + c(1) aJ

+

2 h x

y +

2

= 0

b f H 2 g x

+

2 f y

( ^ )

- )n

+

c f ± ± m )

...(iv)

=0

[replacing 1 by l x +

171

- n

^ from (iii)]

Here the equation (iv) is homogeneous equation of second degree. Above equation (iv) on simplification will be of the form A^ + 2Hxy+ By2 = 0 and will represent the required straight lines. If 9 be the angle between them, then tan 9 =

2 V h 2 - AB

A+B Hence the equation of pair of straight lines passing through the srcin and the points of intersection of a curve and a line is obtained be making the curve homogeneous with the help of the line.

Pair of Straight Lines

179

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. Which of the following pair of straight lines intersect at right angles ? (a) 2x2 = y(x + 2y) (c)2y(x + y)=xy

(b) (x + y)2 = x (y + 3*) (d)y = ±2y

2. If co-ordinate axes are the angle bisectors of the pair of lines ax2 + 2hxy + by 2 = 0, then (a) a = b (b)h = 0

(c) a2 + b = 0 2

(d)2Vf gradient

x - 2mxy — y = 0

2

and x - 2 nxy-y = 0 are such that one of them represents the bisectors of the angles betw een the oth er, then (a) mn + = 10 (b) mn- 1=0 (c) 1/m + 1/n = 0 (d) 1 / m - l / n = 0 4. If the lines joi nin g the srcin to the points of 2

intersection of y = mx + 1 with x + y = 1 are pe rpendicu la r, then (a) m= 1 only (b) m = + 1 (c) m 0= (d) Non e of these

(b) h — a + b

(c) 8n = 9ab

(d) ah 1 = 4ab

the 2

lines

2

(a) 2 tan 0 (c) 2 cot 0

lin es

of

the

2

(c)a-b

(d) (a - b) 2 = 4h2

= 2\h\

6. The pair of lines joining the srcin to the poin ts of in fe rs ec ti on of the curv es 2

2

ax +2hxy + by + 2g x = 0an d 2

a' x + 2h'xy + b'y + 2g'x = 0 will be at right angles to one another if (a)g(a' + b') = g'(a + b) (b)g(a + b)=g'(a' + b') (c)gg' = (.a + b) (a' + b') (d) None of these 7. The equation x2 + 2 ^2 xy + 2y 2 + 4x + 4 42y + 1 = 0 represents a pair of line s. The distance between them 4 (a) 4 (b) 4f

2

2xy

(b) 2 (d) sin 20

10. If a, P > 0 and a
a ( d ) p < a or p > P 11. The image of the pair of lines represented by 2 2 ax + 2hxy + by = 0 by the li ne mirr or y = 0

(b) bx2 — 2hxy + ay 2 = 0

the

lines

tan 0 + y sin 0 = 0 mak e with the x-ax is is

ax +2 hxy + by= 0 bisects the angle be tw ee n positive dire ctions of the axe s, a, b, h satisfy the relation (&)a + b = 2\h\ (b)a + b = -2h

of

2

x (sec 0 - sin 0) -

is (a) ax2 — 2hxy - by 2 = 0

One 2

the

2

pai r

5_ If

of

(a) h = ab

2

2

one

ax2 + 2 hx y + by 2 = 0 is twice that of the other, then:

which

2

of

9. The difference of the tangents of the angles

(d)a + b2 = 0

3. If the two pairs of lines

(c)2 8. The

(c) bx2 + 2hxy + ay 2 = 0 (d ) ax2 - 2hxy + by 2 = 0 12. If the equation ax1 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a pair of parallel lines then the distance between them is

4 4

ac 8 -ac (b) 2 2 , 2 hI.*.+a hl2 +a 2, K1+ a c (c)3 ff +ac (d) 2 a(a + b) a (a+ b) 2 2 13. If the lines repres" ted by x - 2pxy - y •• 0 are rotated about u»c srcin through an angle (a) 2

4

0, one clockwise direction and other in anticlockwise direction, then the equation of the bisectors of the angle between the lines in the new position is (a) px2 +2xy- py 2 = 0 (b) px" + 2xy + py 2 = 0

180


(c)x2 -Ipxy + y 1 = 0 (d) None of these

(c)-2

(d)2

15. The pair o f straight lines joini ng the srcin to

I4_ If the sum of the slopes of the lines given by 2 2 4x + 2Xxy - 7 y = 0 is equal to the product of the slopes, then A. = (a )- 4 (b) 4

2

2

the common points of x +y = 4 y = 3x + c are perpendicular if c 2 = (a) 20

(b) 13

(c) 1/5

(d) 5

and

MULTIPLE CHOICE -II Each question, in this part, has one or more than one correct answer (s). For each question, write the letters a, b, c, d corresponding to the correct answer (s). 2

2

16. If x + ay + 2py = a represents a pair of pe rp endicula r straig ht lines then : (a) a = 1, p = a

(b )a= l,P = - a (c) a = - 1, P = - a (d) a = - 1, P = a

(b) ax2 - 2hxy - by 2 = 0 (c) bx2 - 2hxy + ay 2 = 0 (d) bx2 + 2hxy + ay 2 = 0 22. Products of the perpendiculars fr om (a, P) to the lines

ax2 + 2hxy + by2 = 0 is

17. Type of quadrilater al form ed by the two pairs 2

2

6x - 5xy - 6y = 0 2 2 6x - 5xy - 6y + x + 5y - 1 = 0 is of

lines

(a) square (c) parallel ogram

and

(b) rhom bus (d) rectangle

2

lines x + 4xy - y = 0, then the value of m = 1 +V5~ -1 +V 5 (b) (a) 2 2 1-V5 -1 -V 5 (c) (d) 1 2

20.If

the angle represented by

'

between

2

the

tw o

lines

2x 2 + 5xy + 3y 2 + 6x + ly + 4 = 0 is ta n - 1 (m), then m is equal to (a) 1/5

(b)-l

(c) - 2 / 3 (d) Non e of these 21 If . the pai r of straigh t lines 2 2 ax + 2 hxy + by = 0 is rotated about the srcin throug h 90°, then their equat ions in the new position are given by (a) ax 2 - 2hxy + by 2 = 0

aa - 2/i a P + frp 2

2

2

V4h + {a + b)

a a 2 - 2/i a P + frP 2 V4/i 2 - (a - b)2

aa2 - 2haQ + bQ 2 °

^l4h2 - (a + b) 2

(d) None of these 23. Equat ion of pair of straight l ines dr awn through (1, 1) and perpendicular to the pair

(b )d =l /3 (d)d=3

19. If the line y = mx is one of the bisector of the 2

(a) (b)

18 Tw o of th e stra ight l ines giv en by 3 2 2 3 3x + 3 x y - 3x y + dy = 0 are at right angles if (a )d = - l / 3 (c)d = -3

2

2

2

of lines 3x - Ixy -2y

= 0 is

2

(a) 2x + Ixy - 1 lx + 6 = 0 (b) 2 (x — l ) 2 + 7 (x — 1) (y - 1) - 3y 2 = 0 (c) 2 (x - l) 2 + 7 (x - 1) (y - 1) + 3 (y - l ) 2 = 0 (d) None of these 24. Two pairs of straight lines hav e the equation s

y 2 + xy - 12x 2= 0 and

ax2 + 2hxy + by 2 = 0.

One line will be common among them if (a) a = - 3 (2 h + 3b) (b) a = 8 (h - 2b) + =-3{b + h) (c)a = 2(b h) (d)a 25. The combined equation of three sides of a 2

2

triangle - y )point (2x +and 3y - (b, 6) =1)0.is an If ( - 2 , a ) isis an (x interior exterior point of the triangle then (a) 2
y

b<~

(b )- 2< a< y (d) - 1 < b < 1

Fair of Straight Lines

181


Time : 15 Min

(A) There are 5 parts in this question. Each part has one or more than one correct 1. The equ ati on of ima ge of pai r of lines y = | x - 1 | iny axis is (a) x 2+y2

(a) circle (b) pair of lines (c) a parabola

+ 2* + 1 = 0

(b) x 2 - y 2 + 2x - 1 = 0 (c) x 2 - y 2 + 2 x + l = 0 (d) none of these 2. Mixed term xy is to be removed from the general equation of second degree 2

(d) line segment y = 0, - 2 < x < 2 4. If th e two lines rep res ent ed by 2

V(x - 2) 2 +y2 + V(x + 2) 2 +y2 = 4 rep rese nts

2

2

2

(b) tan a tan p = sec 2 9 + ta n 2 9 (c) tan a - tan p = 2 ta n a 2 + sin 29 (d) ta n P 2 - sin 29 2

2

5. The equati on ax +by +cx + cy = 0 represents a pair of straight lines if (a) a + b= 0 (b) c = 0 (c) a + c= 0 (d) c (a + b) = 0

Record Your Score

Max. Marks

1. First attempt 2. Second attempt 3. Third attem pt

must be 100%

Answers Multiple

Choice-I

1. (a)

2. (b)

3. (a)

4. (b)

5. (b)

6. (a)

7. (c)

8. (c)

9. (b)

10. (d)

11. (d)

12. (b)

14. (c)

15. (a)

20. (a)

13. (a)

Multiple

Choice-ll

16. (c), (d)

17. (a), (b), (c), (d)

18. (d)

19. (a), (c)

21. (c)

22. (d)

23. (d)

24. (a), (b)

25. (a), (d)

2. (d)

3. (d)

Practice 1. (c)

2

x (ta n 9 +c os 9) - 2xy tan 9 +y si n 9 = 0 ma ke angl es a, p wit h the x-axis, th en (a) tan a + tan P = 4 cosec 29

2

ax + 2 hxy + by + 2 gx+ 2/y + c = 0, on e should rotate the axes through an angle 6 given by tan 29 equal to a-b 2h (a) (b) 2h a +b a+b 2h (c) (d) 2h a-b equation 3. The

answer(s). [5x2 = 10]

Test 4. (a), (c), (d)

5. (a), (b), (d)

23 CIRCLE

h)2 + (y- k) 2 =

h k)

.

2fy+ c = g, f , c

g

centre

^ (g 2

0

2hxy+

f2 - c) (c? + f

2

> c).

+ 2gx + 2fy + c

i.e., a = b* i.e. h = 0.

xi +yi

X2 yi

X3 +yi 2gx+

*3

f y + c =•

S-t = x? + y?

c > = h)2

(h + r

if + ( y - yi)

2

1

Y3

k)2 = k+

= ? is

V

(

a

2

*

2 *

Circle

183 2gx + 2fy + c

+ yyi + g

f(y +

at

c = yi + / .

xi

yi

2gx +2fy+ c = f)

f2 y= mx± a

m The

T T

f(y+yi)

+c

x\

s =o Chord of | contact

X2 + y2 + 2gx+2fy+ g(x+ T =

I (AT) =

c = c

+ yi 2 + 2gx-\ + 2/yi +c) =

yz)

-y2)

\S7

184


(X2 ,y2)

(15)

Ort hog ona l ity of two ci rcl es :

(C1C2) 2 = (Ci

=>

(C2P) 2

Zg^gz + Zfifz = c
P(xi, yi)

SSi.=

• re+«t»i + ysm ~2

— e

Circle

185

ri x 2 + r 2xi

ny

CtD =

2

+ r2 yi

J

n

y X1 X2

yi n

= o,

Sz = a

y2

Sz

1).

186


XXI + yyi

(T Q'

Q (h, k)

Circle

187

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letter a, b, c, d whichever is appropriate. 1. The nu mbe r of rational point (s) (a point (a, b) is rational, if a and b both are rational numbers) on the circumference of a circle having centre (n, e) is

(a) p e (3 6, 47)

(b ) p e (1 6,47 )

(c) p e (16, 36)

(d) No ne of these

(a) at mos t one

(b) at least two

and axx + b\y + cj = 0 a2x + bjy + c 2 = 0 cut the coordinate axes in concyclic points, then

(c) exactly two

(d) infinite

(a) a]b]

2. The locus of a point such that the tangents drawn from it to the circle 2

(b) x 2 + y 2 + 6x - 8y - 5 = 0 (c) x 2 + y 2 — 6x + 8y — 5 = 0 (d) x 2 + y 2 — 6 x — 8y + 25 = 0 If

(a) a pair of straight lines (b) a circle (c) a parabola (d) an ellipse 4. The equation of the pair of straight lines paralle l to the y- ax is and wh ic h are ta nge nt s 2

2

— 6x — 4y - 12 = Oi s

(a) x 2 — 4x — 21 = 0 (b) x 2 - 5x + 6 = 0 (c) x2-- 6x - 16 = 0 (d) None of these 5. If a line segm ent AM = a moves in the plane XOY remaining parallel to OX so that the left end point A slides along the circle 2 2 2 x + y = a , the locu s of M is / \ 2 , 2 ,2 (a) x+y (b)x

2 2

- 4a + y 2 = 2 ax

(c) x + y

2

= 2 ay

(d)x 2 + y 2 - lax - lay - 0 6. If (2, 5) is an interior point of the circle 2 2 x + y - 8x - 12 y + p= 0 and the circl e neither cuts nor touches any one of the axes of co-ordinates then

= a2b2

\

b2

(c) ax+a2

= bx + b2

(d) axa2 = b\b2

(a) x 2 + y 2 — 6 x — 8y - 2 5 = 0

to the circle x +y

lines

a2

to

3. The locus of the point (V(3/ J + 2), -Ilk). (h, k) lies on x + y = 1 is

the

a

2

x + y - 6x - 8y = 0 are perp endi cula r each other is

7. If

8. AB is a diameter of a circle and C is any poi nt on the ci rc um fe re nc e of th e circle. Then (a) The area of A ABC is max imum whe n it is isosceles (b) The area of A ABC is minimum when it is isosceles (c) The perimeter of A ABC is max imum w hen it is isosceles (d) None of these 9. The four points of intersection of the lines (2x - y + 1) (x - 2y + 3) = 0 wit h t he axe s lie on a circle whose centre is at the point (a) (- 7/ 4, 5/4) (b) (3/4, 5/4) (c) (9/4, 5/4) (d) (0, 5/4) 10. Origin is a limit ing point of a co-ax ial syste m 2

2

of wh ich x +y - 6 x - 8 y + l = member. The other limiting point is (a) (-2,-4)

is

^ T s ' Y s 4_

(oi-^-^l(d)i-

J_

25 ' 25 11. The centr es of a set of circle s, each of ra dius 2

3, lie on the circle x+y any point in the set is (a) 4 < x2 + y2 < 64 (b)x

2

+ y 2 < 25

(c)x

2

+ y 2 > 25

(d) 3 < x2 + y2 < 9

2

= 2 5 . The locu s of

188


12. Three sides of a triangle have the equations

Lr = y - m r x - C,. = 0; r = 1, 2, 3. Then AZ^L-, + (iL 3 L| + yL, L 2= 0, where X* 0, |i 0, yÔ, is the equation of circumcircle of triangle if (a) X (m2 + m 3) + (i (m 3 +/W))+ y (m, + m 2) = 0 (b) X (m2m3 - 1) + n (m3m, - 1) + y (m,m2 - 1) = 0 (c) both (a) & (b) (d) None of these 13. The abscissaes of two points

A and B are the

roots of the eq uatio n x 2 + 2ax -b 2= 0 and their ordinates are the roots of the equation 2

2

x + 2px-q = 0. The radius of the circle with AB as diameter is b2 +p2 + q2)

(a) 2

(b) V ( a V )

2

x +y + 2gxx + 2fxy= 0 touch

each

other,

then (a)/ig=/gi

(c)/

(b)#i=£?i 2

2

20. The of the chordcoordinates cut off by 2 x -middle 5 y + 1 8point = 0 of by the t he 2

2

circle x + y - 6x + 2y - 54 = 0 are (a) (1 ,4 ) (b) (2, 4) (c )( 4, 1) (d) (1, 1) 21. Let <> | (x, y) = 0 be the equation of a circle. If <{> (0, X) = 0 has equal roots X = 2, 2 and

$ (X, 0) = 0 has roots X = — , 5 then the centre

2 14. If the two circles x + y + 2gx + 2fy = 0 and

2

2

x +y - 1 2 r + 4y + 6 = 0i s given by (a) JC+ y = 0 (b);c + 3y = 0 (c) x = y (d) 3x + 2y = 0

4

(c) 4(b 2 + q2) (d) None of these 2

-a Q 2 2 10 - If a chord of the circle x +y =8 makes a on the equal intercepts of length co-ordinate axes, then (a) I a< I8 (b) I a I < 4 <2 (c) I a< I4 (d) IaI>4 19.One of th e di amete r of th e circle

2

+ g = /, + gi (d) Non e of these

of the circle is (a) (2, 29/10) (b) (29/10 , 2) (c) (- 2, 29/10) (d) None of these 22. Two distinct chords drawn from the point 2

2

(p, q) on the circle x +y = px + qy, where pq * 0, are bisected by the x-axis. Then 2

15. The number of2 integral values o f X for which 2

x + y + Xx+(l-X)y +5 =0 is the equation of a circle whose radius cannot exceed 5, is (a) 14 (b) 18 (c) 16 (d) Non e of these 16. A triangle is formed by the lines whose combined equation is given by (x + y - 4) (xy — 2x — y + 2) = 0. The equation of its circumcircle is 2

(a) x + y - 5x - 3y + 8 = 0 (b) ;c2 + y 2 -

3JC

- 5y + 8 = 0

(c) x + y 2 + 2x + 2y - 3 = 0 (d) None of these 2

2

17. The circle x +y + 4 x - 7 y + 1 2 = 0 cuts an intecept on y-axis of length (a )3 (b) 4

(c)7

(d)l

(2)\p\ 2

= 2\q\

(b)p

(c)p <&q

2

2

= %q2

(d) p > Sq

23. The locus of a point which moves such that the tangents fr om it to the two circles

x 2 + y 2 - 5x — 3 = 0 2

and

2

3x + 3y + 2x + 4y - 6 = 0 are equal, is

(a) 2x 2 + 2y2 + 7x + 4y - 3 = 0 (b) 17x + 4y + 3 = 0 (c) 4x 2 + 4y 2 - 3x + 4y - 9 = 0 (d) 1 3 x - 4 y + 15 = 0 24. The locus of the point of intersection of the tangents to the circle x = r cos 0, y = r sin 8 at points whose parame tric angles diff er by tt / 3 is 2

2

2

(a) x + 2y =24 (2 — VT) (b) 3 (x + y ) = 1 (c) x 2 + y 2 = (2 - VTj 2

2

(d) 3 (x + y ) = 42.

2

Circle

189

25. If one c ircle of a co-axi al system is 2 2 x+y +2gx + 2fy + c = 0 and one limiting point is (a, b) then equation of the radical axis will be (a)(g + a)x+(f+b)y

+ c-a-b

2

= 0

(b) 2 (g + a) x + 2 ( / + b)y + c — a2 — b 2 = 0 (c) 2gx + l f y + c-a-b2 (d) None of these 26

-

2

= 0

(a) x 2 + y 2 + (b) x+y

2

20TIX

- lOy + lOOrc2 = 0

+ 20TIX + lOy + IOO712 =

0

(c) x 2 + y 2 - 207ix - lOy + IOO712 = 0 (d) None of these 32. If the abscissaes and ordinates of two points P and Q are the roots of the equations X2 + 2 ax-b2 = 0 =0 and x+2px-q respectively, then equation of the circle with

2

S = x + y + 2x + 3y+ 1 = 0 and S' = x2 + y2 + 4x + 3y + 2 = 0

PQ as diameter is (a) x 2 + y2 + 2ax + 2py -b2 — q2 = 0

are two circles. Th e point ( - 3, - 2) lies

(b) x+y2

(a) inside S'only

(b) insid e

2

S only

(c) inside S and 5 ' (d) outside

2

28. If (1 + ax) " = 1 + 8x + 24x 2+ . . . and a lin e 2

through P (a, n) cuts the circle A and B, then PA. PB = (a) 4 (b) 8 (c) 16 (d) 32

2

x +y = 4 in

29. One of th e di amete r of the circle circumscribing the rectangle is ABCD 4y = x +7. If A and B are the points ( - 3, 4) and (5, 4) respecti vely, then the area of the rectangle is (a) 16 sq. unit s (b) 24 sq. unit s (c) 32 sq. units (d) No ne of these 30. To which of the following circles, the line y - x + 3 = 0 is nor mal at the poin t 3

J-') 9 1 2 ' -12 J '

x- 3 -

<2 9

(c) x 2 + (y - 3) 2 = 9 (d) (x - 3)

2

+ y

2

+ q2 = 0 2

= 9

31. A circle of radius 5 units touches both the axes and lies in the first quadrant. If the circle makes one complete roll on x-axis along the positive directi on of x-axis, then its equ ation in the new position is

= 0

(x - l)

2

+ (y - 3) 2 =

2

2

x + y - 8 x + 2y + 8 = 0 intersect, in distinct points, then (a) 2 < r < 8 (b)r<2 (c) = r2 (d) r > 2

and two

34. A variable circle always touches the line y - x and passes th roug h the poi nt (0, 0). The common chords of above circle and x 2 + y 2 + 6x + 8y - 7 = 0 will pass throug h a fixed point whose coordinates are (a )| -i f

(b) (-I ,-I )

(c) iH

(d) None of these

35. The locus of the centres of the circles which 2

2

cut the circles x + y + 4x — 6y + 9 = 0 and x 2 + y 2 - 5x + 4y + 2 = 0 ort hogonal ly is (a) 3x + 4y - 5 = 0 (b) 9 x - lOy + 7 = 0 (c) 9x + lOy - 7 = 0 (d) 9 x - lOy + 1 1 = 0 36. If

=

2

(d) x2 + y2 + lax + lpy + b 2 + q2 = 0 33. If t wo circles

r = 2 - 4 r c o s 6 + 6 r sin 9 is (a) (2, 3) (b) ( - 2, 3) (c) ( - 2, - 3) (d) (2, - 3)

(a)

2

(c) x +y - lax - Ipy -b -q

S and &

27. The cent re of the circle

3 +

- lax - 2py + b 2

fro m

any

point

on

x +y2 + 2gx + 2fy + c= 0 drawn

x+y2

to

the

tangen ts the

circle are circle

+ 2gx + 2fy + c sin 2 a +

( g + / ) cos 2 a = 0, then the angle b etween the tangents is (a) 2 a (c) a / 2

(b) a (d) a / 4

37. Th e equations of the circles which touch both the axes and the line x = a are

190

Objective Mathematics 2

(a) x^ + y2 ± ax ± ay+ — = 0 2

(b) x + y2 + ax ± ay+

=0

(c) x + y - ax + ay + — = 0

38 . A, B, C and Z) are t he points of intersect ion with the coordinate axes of the lines ax + by = ab and bx + ay = ab, then (a) A, B, C,D are concyclic (b) A, B, C, D form a parallelogram (c) A, B, C, D form a rhombus (d) None of these 2 o 39. The common chord of x +y - 4 x - 4 y = 0 2

and x +y = 1 6 subtends at the srcin an angle equal to (a) Jt /6 (b) n/4 (c) J I / 3 (d) 71/2 40. The number of common tangents that can be 2

2

x + y - 4x - 6y - 3 = 0

drawn to the circles 2

2

and x +y = 2x + 2y + l = 0 i s (a) 1 (b) 2 (c)3 (d)4 41 . If tne distanc es fro m the srcin of the centres

x + y2 + 2\t x-c2 = 0 ( i = 1, 2, 3) are in G.P., then t he lengt hs of the tangents drawn to them from any point on the circle x + y 2 = c2 are in (a) A.P. (b) G.P. (c) H.P . (d) None of these of

42

three

circles

- If 4/ 2 — 5m 2+ 6/ + 1 = 0 and the lin e lx + my+ 1 = 0 touches a fixed circle, then (a) the centre of the circle is at the point (4, 0) (b) the rad ius of the circle is equal to J (c) the circle passes through srcin (d) None of these

43. A variable chord is draw n throug h the srcin 2

2

to the circle x +y - 2 a x = 0. The lo cus o f the centre of the circle drawn on this chord as diameter is (a)x

2

+ y 2 + ax = 0

(a) 2ax + 2by = a2 + b2 + X 2 (b)ax + by = a2 + b2 + \ 2 (c) x 2 + y2 + lax + 2by + X 2 = 0

(d) None of these

2

(a, b) and cuts the circle x + y2 = A,2 orthogonally, equation of the locus of its centre is

44. If a circle passes through the point

(b) x +y2 + ay = 0

(c) x 2 + y2 — ax = 0 (d) x 2 + y2 — ay = 0

(d) x 2 + y2 - 2ax — 2by + a2+ b2 — X 2 = 0 45 . If O is the srcin and

tangents 2

OP, OQ are distinct

to

2

the

circle

x +y + 2gx + 2fy + c = (), the circumcentre of the triangle OPQ is (a)(-g,-J) (b) (g,f) (c) (-/, - g) (d) Non e of these 46 . The circle passi ng through the distinct points (1,0. (t> 1) and (t, t) for all values of /, passes through the point

(a) (1,1 ) (c) (1 ,- 1)

(b) ( -1 , -1 ) (d) (- 1, 1) .

47. Equation of a circle through the srcin and be longing to the co- axial syste m, of whic h the limiting points are (1, 2), (4, 3) is

(a) x 2 + y 2 - 2x + 4y = 0 (b) x 2 + y 2 - 8x — 6y = 0 (c) 2x 2 + 2y 2 - x — 7y = 0 (d) x 2 + y 2 - 6x — lOy = 0 48 .Equat ion 2

of

the

norma l

to

the

circle

2

x +y - 4 x + 4y - 1 7 = 0 whi ch pass es through (1, 1) is (a) 3x + 2y - 5 = 0 (b) 3x + y - 4 = 0 (c) 3x + 2y - 2 = 0 (d) 3x - y - 8 = 0 49. a , P and y are parametric angles of three points P, Q and R respectively, on the circle x 2 + y 2 = 1 and A is the poin t ( - 1, 0). If the lengths of the chords AP, AQ and AR are in G.P., then cos a / 2 , cos p / 2 and cos y / 2 are in (a) A.P. (c) H.P . 50. The area 2

x+y

2

(b) G.P. (d) Non e of these b ounde d by the circle s 2

= r , r = 1, 2 and the ray s give n by

2x 2 - 3xy — 2y 2 = 0, y > 0 is

Circle

191 (a) — sq. units

(b) — sq. units

, , 371 (c) — sq. units

56. The equation

(d) 7t sq. units

(c)(x-16)

(a) x + y - 4x + 2 = 0

X for which the circle - 8x + 7) = 0 dwindles into a point are V2 (a) 1 ± 3 value s

(c)2± (d) 1±

of

3 3 4 <2

(a) x2 + y 2 = 20 = 0

(c) (x2 + y 2 — 4) + X(x2 + y 2 — 16) = 0

a ' , P' those of ax2 + b'x + c' = 0, the equation of the circle having A (a, a' ) and B (p, p') as diameter is

54. The short est dist ance fr om the point (2, - 7)

to the circle x 2 + y2 - I4x — 10>'— i 51 = 0 is

(c ) bb'{x

2

55. The circle x+y = 4 cuts the line joining the points A (1, 0) and B (3, 4) in two points PA

+ y 2) + a'cx + ac'y + a'b + ab'

(b)cc\x

(d )4

P and Q. Let — - = a

+ y 2 ) + ac'x + a'cy + a'b + ab' = 0

(a) cc'{x

(b) 2

BO

and - TJ = P then xl.

a and p are roots of the quadratic equation

2

(d ) aa'(x2

60. The

= 0

+ y 2 ) + a'bx + ab'y + a'c + ac'

=

2

2

+ (y - a) =c

(a) a = b ± 2c (b) a = b ± V2~c

(c) a = b ± c (d) None of these

0

(x — a)2 + (y — b) 2 = c2

circle and (x-b) then

= 0

+ y2) + a'bx + ab'y + a'c + ac'

2

(a) x + 2x + 7 = 0 (b) 3x 2 + 2x — 21 = 0 (c) 2x 2 + 3x - 27 = 0 (d) None of these

(b) (1/2, 1/2) (d) None of these

59. If a, p are the roots of ax + bx + c = 0 and

(d) None of these

BP

=l

=l

x 2 + y 2 - 4x - 4y - 1 = 0 whic h are far the st and nearest respectively from the point (6, 5) then 22 5 22 IT) (b) Q = 5 '' 5 J 14 11 (c)P = 3 " 5 14

4

2

=l

58. If P and Q are two points on the circle

53. The equati on of the circle passi ng through (2, 0) and (0, 4) and having the minimum radius is

(a) 1 (c) 3

+(y-16)

(a) (1 ,1 ) (c) (-1/2 , -1/ 2)

2<2

(b) x2 + y 2-2x-4y

2

mirror

=l

2

)

the

y = x and passes through the point (0,0). The common chords of above circle and x+y + 6x + 8y - 7 = 0 will pass thr ough a fixed point, whose coordinates are

y2 + 6x + 5 + X (x2 +y2

(b)2±

2

+(y-14)

+ (y-15 2

by 2

57. A variable circle always touches the line

2

(c) x + y + 4x + 2 = 0 (d) None of these x+

2

(d)(x-17)

(b) x2 + y 2 + 4x— 2 = 0

52. The

2

(b)(x-15)

2

2

2

(x - 3) + (y - 2) = 1 x + y = 19 is (a)(x-14) 2 + (y- 13)

51. The equation of the circle touching the lines I y I= x at a distanc e V2 units fro m the srcin is 2

of the imag e of the circl e

2

2

touch each other

192


MULTIPLE CHOICE -II Each question in this part has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). 61. The equation of the circle which touches the x y axis of coordinates and the line —+ . = 1 3 4 and whose centre lies in the first quadrant is X

2 ,

+ y

2

-

• 2Xx - 2Xy + X = 0, wh ere

equal to (a)l (c)3

X is

62. If P is a point on the circle x + y = 9, Q is a point on the line lx + y + 3 = 0, and the line x - y + 1 = 0 is the per pendicular bisector of PQ, then the coordinates of P are

(hM12

= 0,x2 + y2

and x +y2 +lx-9y

+ 29

+ 5x - 5y + 9 = 0 2

2

(a)x

+ y - I6x-

0 is

18y - 4 = 0

(b)jc 2 + y 2 -7jc+ lly + 6 = 0 (c) x 2 + y 2 + 2x — 8y + 9 = 0 (d) None of these

(b)2 (d)6

(a) (3, 0)

x2 + y2-2x+3y-7

67. A line is drawn through a fixed point 2

P (a, (3)

2

to cut the circle x + y = r at A and B, then PA. PB is equal to (a) (a + |3) 2 - r 2 (b)a 2 + p 2 + r 2 (c) ( a - P) 2+ r

21

2

(d) Non e of these

68. If a is the angle subtended at P (x^ yj) by the

(c) (0, 3) 63.If

a

circl e

^.

72 21_ (d) I I 25 ' 25 passes th rou gh the and

touches

x + y= 1

circle poi nt and (b) cot a/2 =

x - y = 1, then the centr e of the circle is (a) (4, 0) (b) (4, 2) (c) (6, 0) (e) the (7, 9) 64. The tangents drawn from srcin to the circle 2

x+y

2

-2px

(a)p = q (c )q = ~P

S = x + y + 2gx + 2fy + c = 0, then Js7 (a) cot a =

2

— 2qy + q = 0 are perpen dicul ar if 2 2 (b ) p = q (d) p2

65. An equation of a circl e touching the axes o f coordinates and the line x cos a + y sin a = 2 can be

(a) x + y2 - 2gx -2gy + g 2 = 0 where g = 2/(cos a + sin a + 1 ) (b) x 2 +y2 - 2gx-2gy + g=0 where g = 2/ (c os a + sin a - 1) (c) x 2 + y - 2gx + 2gy + g2 = 0 where g = 2/(cos a - sin a + 1) (d) x+y 2 - 2gx + 2gy + g = 0 where g = 2/(cos a - sin a — 1)

66. Equation of the circle cutting orthogonally the three circles

4g2+f2-c

2 (c) tan a = — (d) a = 2 tan 69.The 2

two 2

-

I

circles

2,2,

,,

x +y +ax = 0

and

2

x + y =c touch each other if (a)a + c = 0 (b) a - c = 0 / \ 2 2 (d) None of these (c) a = c 70. The equation of a common tangent to the

circles 2

x + y2 + 1 4x- 4y + 28 = 0

and

2

x + y — 14x + 4y - 28 = 0 is (a) x - 7 = 0 (b) y — 7 = 0 (c) 28 x + 45 y+ 371 = 0 (d) lx-2y+ 14 = 0

71, If A and B are two points on the circle 2

2

x + y - 4x+ 6y - 3 = 0 whic h are far hes t and nearest respectively from the point (7, 2) then

Circle

193 (a) A = (2 - 2 <2., - 3 - 2 V2) (b) A = (2 + 2 V2, - 3 + 2 V2 )

78. The equ ation 2

(C )B = (2 + 2 A /2,-3 + 2^2)

equat ions 2

of 2

f our

circles

are

2

(x + a) + (y ± a) = a . The radius of a circle touching all the four circles is (a ) (V I- l) a

(b) 2 <2 a

(b)(V2 + l ) a

(d) (2 + <2)a 2

73. The equation of a circle Q

2

2

is x + y = 4. The

is the curve C

3.

Then

(a) C 3 is a circle (b) The area enclosed by the curve C

3

is 871

(c) C 2 and C 3 are circles with the same centre

(a)x

+ y - 6 x + 6y + 9 = 0

(d) x + y 2 - 30x + 30y + 215 = 0 75. If a circle pa sses throu gh the points of intersection of the coordinate axes with the lines Xx - y + 1 = 0 and x - 2y + 3 = 0, then the value of X is (a) 2 (b) 1/3 (c) 6 (d) 3 76. The equation of the tangents drawn from the circle

= 0

(b) y — 0 (c) (h2 - r 2 ) .v - 2 rh y = 0 (d) (h2 - r2) x + 2rhy =-0

77.Equati on

of a circle with ce ntre (4, 2

touching the circle

2

x + y = 1 is

(a) x2 + y 2 - 8x - 6y — 9 = 0 (b) x 2 + ) 2 - 8x - 6y + 11 = 0 (c )x + y 2 - 8x - 6) >- 11 =0 (d) a 2 + y 2 - 8.v - 6y + 9 = 0

2

pe rp en di cu la r if h=r (a)

2

- 2 r x — 2hy + =h 0

are

(b)h = -r

(c) r2 + h2= 1

( d ) r 2 = h2

80. The equation of the circle which touches the x y axes of the coordinates and the line —+ , = 1 3 4 and whose centre lies in the first quadrant is

x+y2

- 2 cx - Icy + c 2 = 0, where c is

(a )l

(b) 2 (d) 6

of

2

2

2

2

2

2

+ y + 2,?x + 2/y + c = 0 cuts

the

2

circles

2

x + y - 4 = 0,

x +y -6x-8y+10 =0

(c) x 2 + y 2 + 30x - 30y + 225 = 0.

JC

2

x+y

each

(b) x 2 + y 2 + 6x - 6y + 9 = 0

(a)

(d) 7x + 24y = 230 79. The tangents drawn from the srcin to the

81. If the circle x

2

srcin to the x~ + y 2 — 2 rx — 2 hy + h = 0, are

(c) 24x — 7y + 125 = 0

(c )3

(d) None of these 74. The equation of circle passing through (3, -6) and touching both the axes is 2

(b) 3x + 4y = 38

ci rcle

locus of the intersection of orthogonal tangents to the circle is the curve C 2 and the locus of the intersection of perpendicular tangents to the curve C

= 25 passing throu gh ( - 2,1 1) is

(a) 4x + 3y = 25

(d) B = (2 - 2 VI , - 3 + 2 <2 j 72. The

of a tangen t to the circle

2

x+y

3)

and

x + y + 2x - 4y - 2 = 0 at the extre mit ies of a diameter, then (a) c = - 4 (b) g+/ =c -l (c) g2

— c = 17

(d) gf— 6 82. A line meets the coor dinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the srcin from the points A and B respectively, the diameter of the circle is (a) m(m + n)

(b) m + n

(c) n (m + n)

(d) ^(m

+ n)

83. From the point A (0 ,3 ) on the x 2 + 4x + (y - 3) 2 = 0, a chor d AB is and extended to a point M, such AM = 2 AB . An equation of the locus of (a) x 2 + 6x + (y - 2) 2 = 0 (b) x 2 + 8x + (y - 3) 2 = 0 (c)x

2 2

+ y2 + 8 x - 6 y + 9 = 0

(d) x + y 2 + 6x — 4y + 4 = 0

circle drawn that M is

194


84. If chord of the circle 2 "> x + v - 4x - 2y - c - 0 is trise cted at th e points (1/3, 1/3) and (8/ 3, 8/3), then (a) c = 10

(b) c = 20

(c) c = 15

(d) c 2 - 40c + 400 = 0

x + y - 6x - 4y —3 = 0, then limiting point is (a) (2, 4) (b )( -5 ,- 6) (c) (3, 5)

2

[ 1 +1 J being a pa ra mete r (a) circle (c) Ellipse

t

represents

2

1 +t (b) parab ola (d) Hype rbo la

(a) (x - l) 2 + (y — 2) 2 = 1

(d) x 2 + y 2 — 3x — 4y + 7 = 0 89. Length of the tangent drawn from any point of the circle x + y 2 + 2gx + 2fy + c-0 to the

C, =x2 +y2 — 2x - 4y - 4 = 0

C2 = x2 + y2+ 2x + 4y + 4 = 0

and the line L = x + 2y + 2 = 0. Then (a) L is the radical axis of C, and C

2

(b) L is the common tangent of C\ and C (c) L is the common chord of

(d) (- 2, 4)

(b) (x - 2) 2 + (y - l) 2 = 1 (c) x 2 + y 2 — 2x — 4y + 4 = 0

86. Consider the circles

and

othe r

88. Equation of the circle having diam eters x - 2 y + 3 - 0 , 4x — 3y + 2 = 0 and rad ius equal to 1 is

85. The locus of the point of intersection of the lines 2 (l-t ) A 2 at x = a\ ——r a n d=y r

the

2

Ct and C2

(d) L is perp. to the line joining centres of C] and C 2 87. If (2, 1) is a limiting point of a co-axial system of circles containing

2fy + dc =0, (d>c)is circle x 2 + y2 + 2gx +•2fy (a) Vcâi" c (b) (c) (d) V F s 90. A region in the x-y plane is bounded by the curve y = V(25 - x2) and the li ne y = 0. If the poi nt (a, a + 1) lies in the interio r of the region then (a) ae

(- 4, 3)

(b)fle

(-°o,-i)

U

(3 ,oo )

(c) a e ( -1 ,3 ) (d) None of these


M.M. 20

(A) There are 10 parts in this question. Each part has one or more than one correct I

If point P (x, y) is called a lattice point if x,y e /. The n the tot al num ber of lattice po in ts in th e in te ri or of the cir cle 2

2

2

x +y = a , a 0 can not be (a) 202 (b) 203 (c) 204 (d) 205 The equation of the circle having its centre on the line x + 2y - 3 = 0 and pas sin g through the points of intersection of the x2+y2 - 2x -4y + 1 = 0

circles

x2 +y2 - 4x - 2y + 1 = 0 is 2

2

(a)x +y -6x

+7 = 0

fbj x 2 + y 2 - 3x + 4 = 0

an d

answer(s). [10 x 2 = 20]

(c) x2 +y2 - 2x - 2y + 1 = 0 (d) x 2 +y2 + 2x - 4y + 4 = 0 Th e locus of a cent re of a circle w hi ch touches externally the circle x2 + y2 - 6x - 6y +14 = 0 and also tou ches the y-axis is given by the equation (a) x - 6x - lOy + 14 = 0 (b) x2 - lOx - 6y + 14 = 0 (c) y 2 - 6x - lOy + 14 = 0 (d) y 2 - lOx - 6y + 14 = 0 4. lif{x+y)=f{x).f(y) for all x and y, f{\) = 2 and a„ = f (n), n e N, then the equation of

Circle

195 the circle having (a 1 ; a 2 ) and (a 3 , a 4 ) as the ends of its one diameter is

at points A and B. A line through point A meet one circle at P and a parallel line through B meet the other circle at Q. Then the locus of the mid point of PQ is

(a) (x - 2) (x - 8) + (y - 4) (y - 16) = 0 (b) (x - 4) (x - 8) + (y - 2) (y - 16) = 0 (c) (x - 2) (x - 16) + (y - 4) (y - 8) = 0 (d) (x - 6) (x - 8) + (y - 5) (y - 6) = 0 5. A circle of th e co-axial system wit h l imit ing po in ts (0, 0) a nd (1, 0) is (a)x

2

2

2

2

2

+ y - 2x = 0 (b) x

2

2

squ ar e

is

i nscrib ed

(b) (x - 5) 2 + (y - 0) 2 = 25

2

+y - 6x + 3 = 0 2

(c)x + y = 1 (d) x +y - 2x + 1 = 0 6. If a variable circle touches externally two given circles then the locus of the centre of the variable circle is (a) a str aig ht line (b) a parab ola (c) an ellips e (d) a hyper bola 7. A

(a) (x + 5) 2 + (y + 0)2 = 25

in

t he

x + y - lOx - 6y + 30 = 0 . On e si de of th e square is parallel to y = x + 3, t he n one vertex of the square is

+ y 2 + lOx = 0 + y 2 - lOx = 0 2

2

(a) (x + 2) 2 + ( y - 3) 2

2

= 6-25

2

(b) (x - 2) + (y + 3) = 6-25 (c) (x + 2) 2 + (y - 3) 2 = 18-75 (d) (x + 2) 2 + (y + 3) 2 = 18-75 10. The poi nt ([P +1 ], [P]), (wh ere [.] de not es the greatest integer function) lying inside 2

the region bounded by the circle x (a) P € [- 1, 0) u [0, 1) u [1, 2) (b) P e [ - 1, 2) - {0, 1} (c) P 6 (-1, 2)

+ 2 4 x - 8 1 = 0 interse ct eac h other

(d) None of these

circles

x

2

+ / - 4 x - 8 1 = 0,


must be 100%

Answers Multiple

2

+y - 2x

- 15 = 0 and x 2 +y2 - 1x - 7 = 0 th en

2

8. The

x+y

(d)x

2

th e circle x +y +4 x - 6 y - 12 = 0 which subtend an angle of tc/3 radians at its circumference is

(a) (3, 3) (b) (7, 3) (c) (6, 3 - V3) (d) (6, 3 + V3) 2

2

9. The locus of th e mid poi nts of th e chords of

circle

2

(c)x

Choice-I

1. (a)

2. (a)

3.(b)

4. (c)

5. (b)

6. (a)

7. (d) 13. (a)

8. (a) 14. (a)

9. (a) 15. (c)

10. (b) 16. (b)

11. (a) 17. (d)

12. (c) 18. (c)

19. (b)

20. (a)

21. (b)

22. (d)

23. (b)

24. (d)

25. (b)

26. (a)

27. (b)

28. (c)

29. (c)

30. (d)

31. (d)

32. (a)

33. (a)

34. (c)

35. (b)

36. (a)

37. (c)

38. (a)

39. (d)

40. (c)

41. (b)

42. (b)


196 43. (c)

44. (a)

45. (d)

46. (a)

47. (c)

48. (b)

49. (b)

50. (c)

51. (a)

52. (c)

53. (b)

54. (b)

55. (b)

56. (d)

57. (b)

58. (b)

59. (d)

60. (b)

61. (a), (d)

62. (a), (d)

63. (a), (c)

64. (a), (b), (c)

65. (a), (b), (c), (d)

66. (a)

67. (d)

68. (b), (d)

69. (a), (b), (c)

70. (b), (c)

71. (a), (c)

72. (a), (c)

73. (a), (c)

74. (a), (d)

75. (a), (b)

76. (a), (c)

77. (c), (d)

78. (a), (c)

79. (a), (b)

80. (a), (d)

81. (a), (b), (c), (d)

82. (b)

83. (b), (c)

84. (b), (d)

85. (a)

86. (a), (c), (d)

88. (a), (c)

89. (b)

90. (c)

1. (a), (b), (c)

2. (a)

3. (d)

4. (a)

7. (a) (b)

8. (a) (c)

9. (a)

10. (d)

87. (b)

5. (d)

6. (d)

CONIC SECTIONS-PARABOLA

Eccentricity e < e

2hxy+

abc

& ab-h2

2gx+2fy+

fgh - af

2

-b^-ct?

=

& ab-h2*0 & ab < h 2 & ab > h 2

, h =0,a=

b

ab - h 2 ab - h 2 > ab - h 2 < *

ab - h 2 <

b

+ 2hxy+by^+ H

= 2ax 2hy+2g,^ +

2gx + 2fy+c =2hx+2by+2f

198

Objective Mathematics o

o 0,

(x,y) = (xi ,yi).

M

P J

L

II « + K

X' -

z

A

S (a, 0)

axis

L'

r

Y M c •

Sj(-a,0) J

Y

\ \

S (0. a) J

L'V

\

// L

N

/

\

A

y+a=0 Y'

P

I I I I h M

A

o II M I *

z

Conic Sections-Parabola

199

-f.

~2 "

TT

- a) + ( y - br =

Ix+my+n

=>

V

.to

o

=>

.2 , . .2 ( lx+my+n ) ( x - a) + ( y - b) = ±— 2

2

I + rrf

+ ^ y 2 - 2/mxy +x ter m + y term + cons tant = 0 This is of the form

(.mx-ly)2 + 2g x+ 2/ y+ c = 0. This equation is the general equation of parabola.

: (i) Equation of the parabola with axis parallel to the x-axis is of the form (ii) Equation of the parabola with axis parallel to the y-axis is of the form y =

The parametric equations of the parabola y 2 = 4axare where f is the param eter. Since the point (at2 , 2 at ) satisfies the equation y 2 = 4ax, therefore the parameteric co-ordiantes of any point on the parabola are Also the point (at2 , 2at) is reffered as f-point on the parabola.

Now

willlie

or

Let be any point , the parabola according as

Let the parabola be y 2 = 4a x and the given line be y= mx + c . m

The equation of the tangent at any point (xi , yi) on the parabola y yyi = 2a(x+xi) 2a Slope of tangent is — • (Note)

y n t

2

m

= 4ax is

: Equation of tangent at any point't' is ty = x + at2 Slope of tangent is -: Co-ordinates of the point of intersection of tangents at 'fi ' and 'fc ' is {afi tz , a (fi + fc)} 3 : If the chord joining 'fi' and 'f 2 ' to be a focal chord, then fi tz = - 1.

(a_

Hence if one extremity of a focal chord is (atf , 2a fi ), then the other extremity (at2 , 2afc) bec ome s 2a

f t 2 '"*


200

y

yi =

yi ,

yi

fi

SSi = T

2


201

p (*i, yi)

Pole

Polar

(X2,y2),

Y,

(X2,y2)

202


MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. If P and Q are the points

(atx , 2at{)

and

(at2 . 2at 2) and normals at P and Q meet on the parabola y

= 4ax, then txt2 equals

(a) 2

(b) - 1

(c)-2 (d) - 4 2. The locus of the points of trisection of the double ordinates of the parabola (;a)y' (c) 9 y

y

= 4 ax is

(b) 9 / = 4 ax

= ax = ax

(d) y

- 9 ax

3. If the tangents at P and Q on a parabola meet in T, then SP, ST and SQ are in (a) A.P.

(b) G.P.

(c) H.P .

(d) None of these

If the normals at two points P and Q of a parabola y = 4 ax intersect at a third point R on the curve, then the product of ordinates of P and Q is (a) 4a 2

2

(b) 2a 2

2

(c) - 4a (d) 8a 5_ The point (— 2m, m + 1) is an interior point of the smaller region bounded by the circle 2 2 2 x + y =4 a nd the parabol a y = 4x. Then m belongs to the interval (a) - 5 - 2 V6 < m < 1 (b) 0 < m < 4 , , , 3 (c) - 1 < m < ~ (d) - 1 < m < - 5 + 2 VfF 6. AB, AC are tangents to a parabola v = 4ax, are Pi'Pi'Pi 'he lengths of the perpendicu lars fr om A,B,C on any tang ent to the curve, then p2,px , p3 are in (a) A.P. (c )H .P .

(b) G.P. (d) None of these

7. If the line y - VTx + 3 = 0 cuts the parabola y = x + 2 at A and B, then PA .PB is equal to P = 2) [where (V3~, 0)] 4 (V3~+ 4 (2 - VT) (a) (b)

3

(c)

4V3" 2

(d)

2(V3~+2) 3

8. The normals at three points P, Q, R of the parabol a y = 4ax meet in (h, k). The centroid of triangle PQR lies on (a) x = 0 (c)x = -a

(b) y = 0 (d)y = a

A and B on the parabola at y = 4 ax intersect at point C then ordinates of A, C and B are

9.If t angents

(a) Alwa ys in A.P. (c) Alwa ys in H.P. 10. The

condition 2

(b) Alwa ys in G.P. (d) Non e of these

that

the

parabo las

2

y =4c[xd) and y = 4 ax have a common normal other then x-axis (a > 0, c > 0) is (a) 2a < 2c + d (b) 2c < 2 a + d (c) 2 d < 2a + c (d) 2d < 2c + a 11. A ray of light moving parallel to the x-axis

gets reflected from a parabolic mirror whose (y — 2)=4 ( x+ l ) . equation is Af te r reflection, the ray must pass through the point (a) (-2, 0) (c) (0, 2)

(b) (-1,2) (d) (2, 0)

P, Q, R of the = 4ax meet in a point O and S be its focus, then I SP i . I SQ I . I SR I =

12. If the normals at three points,

parabol ay (a) a2

(c) a (SO?

(b) a (SO? (d) None of these

13. The set of points on the axis of the parabola

y - 4x - 2y + 5 = 0 fro m whic h all the thre e normals io the parabola are real is (a) (k, 0),k>\ (b) (k, 1); k> 3 . (c) (k, 2); k> 6 (d) (k, 3); k > 8 14. The orthocentre of a triangle formed by any three tangents to a parabola lies on (a) Focus (b) Directri x (c) Vertex (d) Focal chord


203

15. The vertex of a parabola is the point (a, b) and latus rectum is of length I. If the axis of the parabola is along the positive direction of y-axis then its equation is

(a)(x-af

=

(b )(x-a)2

^(y-2b)

=

22. The equation to the line touching both the 2

pa rabo la y = 4ax cuts the curve again at the point whose pa ra me te r is

^(y-b)

)(x-a) =l(y-b)

( a ) -1/t

(d) None of these 16. The equation V(x-3)

2

+ (j' -l)

+ V(x+3)

2

+ (>'- l)

2

= 6

represents (a) an ellipse (b) a pair of straight lines (c) a circle (d) a straig ht line joi nin g the point ( - 3, 1) to the point (3, 1) 17. The condit ion that the straight line

lx+my + n = 0 touches the parabola x = 4ay is (a) bn = am (c) In = am

2

2

(b) al - mn = 0

2

2

(-1, 1) and directrix is 4x + 3y - 24 = 0 is (a) (0, 3/2) (b) (0, 5/2) (c) (1, 3/2 ) (d) (1,5 /2) 19. The slop e of a chord of the parabo la y = 4 ax, which is normal at one end and which subtends a right angle at the srcin, is (b) \
2

2

(b) acose c a sec 2 (d) acos a

2

a

(a) a = 2b 2

(c) a = 2b

(b) 2a = b (d) 2a = b

24. If >>],y2 are the ordinates of two points

P and Q on the parabola and y 3 is the ordinate of the point of intersection of tangents at P and Q then (a) y\> yi< y?>ar e in AP - (b) >1, y?,, yi are in A.P. (c) y,, y 2, y 3 are in G.P. (d) y x, y3, y2 are in G.P. 2

focal chord of the parabola y = 4ax is (a) 4a (c) a

(b) 2a (d) a 2

27. A double ordinate o f the parabola y = 8 px is of length 16p. The angel subtended by it at the vertex of the parabola is

(a) 7t/4 (c) 71

(b) (d)

TT /2 71 /3

y = 4x + 8 from which the 3 normals to the parabola are all real and di ff eren t is (a) {(*,0) :k<-2) (c) {((),k): k>-2) 29. If

y+

(b) {(it, 0 ) : k>-2] (d) Non e of these

b = OT,(x + a) and y + b = m 2 (x + a)

y = 4x,

(a) m | + m2 0= (c) m j m2 = —I

(b)

mt m2 = 1 • (d) None of these

30. The length of Ih c latus rcctum of the para bola 16 9 {(x

2

2

25. The equation ax + 4xy +y +ax + 3y + 2 = 0 represents a parabola if a is (a)-4 (b)4 (c) 0 (d) 8

arc two tangents to th e parabola y = 4ax then

21. If (a, b) is the mid-point of chord passing through the vertex of the parabola then

(d)t + ~

28. The set of points on the axis of the parabola

20. Let a be the angle which a tangent to the para bola y = 4 ax makes with its axis, the distance between the tangent and a parallel normal will be (a) asin a cos a 2 (c) a tan a

j

26. The Harmonic mean of the segments of a

(b) am = In

18. The vertex of the parabola whose focus is

(a)l /<2 (c) 2

(b )- ^f + y

(c)-2 1 + - { 2

(at , 2 at ) on the

23. The normal at the point

2

(c

2

para bolas y = 4x and x = - 32y is (a) x + 2y + 4 = 0 (b) 2x + )' - 4 = 0 (c)x-2y-4 =0 (d) x — 2y+ 4 = 0

(a) 14/13 (c) 28/13

!) 2 + ( Y - 3 ) 2 } = (5 JC - 12 y+ 1 7) 2 i s



204 31. The poi nts on the axis of the parabo la 2 3y + 4y - 6x + 8 = 0 f ro m whe n 3 d ist inct normals can be drawn is given by

(a)[ a, ^ |; a > 19/9 2

(b)|

line y = x0 has the greates t area if x 0 = (a)0

(b)l

(c) 2

'Cd) 3

2 34. If the normal at P 'f ' on y = 4 ax meets the curve again at Q, the point on the curve, the normal at which also passes through Q has co-ordina tes ( , ). 2a 2a ^ 4 a 2a

19

1 ^ 7 '3]'a>9 (d) None of these

(c)f

is x 0 (X0 G [1, 2]), the y-axis and the straight

a

(a)

32. Let the line Ix + my = 1 cut the parabola

y = 4 ax in the points A and B. Normals at A and B meet at point C. Normal from C other than these two meet the parabola at D then the coordinate of D are ., , f 4am 4a (a) (a, 2a) I V r 2 am la 4 am 4 am (d) (c) ~T I ~ T 12 33. The triangle formed by the tangent to the 2 parabola y = x at the point whose abscis sa

(c)

2

t 4a

t 4a t

t 4a

(d)

t2

t 8a t

35. Two pa rabol as C and D intersect at two differen t points, where C is y = x - 3 and D is y = kx'. The intersection a t which th e x value is positive is designated point A, and x = a at this intersection, the tangent line I at A to the curve D intersects curve C at point B, other than A. If x- value of point B is 1 then a = (a) 1 (c )3

(b) 2 (d) 4

MULTIPLE CHOICE -II Each question in this part has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s): 36. Consider a circle with its centre lying on the focu s of the parabo la y = 2 px such that it touches the directrix of the parabola. Then a point of int ersec tion of the cir cle and the parabola is

(a) (c)

(b)

f.P

the

parabolas

(d) its latus rectum = 2a 39. The equation of a tangent to the parabola y = 8x which make s an angle 45 (a) 2x + y + 1= 0

with the

(b) y = 2x + 1

(c) x — 2y + 8 = 0 2 *

y= 4 ( x + l )

(d) x + 2y - 8 = 0

normal y - nvx — 2am - arn to the 2 parab ola y = 4 a x subte nds a right ang le at the vertix if (a) m = 1 (b) m = \r2

The

37. The locus of point of intersection of tangents to

(c) its latus rectum = a

line y = 3x + 5 is

E 2 '

(d)

2 '

(b) it is a parabola

and

y = 8 (x + 2) which are perpendicular to each other is

38. The

(a) x + 7 = 0

(b) x — y = 4

(c) x + 3 = 0

(d) y — x = 12

equati on

of

a

y 2 + 2ax + 2 by + c = 0. Then (a) it is an ellipse

locus

(c) m = - V2 41. The is

straight

(d) m = <2 1 line

x +y =k

touches

paraboa y = x — x 2 if (a) 0 k= ~(c) = 1k

(b) (d)

k=- 1 k takes any value

the


205

42. A tangent to a parabola y = 4 ax is inclined at 71/3 with the axis of the parabola. The point of contact is

(a) (a/3 - 2a/-IT) (b) (3a, - 2 -ITa) (c) (3a, 2-IT a) (d) (a/3, 2a/-IT) 43. If the normals f rom any point to the parabola x = 4 y cuts the line y = 2 in points whose abscissae are in A.P., then the slopes of the tangents at the three conormal points are in (a) A.P. (b) G.P. (c) H.P.

(d) Non e of these

44. If the tangent at P on y2 = 4ax meets the tangent at the vertex in Q, and S is the focus of the parabola, then Z.SQP = (a) TC /3 (c) 7t/2

(b) rc/4 (d) 27t/3 2

(a,i

(c) a

(b) (-16, 8) (d) (16, 8)

48. A line L passing through the focus of the para bo la y = 4 (x — 1) intersects the parabola in two distinct points. If 'm' be the slope of the line L then (a) m e ( - 1 , 1) (b) m e ( - oo, - l ) u ( l , ° o ) (c) me/? (d) None of these

x = ay2 + by + c is

+ —^r =

(a) a / 4

(b) a / 3

(c) 1/ a

(d) l / 4 a

50. P is a point which moves in the x-y plane such that the point P is nearer to the centre of a square than any of the sides. The four vertices of the square are (±a,±a). The

(b)

a

(a) (- 16 ,- 8) (c) (16, - 8 )

49. The length of the latus rectu m of the.par abola

45. A focal chord of y = 4 ax meets in P and Q. If S is the fo cus , then

Let PQ be a chord of the parabola y 2 = 4x. A circle drawn with PQ as a diameter passes through the vertex V of the parabola. If Area of A PVQ = 20 unit^then the co-ordinates of P are

(d) Non e of these 2

y = 6x 46. The diameter of the parabola corresponding to the system of parallel cho rd s 3x - y + c = Q, is (a) y - 1= 0 (b) y - 2 = 0 (c) y + 1= 0 (d) y + 2 = 0

P will region in which bounded par ts of parabolas of move whic his one has by the equation (a) y 2 = a + 2 ax 2

(c)y +2 ax = a

2

(b) x = a + 2ay (d) None of these

Practice Test M.M : 20

Time : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] 2 2 1. If th e no rm al s from an y point to t he distance from the circle x + (y + 6) 1, pa rab ol a x = 4y cut s th e line y = 2 in are po in ts wh ose ab sc is sa e are in A.P ., then th e (a) (2, - 4) (b) (18, - 12) slopes of the tangents at the three (c) (2, 4) (d) No ne of th es e co-normal points are (a) A.P. (c) H.P . 2. The c oor dina tes 2

in (b) G.P. (d) Non e of th es e of th e poin t on

pa rab ol a y = 8x, which is

at

3. The figure shows the graph of

t he

mi ni mu m

th e p arab ola

y = ax + bx + c then (a) a > 0 (b) 6 < 0 ,2 (d) b - Aac > 0 (c)c > 0

206

Objective Mathematics (a) - 2 < T < 2

4. T he equatio n of th e parabol a whose vertex and focus lie on the axis of x at distances a and cij from the srcin respectively is

(b) T € ( -

2

(a)y

= 4 (a

x

(d) T2 > 8

- a) x

2

(b) y

= 4 (a 1-a) (x -a)

(c) y 2 = 4 (a1-a) (x - a ^ (d) None of these 2 5. If (a , a - 2) be a point inte rio r to th e regio n of the parabola y = 2x bounded by the chord joining the points (2, 2) and (8, -4) then P belongs to the interval (a)-2+ 2 V2" - 2 + 2 V2" (c) a > - 2 - 2 V2~ (d) None of these 6. If a circle and a parabola intersect in 4 po in ts t he n th e al ge br ai c sum of the ordinates is (a) proportional to arithmatic mean of the radius and latus rectum (b) zero (c) equal to the ratio of arithmatic mean and

8. P is the point't' on the parabola y = 4ax an d PQ is a focal chord. PT is the tangent at P and QN is the normal at Q. If the angle be tw ee n PT an d QN be a and the distance be tw ee n PT an d QN be d then (a) 0 < a < 90° (b) a = 0° (c) d = 0

Q

2

(a)

(aa' - bb' f , 2

,2,3 /2

(bb' - aa')2 2

(b = 4ax

2

(b)

(a +a )

2

y

1 VT + t

^ For parabo la x + y + 2xy - 6x - 2y + 3 the focus is (a) (1, - 1) (b) ( - 1, 1) (c) (3, 1) (d) Non e of th es e 10. The lat us rectum of the parab ola x = at2 + bt +c,y = a't 2 + b't + c' is

(c)

(d) None of these 7. If the normal to the parabola

VT+ t2 +

(d) d = a

latus rectum

the point

- 8) u (8,

(c) T2 < 8

2 3/2 +

b' )

(ab' - a'b f , 2 ,

,2,3 /2

(a + a ) (d)

(a'b - ab')2 2

2 3/2

(ib + b' )

at

(at , 2 at) cuts the parabol a again

a t (aT2, 2 a T ) then Max. Marks 1. First attempt 2. Second attempt 3. Third attempt

must be 100%

Answers

1. (a) 7. (a)

2. (b) 8. (b)

3. (b) 9. (a)

4. (d) 10. (a)

5. (d) 11. (c)

6. (b) 12. (c)

13. (b)

14. (b)

15. (b)

16. (d)

17. (b)

18. (d)

19. (b)

20. (b)

21. (d)

22. (d)

23. (b)

24. (b)

25. (b)

26. (b)

27. (b)

28. (d)

29. (c)

30. (c)

31. (b)

32. (d)

33. (c)

34. (c)

35. (c)

0,


36. (a), (b)

37. (c)

38. (b), (d)

39. (a), (c)

40. (b), (c)

41. (c)

42. (a), (d) 48. (d)

43. (b) 49. (c)

44. (c) 50. (a), (b), (c)

45. (a)

46. (a)

47. (c), (d)

2. (a) 8. (b)

3. (b.) (c), (d) 9. (d)

4. (b ) 10. (b)

5. (a)

Mb) 7. (d)

6. (b)

ELLIPSE

a

b

(± a , 0).

,

0).

f

I

a

f

J I

/

a

J

K

tT a

)

Ellipse

209

4

+ 4

a

b

=1

= L1L1' =

F(*,y)

,2

„

, ,2

2 2

2

{I + m )

4 . 4 =1

0

X2

"2 + 2

a

b

2

M

210

Objec tive Mathe matic s

a2

fa2

2

(a^m +1?) £ + 2mca 2x x

i.e.,

+ a2

- b2) = 0

a2rrP y = mx ± c 2 = a 2 ™ 2 + fa2.

± a2m

= mx ± ^{a2rr? +

2

^ a rrf + fa

± fa2 2

2

' ^ a m 2 + fa2

lx+ my+

a

fa

aryi

v - COS w . ,

rmol sat an\/ r»r\int ( Vi

\ r»n tho ollinco

= a

-

x2

tr

ty = a 2 - b2

ax i.e.,

2

J

fa""|

a

2

tr

]

— [ 2

_u

^

a

b

Ellipse

211

S

1

. 4

+

4 _ 1 = o

XX1

4 * 4 -i. SSi =

x?

v?

212


t?

yyi

1

b2

ePm

y = £2 a

j(2 a

tr

to

2rm,

ne

x2

f

a

tr

I.

a + P + y + 5 = (2n + 1) n , n e I

y (P + y)

(y + a)

(a + P) = 0.

Ellipse

213

19. Reflection Property aof n Ellipse : If an incoming light ray passes through one focus (S) strike the concave side of the reflected towards other focus (S') & Z Z

ellipse then it will get

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. Let P be a variable point on the ellipse 2

25

The equation

2

+ 2- = \ 16

w ith

foci at S and 5'. If A be

PSS*, then the maximum

(a) 24 sq. unit s

(b) 12 sq. uni ts

(c) 36 sq. units

(d) Non e of these

(a)< a4

(a) b/a

(b) 2a/b

(c) cf/b2

(d)

b2/a2

3. The line Ix + my + n = 0 is a normal to the 2 1 elli pse ^ + = 1 if a b b (az~br (a)+ — r = m

a > 4

(b)

(c) 4 < a< 10

2. The area of a triangle inscribed in an ellipse be ars a const ant ratio to the area of the triangle formed by joining points on the auxiliary circle corresponding to the vertices of the first triangle. This ratio is

a' (c) — /" id) None

= 1 represen ts

4-a

an ellipse if

the area of triangle value of A is

,, v (I (b)— +

10-a

a > 10

(d)

5. The set of values of a for which (13.x- l) 2 + (13 y- 2) represents an ellipse, is

= a (5.v + 1 2 y - 1)

2

:

(a) 1 < a < 2 (b) 0 < a < 1 (c) 2 < a < 3 (d) None of these 6. The length of the commo n cho rd of the ellipse

(•v-ir 9

(y-2)4

= 1

and

the

circ le (.v - 1)" + ( y - 2)" = 1 is (a) zero

(b) one

(c) thr ee

(d") eight

7. If CF is the perpendicular from the centre C

m b2 (a2-l>2)2 — = iiiII of these

of the ellipse

= 1 on the tansrent at

any point P. and G is the point when the normal at P meets the major axis, -then CF.PG

=


214

13. If CP and CD are semi-conjugate diameters

(b) ab

(a) a 2

(c )b

(d ) b

2

3

8. Tangents are drawn from the points on the line x - y - 5 = 0 to x + 4y 2 = 4, then all the chords of contact pass through a fixed point, wh ose coord ina te are (a) (b) Wl

(d) None of these 9. An ellipse has OB as semi-minor axis. F and F' are its focii and the angle FBF' is a right angle. Then eccentricity of the ellipse is (b) 1/2 (d) None of these

10. The set of positive value of m for which a line with slope m is a common tangent to ellipse 2

2

x y 2 —r + ^ r = 1 and para bola y = 4 ax is given a b

by (a) (2,0)

(c)

(c) a - b

1

a2+b2

(d) 4a2 + b 2

14. A man running round a race course notes that the sum of the distances of two flag-posts from him is always 10 metres and the distance between the flag-posts is 8 metres.

(a) 15:t (c) 187t

(b) 12n (d) 8it

15. If the normal at the point P (<(>) to the ellipse

= 1 intersects it again at the point 14 5 Q (2<()), then cos
2

x y —z + ^ = ' > then

(b)(3, 5) (d) None of these of the ellipse

ax2 + by2 + 2fx + 2 gy + c = 0 if axis of ellipse parallel to x-axis is

(b)

(b)

2

a focal chord of the ellipse

(c)(0, 1) 11. The eccentr icity

(a)

(a) a + b

The area metres is of the path he encloses in square

7--T

(a) 1/ VT (c) 1/ V f

2

of the elli pse ^ + ^ = 1, then CP 2 + CD2 = a b

W

tan — a tan P^ .is equal to e-1 e+ 1 e-1 (d) e+3

1 -e 1+e e+1 (c) e-1

(b)

(a)

2

17. The ecc ertricity

+ b

a" tf whose latusrectum is half of its minor axis is

+b

1

(a)

= 1

(b)

12

None of these 12. The minimum length of the intercept of any 2

tangent on the ellipse

of an ellipse

2

_ 1 between L. =

(c)

(d) None of these

T

18. The distances fro m the foci of P (a , b) on the 2

the 2a coordinate axes is (b) 2b (a) (c )a-b (d )a + b

2

ellipse

+

(a)4±|

b

(b )5 ±j a

= 1 are

215

Ellipse (c)5

±~b

(d) None of these 19. If the normal at an end of a latus rectum of 2

2

x y the ellipse — 2 + 2 = 1 passes through a

one

b

extremity of the minor axis, then eccentricity of the ellipse is given by

the

(a) e 4 + e1 - 1 = 0 2

(b) e +<2e — 4 = 0 (c)e = (d) e = 3je 20. If A and B are two fixed points and P is a variable point such that PA + PB = 4, the locus of P is (a) a parabola (b) an ellipse (c) a hyperbola (d) None of these 21. The area of the parallelogram formed by the tangents at the ends of conjugate diameters of an ellipse is (a) constant and is equal to the product of the axes (b) can not be constant (c) constant and is equal to the two lines of the produ ct of the axe s (d) None of these 22. If

C 2

be

t he

ce ntre

of

the

elli pse

2

9x + 16v = 144 and S is one focus, the ratio of CS to major axis is (a) VT: 16_ (b) :4 (c) < 5 : VT (d) None of these 23. The (JC

+ y-

(a) (0, 0) (c) (0, I)

centre 2)

2

(x -

of Y)

2

the

ellipse

, .

(b)(1, 1) (d) (1, 0)

24. The radius of the circle passing through the 2

2

JC y foci of the ellipse T7 + "ir = 1. and having its 16 9 centre (0, 3) is (a) 4 (b) 3 (c) V 12 (d) 7/2

25. The length of the latus rect um of an ellips e is one third of the major axis, its eccentricity would be (a) 2/3 (b) 1/V3" (c) 1/V 2 (d) <273

26. If the length of the maj or axis of an elli pse is three times the length of its minor axis, its eccentricity is (a) 1/3 (b) 1/V3~ (c) 1/V2" (d) 2 / V 2 / 3 27. An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respe ctive ly in cms. are (a) 6, 2 <5 (b) 6, V5 (c) 4, 2 VJ (d) None of these 28. The locus of mid-points of a focal chord of 2

2

JC y the elli pse — + = 1 is a b

2

2

, . x v ex (a)—+ —=2 — a b a 2 ex x •> (b) — a - ^b2 = / V 2 2 2 (c).v +y =a

— a , 2 +b

(d) None of these 29. The locus of the point of intersection of 2

tangents

to the ellipse

a'

2

+

b~

which

meet at right angles, is (a) a circle (b) a parabola (c) an ellipse (d) a hyperbola 30. The number of real tangents that can be 1 2 drawn to the elli pse 3.v + 5y" = 32 pas sing through (3, 5) is (a) 0 (b ) l (c)2 (d)4 31. Equation to the ellipse whose centre is (-2 . 3) and whose semi-axes are 3 and 2 and major axis is parallel to the .v-axis, is given by


216 (a) 4x + 9y 2 + 16x - 54y - 61 = 0

(a) n/4'

(b) n/2

(b) Ax + 9y 2 - \6x + 54y + 61 = 0

(c) n/3

(d) 71/8

(c) Ax + 9y 2 + I6x - 54y + 61 = 0 (d) None of these

Eccentricity of the ellipse, in which the angle between the straight lines joinin g the foci to an extremity of minor axis is n/2, is given by (a) 1/2 (b) 1/V2

foci

of

ellipse

the

(c) 1/3

(c) (-1,-2) and (-2,-1) (d) (-1, -2) and (-1,-6)

(a)ZPOS

(b) ZPSA

Tangents drawn from a point on the circle

(c) ZPAS

(d) None of these

2 2 x +y = 4 1 to the ellipse+

x

2

(d)

1/43

25 {x + l) 2 + 9 (y + 2) 2 = 225, are at (a) (- 1,2 ) and (- 1, -6 ) (b) (-2, 1) and (-2, 6)

If O is the centre, OA the semi-major axis and S the focus of an ellipse, the ecentric angle of any point P is

2

y th

en

tangents are at angle

MULTIPLE CHOICE

= 11

Each question in this part has one or more than one correct answer(s). For each question write the letters a, b, c, d corresponding to the correct answer(s). (c) perpendicular to the major axis (d) parallel to the minor axis.

The locus of extremities of the latus rectum of the family of ellipse (a) x —ay = a

/

2

2

(b) x - ay = b

2

/ \ 2 ,

2

22

b x +y

2

22

= a b is If

of

the is

ellipse 1/2

then

(a) 7t/12 (b) 7t/6

The distance of the point (V^Tcos 0, V2~sin 0) 2

2

x y on the ell ips e — + = 1 fr om the cent re is 2 6 2 if (a) 0 = 71/2 (b) 0 = 3 T I / 2

(c) 0 = 571/2 (d) 0 = 77t/2 38. The sum of the square of perpendiculars on 2

2

x v —^ + ^ = 1 from a b two points on the minor axis, each at a distanct ae from the centre, is any tangent to the ellipse

(b) 2b

b2

rectum

a (0 < a < 7t) is equal to

(c)x 2 +ay = a 2 (d) x + ay = b

(a) 2a

latus

x tan 2 a + y 2 seca2 = 1

(c) a + (d) a2-b2 39. A latus rectum of an ellipse is a line (a) passing through a focus (b) through the centre

(c) 5 TT /12

(d) None of these In the ellipse 2 5 / + 9y 2 - 1 50A:- 90y + 225 = 0 (a) foci are at (3, 1), (3, 9) (b) e = 4/5 (c) centre is (5, 3) (d) major axis is 6 Equation of tangent to 2

(d)y = -x-4(U) The equation of tangent 2

the

ellipse

2

jc/9+y/4=l which intercepts on the axes is (a) y = x + 4(13) (b)y = -x + 4(U) (c)y =x- 4(13)

cut

off

to the

equal

ellipse

2

x + 3y = 3 which is perpend icular to the line 4y =x — 5 is

217

Ellipse (c) sec a + sec P sin a + sin P (d) sin (a + P)

(a) 4x + y + 7 = 0 (b)4x + y - 7 = 0 (c) 4JC + y + 3 = 0 (d) 4x + y - 3 = 0 If P (9) and <2 x

2

y

2

elli pse — + a

PQ is 2 , . x

a a

= 1, locu s of the mid- point of b

y

2

Z 2

2

2

x + 3y = 37 be paral lel to the line : (a) (5, 2) (b) (2, 5) (c) (1, 3) (d) (- 5, - 2) The le ngth of the ch ord of the e llips e +^

, 1

b 2

The points where the normals to the ellipse + 9 j are tw o poi nts one the

z

= 1 wher e mid- poin t is ^- , - j s (b) V8161

(a)-J _

2

10

'10

/z= 4 b

(c)

.2

* 2 ' ' ^.2= 2 a i> (d) None of these

V8061

(d) None of these

10

x

For the elli pse

y

+ a

the equat ion of b

An ellipse slides between two perpendicular straight lines. Then the locus of its centre is

the diameter conjugate to

(a) a parabola (b) an ellipse (c) a hyperbola (d) a circle

(b) bx — ay = 0 (c) a'y + b x = 0 (d)ay -b 3x

If a, P are eccentric angles of the extremities of a focal chord of an ellipse, then eccentricity of the ellipse is

ax — by = 0 is

(a) bx + ay = 0

= 0

The parametric representation of a point on the ellipse whose foci are (—1,0) and (7, 0) and eccentricity is 1/2, is (a) (3+ 8 cos 9, 4 V I sin 9) (b) (8 cos 9, 4 V I sin 9) (c) (3 + 4 VI cos 9, 8 sin 9)

(a)- cos a + cos P cos (a + P) sin a - s i n P (b)sin ( a - P)

(d) None of these

Practice Test

M.M : 20

Tim e : 30 Min.

(A) There are 10 parts in this question. Each part

one or more than one correct

answer(s). [10 x 2 = 20]y

1. Let

, F 2 be two focii of the ellipse and

PT and PN be the tangent and the normal respectively to the ellipse at point P. Then (a) PN bisects

ZF]PF2

(b j PT bisects

ZFXPF2

(c) PT bisects angle (180* (dj None of these

2

Let E be th e ellips e ^ 2

2

+

C be

the circle x + y = 9. Let P and Q be the po nt s (1, 2) and (2, 1) res pe ct iv el y. Th en fa ) Q lies inside C but outside

/b\PF.j)

= 1 and

2

fb ) Q lies outside both

Ic) P lies inside both

E

C and E C and E

(d) P lies inside C but outside

E

218

Objective Mathematics P (a cos 0, b sin 0) of

3. The tangent at a point 2

an ellipse

+ ^ a

= 1 mee ts its auxi liar y (c)

b

circle in two points, the chord joining which subtends a right angle at the centre, then the eccentricity of the ellipse is (a) (1 + sin

2

1 <(i)~

(b) (1 + sin ()>) (c) (1 + sin

2

(a)

2

-1 /2

<) >)'

3/2

3 V2" 7

(b)

Vf

2 V3

(d) None of these

7

7. The eccentricity of an ellipse whose pair of a conjugate diam eter are y = x and 3y = -2xis (a) 2/3 (b) 1/3 (c) 1/V3" (d) Non e of th es e 8. The eccentric angles of the extremities of

2

2 (d) (1 + sin <(>)" 4. If (5, 12) and (24, 7) ar e th e focii of a conic pas si ng through t he orig in th e n the eccent ricity of conic is

(a) (c)

V386~

(b)

38 V386

(d)

13

12

(b)tan

>/386 25

x

v

x

-1

1

a

1

2

—+ ^

(c) tan" 1

,

2

1 on the axis of* v

(d) tan

ae

-1

["fee 9. A la tu s rect um of an ellipse is a lin e (a) passing through a focus (b) perpen dicula r to the maj or axis (c) parallel to the minor axis (d) through the centre 10 The tangents from which of the following

o - tr5 = 1 on th e axis and the straight line — of y an d who se axes lie alon g the axe s of coordinates, is

po in ts to th e ell ipse 5x 2 +4y 2= 20 ar e pe rp en di cu la r (a) (1, 2 V2) (b) (2 V2, 1) (c) (2, VS) (d) (V5, 2)


must be 100%

Answers 1- (b) 7. (c) 13. (b)

2. (a) 8. (b) 14. (a)

= 1 are

1

(a) tan "

V386~

5. AB is a d iam et er of x + 9y =25. The eccentric angle of A is n/6 then the eccentric angle of B is (a) 5 n / 6 (b) 5n/6 (c )- 2ji /3 (d) None of these 6. The eccentricity of the ellipse which meets the straight line

.2— la tu s rec tum to the elli pse —^ a given by

3- (b) 9. (c) 15. (a)

4. (a) 10. (c) 16. (b)

5. (b) 11. (a) 17. (d)

6. (a) 12. (d) 18. (c)

219

Ellipse 19. (a) 25. (d) 31. (c)

20. (b) 26. (d) 32. (a)

21. (a) 27. (a) 33. (b)

22. (d) 28. (a) 34. (b)

23. (b) 29. (a) 35. (d)

24. (a) 30. (c)

36. (a), (c)

37. (a), (b), (c), (d)

38. (a)

39. (a), (c), (d)

40. (a), (c)

41. (a), (b)

42. (a), (b), (c), (d)

43. (a), (c)

44. (a)

45. (d)

46. (d)

47. (a, d)

49. (c)

50. (a)

48. (d)

1. (a), (c)

2. (d)

3. (b)

7. (c)

8. (c)

9. (a), (b), (c)

4. (a), (b) 10. (a), (b), (c), (d)

5. (b)

6. (d)

HYPERBOLA 2

2

x - î/ = 1 , whe re a & The gener al form of sta nda rd Hyperbola is : —a tr

are constants. (Fig. 1)

X

lB

B S b

| j

M' 'J

4

! (a, 0) / zj AI

' " " T o . 0) C

z' (- ae, 0) S' ) A' / ( - a, 0)

axis \ S (ae, 0)

®

^ ^

B I II H

B'

1

1 Fig. 1

(i) and AS = (ii) Co-ordinate of centre C(0, 0). (iii) = 2a is the (iv) 6 6 ' = is the (v) Co-ordinates of a nd (vi) Relation in a , (vii) Co-ordinates of the

is

of

and 6 ' ar e (0, ±

= a 2 (e 2 - 1) and S ' are (±

(viii) Co-ordinates of the (ix) Equation of

of the Hyperbola. of the Hyperbola. ' are (± a , 0) &

, 0) ar e ^±

,0

x = + a/e

(x) Equation of

and length

/_iLi'

2 tf 3. i, 2

(xi) (xii)

L-1-.-I

the latus rectum, are

, Li

an d Transverse axis.

a

and Li'

-

Hyperbola

221

Let (a , b ) be the focus S, and lx+ my + n = 0 is the equation of directrix, let P (x, y) be a ny point on the hyp erbo la, (Fig. 2) then by definition. => SP = e PM (e > 1)

,2 , . ,2 (x-af + (y - bf =

=>

(/2 + m2) {(x- a)

=>

2

—

e2(lx+mv+nf — 2 (r + rrr)

S (a, b)

e2(lx+my+n)2.

+ (y- b) 2} =

Fig. 2

X2

The parametric equations of the hyperbola

p (x y)

are x = a s ec <

y = btan is the para mete r. Since the point (a s ec 0 , b tan ) satisfies the equation x2 v2 — = 1, ther efore the paramet ric co-or dinat es of any point on the hyperbola is (a s e c (|>, b ta n <> | ) also a b the point (a sec 0 , b tan 0) is reffered as <> | point on the hyperbola. <> j e [0, 2K)

T

he circle described on transverse axis of the hyperbola as diameter is called auxiliary circle and so its equation is x2 + y2 = a 2 Let P be any point on the hyperbola. Draw perpendicular PN to x-axis. The tangent from N to the auxiliary circle touches at Q, P an d Q are called corresponding points on hyperbola and auxiliary circle and
x

'

The point P ( x1 , yi) lies outside, on or inside the hyperbola 4 - 4

= 1,. Si =

xf

-

- 1 > 0, = 0, < 0.

= 1 and th e given line be y = mx+ c atf Solving the line and hyperbola we get Let the hyperbola be

(mx + c) 2 a2 ~ 2

2

2

b2 2

= 1 2

2

2

i.e. Above equation being a quadratic (a rrfin -x.b ) x + 2 mca x+ a (c + b ) = 0. discriminant = b2 {( a2 m 2 - b2) - c2} a2m2 - b2 > c 2 , in one point if Hence the line intersects the hyperbola is 2 distinct points if 2 2 2 2 2 2 2 a rr?-b < c = a m - b and does not intersect if c . y = mx ± V ( a 2 m 2 - b2) touches the hyperbola and condition for tangency c

2

= a 2™ 2

b2.

222


y= mx±

a 2m sr

b

x

Ix+ my + n =

•.

x2

/

ar

tr

tfx-i

sry\ X

V

a

b

y

31

x2

/

tr

by

x2

f b

b2

c

_

51 =

x

7

?

y?

" 7

i

"

1=

0

n

x2

v2

a

tr

Hyperbola

223

SSi =

224

Objective Mathematics XXF

yyi = 1.

mx+c

= mix be b2 m/77i =—p a

a

f

x2

4 - 4

1

e-p e# =

1

~2

1 +

~2 ei

=

1

b

= 1. ,.

Hyperbola

225

x

2

b,—z

f tr V2

*>) =

*

2

-1 4 - 1

b b

xy y> = j'

ct

, -y j

x yt2 - 2ct yt - ct 4 + c

xxi - yyi = xf -

y?

tr

(?. t

226


If an incoming light ray passing through one focus (S) strike convex side of the hyperbola then it will get reflected towards other focus (S'). fig. 10. Z TPS' = ZLPM = a. M .Light ray

\

/

L

Fig. 10

MULTIP LE CHOICE - I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. X

2

"V

(c)l

2

If the foc i of the ellip se — + j 25 £

x 144 ,2 • the value of b is hyperbola

81

(a) 3 ( c) 9

25

1 and the

2

coincide, then

(b) 16 (d) 12 2

2. If chords of the hyperbola the parabola

y

2

x —y

touch

= 4 ax then the locus of the

middle points of these chords is the curve

(a)y\x

+ a) = x = x3

2

+ 2a) = 3x 3

2

3

(c)y (x

)y (x-2a)

= 2r

normals fr om a 2

po in t P on hyperbola xy = c is constant k (k > 0), then the locus of P is (a) y 2 = k2c (b ) x = kc2 (c )y2

= ck2

(d )x 2

2

= ck

2

(a - 2) x + ay =4 represents a equals rectangular hyperbola then (a )0 (b) 2

4. If

2

2

x y = —z * SUC'1 ^ a t OPQ is an equilateral 2 a b triangle, O being the centre of the hyperbola. e of the hyperbola Then the eccentricity satisfies (a) 1 < e <

3. If the sum of the slopes of the

a

2

x — -f— y = 1 fr om wh er e two pe rp en di cu la r — 25 36 tangents can be drawn to the hyperbola is/are (a) 1 (b) 2 (c) infinit e (d) zero 6. If PQ is a double ordinate of the hyperbola

2

= a

(d) 3

5. The number of point(s) outside the hyperbol

(d)e

(c) e = V 3 / 2 The equations hyperbola

2

> vr

of the asymptotes of the

2

2x + 5xy + 2y — 1l x - 7y - 4 = Oa re (a) 2x2 + 5xy + 2y2

ll ;c -7 y- 5 = 0

(b) 2x + 4xy + 2y 2 - 7x - 1 ly + 5 = 0 (c) 2x 2 + 5xy + 2y 2 - 1 lx - 7y + 5 = 0 (d) None the these

227

Hyperbola 8. The normal at P to a hyperbola of eccentricity e, intersects its transverse and conjugate axes at L and M respectively. If locus of the mid point of LM is hyperbola, then eccentricity of the hyperbola is

I -v

e +

1

/U\

(d) e

g

10. The eccentaic ity of the hyperbola asymptotes are 3JC + 4y= 2 4x - 3y + 5 = 0 is

R.

whose and

(b)2 (d) Non e of these

a variable straight line x cos a + y sin a = p, which is a chord of the 2

V

2

hyperbola —~ - 2 = 1 (b > a), subtend a right a b angle at the centre of the hyperbola then it always touches a fixed circle whose radius is ab / \ a 27 ab ,... ab (c) (d> b^b + a) 12. An ellipse has eccentiricity 1/2 and one focus at the point P (1/2, 1). Its one directrix is the common tangent nearer to the point P, to the 2 2 circle x+ y = 1 and the hyperbola 2

2

x -y = 1 . The equation of the ellipse in standard form is (a) 9x + 12y 2 = 108 (b) 9 (x— 1/3)

2 2

+ 12 (y - l) 2 = 1

(c) 9 ( x - 1/3) + 4 (y - l) (d) None of these

2

2

2-

= a is

(a) m 6 - 4m + 2m - 1 = 0 6

Let £] be the eccentricity when k = 4 and e2 be the ecc entricity whe n k = 9 then e, - e2 = (a)-l (b)0 (c) 2 (d) 3

X

well as a tangent to rectangular hyperbola

x -y

(c) m -2m

9. Consider the set of hyperbola xy = k,ke

If 11.

m will be normal to parabola y = 4 ax as

(b) m + 3m + 2m

(d) None of these

(a) 1 (c) V2"

14. The condition that a straight line with slope

2

= 36

13. The equation of the line passing through the centre of a rectangular hyperbola is x - y - 1= 0 . If one o f its asymp tote is 3 x - 4 y - 6 = 0, the equa tion of the other asymptote is

(a) 4* - 3y + 8 = 0 (b) 4x + 3y + 17 = 0 (c) 3x - 2y + 15 = 0 (d) None of the se

+1=0

=0

6 2 (d) m + 4m + 3m +1 =0

15. If e is the eccentricity of the hyperbola 2

~

2

-

= 1 and 0 is angle betwee n the

a

b

asymptotes, then cos 0 / 2 =

(a) 1-e e (c) l/e

(b . )-1 -l

e

(d) None of these

16. If H (x,y) = 0 represent the equ ation of a hyperbola and A (x, y) = 0, C (x, y) = 0 the equations of its asymptotes and the conjugate hyperbola respectively then for any point ( a , p) in the plane; H ( a , P), A ( a , P) and C ( a , P) are in (a) A.P.

(b) G.P.

(c)H .P.

(d) None of these

17. The eccentricity of the conic 4 (2y -x-3)2

-9 (2x + y—I)

(a) 2

2

= 80 is

(b) 1/2

(c) Vn/3

(d) 2.5

18. If e and e' be the eccentricities of a hyperbola ,. . 1 1 and its conjugate, then - j + =

e e (b)l (d) Non e of these

(a)0 (c) 2

19. The line x cos a + y sin a = p touches the

x

2

hyperbola —z -

a

y

2

b

^ = 1 if

2

2

2

2

2

2

2

2

(a) a cos a -b sin a

=p

2

(b)a cos a -b sin a = p 2 2 2 2 (c) a cos a + b sin a = p 2

2

2

2

(d) a co s a + b sin a = p . 2

2

20. The diameter of 16x - 9 y = 144, which is conjugate to x = 2y, is 15* (a)y = -

_

(b) y

=

32x

228

Objective Mathematics 16y

(c)x

(d)* =

32y

23. The locus of the middle points of chords of 2

21. (a sec 9, b tan 9) and ( a sec <}>, b tan ) are the X

2

V

2

ends of a focal chord of —r —~ = 1, then L iZ a b tan 9 / 2 tan <|>/2 equals to e-\ 1 -e (a) e + 1 (b) 1 +e 1 +e e+l 1 e(c) — e 22. The equation of the hyperbola whose foci are (6, 5), ( - 4, 5) and ecountricity 5 / 4 is

2

hyperbola 3JC~ - 2y + Ax - 6y = 0 parallel to y = 2x is (a) 3JC - 4y = 4 •(b) 3y - Ax + 4 = 0 (c) Ax - Ay = 3 (d) 3JC - Ay = 2 24. Area of the triangle formed by the lines x- )' = 0, jc + y = 0 and any tangent to the hyperbola JC2 -y~ 2= a 2 is 1 (a) I a I (b) j I a I (c) a

2

(d)Ia

2

C is cut by any circle of radius r in four points P, Q, R and S. Then CP2 + CQ2 + CR2 +

25. A rectangular hyperbola whose centre is 2 ( b )

2 = 1

T6"i

CS2 =

2

(x-l) _(y-5)2 16 9 (d) None of these

(c)

(a ) r

(b) 2r

(c) 3r 2

(d) 4r 2

MULTIPLE CHOICE -II Each question in this part has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). 2

2 6 . Th e equa tion 16

2

JC - 3 y - 32JC- 12y - 4 4 = 0

represents a hyperbola with

(a) length of the transverse axis = 2 <3 (b) length of the conjugate axis ='8 (c) Centre at (1,-2) (d) eccentricity = VHT 27. The equation of a tangent to the hyperbola 2

2

3JC - y = 3 , parallel to the line y = 2JC + 4 is (a) y = 2JC + 3 (b) y = 2JC + 1 (d) y = 2x + 2 (c)y = 2x- 1 28. Equation of a tangent passing through (2, 8) 2 2 to the hyp erb ola 5x —y = 5 is

(a) 3JC— y + 2 = 0 (b) 3jc + y + 1 4 = 0 (c) 23x - 3y - 22 = 0 (d) 3JC - 23y + 178

(a)/n, +m 2 = 24/11 (b)m,m 2 = 20/11 (c) m, +m

(a) a > 0, > b0 (b) a > 0, b < 0 (c) a < 0, b 0> (d) a < 0, b < 0 30. If m\ and m 2 are the slopes of the tangents to 2

2

the hyperbola x /25-y / 1 6 = 1 which pass through the point (6, 2) then

= 48/11

31. Product of the lengths of the perpendiculars drawn from foci on any tangent to the 2

hyperbola JC/a

2

2

-y /b

2

2

= 1 is

(a )\b '

(b) r

(c) a2

(d) | a

32. The locus of the point of intersection of two perpen dicula r tan gents to the hyp erbol a 2/2 2 2 ,• JC/a —y/b = 1 is (a) director circle t

29. If the line ax + by + c = 0 is a nor mal to the hyperbola xy = 1, then

2

(d)m 1 m 2 = 11/20

2

2

(b) x + y = a

2

\

b ,(a)x 2 +y b (c) 2x +y , 2 =a 2 - ,2 . 2 =a 2 + , ,2 33. The locus of the point of intersection of the line <3x - y - 4 <3k =0 and <3kx + ky-A<3=0 is a hyperbola of eccentricity (a) 1 (c) 2-5

(b) 2 (d) <3

Hyperbola

229

34. If a triangle is inscribed in a rectangular hyperbola, its orthocentre lies (a) inside the curve (b) outside the curve (c) on the cur ve (d) Non e of thes e 35. Equation of the hyperbola passing through the point (1, -1) and having asymptotes x + 2y + 3 = 0 and 3x + 4v + 5 = 0 is

(c) c

(d) - c

38. If x = 9 is the chor d of cont act 2

hyperbola x - y = 9 , then the equation o f the corresponding pair of tangents is (a) 9x 2 - 8y2 + 18x - 9 = 0 (b) 9x 2 - 8y2 - 18x + 9 = 0

(a) 3x 2 - 1 Oxy + 8y 2 - 14x + 22y + 7 = 0

(c) 9x 2 - 8y 2 - 18x - 9 = 0

(b) 3x 2 + 1 Oxy + 8y2 - 14x + 22y + 7 = 0

(d) 9x 2 — 8y2 + 18x + 9 = 0

(c) 3x 2 - lOxy - 8y 2 + 14x + 22y + 7 = 0 (d) 3x 2 + 1 Oxy + 8y2 + 14x + 22y + 7 = 0 36. The equation of -2

drawn to

tangent parallel to y = x

?

- ^ r = 1 is 3 2 (a) x — y + 1= 0 (b) x — y — 2 = 0 (c) x + y - 1= 0 (d) x — y — 1= 0

37. The norma l to the recta ngular

hyperbo la

39. Tangents drawn from a point 2on the circle 2 2 2 x y x + y = 9 to the hyper bola — - r r = 1. then 25 16 tangents are at angle (a) n/4 (b) n/2 (c) Jt/ 3 (d) 2t i/ 3 40. If eand e, are the eccen trici ties of the hyperb ola

2

xy = c and

7

2

**

x" - y = c~,

then

e + ex

xv = c~ at the point *fj' meets the curve again

(a) 1

at the point 'r 2*. Then the value of tj t2 is

(c)6

(a) 1

of the

2

(b)4 (d) 8

(b) - 1

Practice Test M.M : 20

Time : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] 1. The poi nts of int erse cti on of the curves 3. The asymptote s of th e hyperbola whose parametric equations are xy = hx + ky are 2 2 h, y = k (a )x = k, y = h (b)x = x = t + 1, y = 2 1 and x = 2s, y = — is s (c)x = h, y = h (d)x = k, y = k given by 4. If P (xj, yO, Q (x 2, y 2 ), R (x 3, y 3 ) and (a) (1, - 3) (b) (2, 2) S (x4, y 4 ) and 4 concyclic points on the (c) ( - 2, 4) (d) (1, 2) 2 rectangular hyperbola xy = c , the co-ordi2. The equations to the common tangents to PQR are nates of the orthocentre of the A .2 ,2 th e two hype rbol as '— - ^ = 1 and (a)(x 4 , - y 4 ) (b)(x 4 ,y 4 ) a b (c)(-x 4 , - y 4 ) (d)(-x 4 , y 4 ) 2 2 iL2 _ ^L 1 are 5. The equation of a hyperbola, conjugate to the ,2 a b 2 2 hyperbola x + 3ry + 2v + 2x + 3y = 0 is 2 2 4(b -a ) (a )y = + .v + (b) y = ± x + V(a 2 - b2) (c)y = ± x ± (a2 (d )v = ± .v ± V ^

- f t

2

2

)

+ ft2)

(a) .r 2 + 3xy + 2 v 2 + 2x + 3y + 1 = 0 (b) x 2 + 3lvv + 2y2 + 2 t + 3y + 2 = 0 (.c)x 2 + 3xv + 2y 2 + 2r + 3v + 3 = 0 t,d) x 2 + 3.vy + 2y2 + 2x + 3y + 4 = 0


230 2

y = 20x 2

(a)*!?!

+ x2y2

= 0

(c)x a *2 + y\V2

2

2

x

2

MP

y x +y

2

y

P

2

NP

+y -

= 4a (x + b) 2

2

= ab

b2x2 + a2y2 = a2 + b 2 y =x

2

0

=

2

2

by

2

y =

= (a + by

Answers

3y

2

TRIGONOMETRY

TRIGONOMETRICAL, RATIOS AND IDENTITIES rm

mz =

n

I n

(rm

I (rm +

n e I

j

n

0"

7.5"

sin

0

cos

1

V8 - 2V6 - 2V2 4 -IB + 2V6 + 2-12 4

tan

0

(>/3-V2)(V2-1)

cot

oo

(-I3+-I2) (V2 + 1)

se c

1

V(16 - 10V2 +8V3 -6V6)

cosec

oo

V(16+ 10V2 +8V3 +6V6)

15" V 3V3-1 -1 2-l2 V3 + 1 2-l2 2-V3

(V6-V2)

18 '

22.5"

V5-1 4 Vl0 + 2>/5 4

V2-2 2 -I2 + -I2 2 V2

V25-10V5 5 V( 5 + 2V5)

v

(-J6+-/2)

V5"+

V2

30" 1

1 2 V3 2 1 73

36". V10-2V5 4 V5 4 V5 - 2-15

45"

60"

67.5'

75"

90'

1 V2 1 72 1

V3 2.

-12 +-12 2 -12- -12 2 -12+1

V3+ 1 2-l2 V3-1 2V2 2 + V3

11

V3

V4 - 2V2

2

V4 + 2V2

2

2 V3

1 -13

-12

-12-1

2

V4 + 2V2

2

-14 + 2-12

2--I3

-16 + -12

0 OO

0

00

^ 1

Trigonometrical, Ratios and Identities

233

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. x = r sin 0 cos <(>, y = r sin 0 sin ()> and z = rcos 0 then the value of x + y 1 + z is independent of (a) 0, <(> (b) r, 0 (c) r, <{> (d) r 2. If 0° < 0 < 180° then

(b) 1/2 (d) 5/4

(a) 3/4 (c)2

1. If

7. t a n

7

| =

. . 2V 2 -( 1 +V T )

(d) 2 V2"+ VJ"

then The8.

(d) None of these 3. If tan a / 2 and tan p / 2 are the roots of the equation 8x 2 - 26x +1 5 = 0 then cos ( a + P) is equal to , ,

627 725

.. . 627 725 (c) - 1 (d) Non e of these 4. If a sec a - c tan a = d b sec a + d tan a = c then and (a) a +c

2

2

2

=b +d

2

2

2

(a) sin (c) sin

n r n 0 = £ br sin 0 r=0

(b)b0=

(d)b0 = 0,b]

0,b

6. If 0 is an acu te angl e and tan 0 = value of

cosec

2

2

0 - sec 0 . is cosec 2 0 + sec 2 0

„ , (a+ P = 0 (b) cos ^

•0

11.

,

•0

71

1 + cos —

1 + cos

371

571

1 + COS

In 1 + cos -—- is equal to 8

for all real 0 then

(c)b0 = -l,b ]=n

a+P

2

(b) a + d = b +c

(a)*b= 1,*,=3

maxim um value of sin (x + 7t/6) + cos (x + 71/6) in the interval (0, TI /2) is alttained at (a) 7t/ 12 (b) 71 /6 (c) n/3 (d) tc /2

9. The minim um value of the expression sin a + sin P + sin y, whe re a , p, 7 are real numbers satisfyi ng a + P + 7 = Jt is (a) +ve (b) - v e (c) zero (d) - 3 10. If sin a = sin P and cos a = cos P, then

(c) a2 + b2 = c2 + d2 (d) ab = cd 5. Let n be an odd integer. If sin

... 1 +V3 "

]=n

(a) 1/2 -3«-3

, then the

(b) cos

71 /8

(c) 1/8 12. If A + C = B, then tan A tan « tan C =

(a) tan A + tan B + tan C (b) tan B - tan C - tan A

234


(c) tan A + tan C - tan B (d) - (tan A tan B + tan Q 13. If A lies in the third quadran t 3 tan A - 4 = 0, then 5 sin 2A + 3 sin A + 4 cos A = 24 (a)0 (b)5 , ,24 ( d) f (c )y

__ . , jc 2n 3K 4K The value of cos — c os — cos — cos — and

5K

(a) 3 (c) 1 15. If sin x + sin

2

(C)

(a) < 1

x=l,

(b)0 (d) 2 2

2

2

a sin x + b cos x = c,b sin y

(c) = 1 x 23. If cos a

(c)

-)c

<7/2,

(d) None of these

67C 1Q Tl,e val Iue of2K fcos — +4Ucos — +, cos — ' 19. Th is

(a) 1 (c) 1/2

(a) 2 + 4 sin a (c)2

2

cos Ia - —

(

cos

2K

a + —

then x + y + z = (a) 1

(b) 0

(c)-l

(d) None of these 3K

2 '

n " If L cos 9, = n, then £ sin 9; = 1=1 i=l (a) n - 1 (b) 0 (c) n (d) n + 1

26. If cos a + cos P = 0 = sin a + sin p, cos 2a + cos 2p =

(a) - 2 sin ( a + P) (c) 2 sin ( a + p)

then

(b) - 2 cos (a + P) (d) 2 cos ( a + p)

27. If Xj > 0 for 1 < i < n and xx + x2 + ... + x„ = K

sin X\ + sin x 2 + sin x 3 + ... + sin x n =

k a is equal to 4 ~ 2 (b) 2 - 4 sin a (d) None of these

2 a + 4 cos

2 K )

then the greatest value of the sum

(b)-l (d) -1/2

3tc 20. If 7t< a < ^ , then the expres sion

<4 sin 4 a + si n

i (

(b) 4 sin A sin B sin C (c) 1 + 2 cos A cos B cos C (d) 1 — 4 sin A sin B sin C

<7+2 /u. (b) — - —

<1

(d) Non e of these z

then cos 2A + cos 2 B + cos 2 C = (a) 1 — 4 cos A cos B cos C

a + a cos y = d and a tan x = b tan y, then — =• b2 is equal to (a-d)(c-a) (b-c)(d-b) (a) (b) (b-c) (d-b) 0 a-d)(d-b ) (b-c) (b-d) (d-a) (c-a) (c) (d) (a- c) (a- d) (,b-c)(d-b) 18. fI 0 < a < 7 i / 6 and sin a + cos a = then tan a / 2 =

(b) > 1

24. I f A + B+C =

2

2

, , <7 - 2 (a) — ; —

(d) None of these

sin (a + p + y) sin a + sin P + sin y

then cos 8 x + 2 cos 6 x + cos 4 x = (a) - 1 (b) 0 (c) 1 (d )2 27t 4k 16. If x = y cos ~r = z cos — , then xy + yz + zx = 3 (a)-l (c)l 17. if

2 1

22. If a, (3, y e ^ 0, — J, then the value of

+ cos 9 3 = (b) 2 (d)0

2

7K .

( 4

14. If sin 0, + sin 9 2 + sin 9 3 = 3,

then cos 0, + cos 0

6K

cos — cos — cos — is

2

(a)n

(b) K

(c) n sin — ' n 6

(d) 0

28. If A = sin 9 + cos

(a) A > 1 (c) 1 < 2a <3

14

9, then for all values of 9, (b) 0 < A< 1 (d) Non e of these


235

29 . If

sin a = - 3 / 5 and lies in the third quadr ant, then the value o f cos a / 2 is (a) 1/5 (c) - 1 / 5

(b) - l W l O (d) 1/VTo

valu es of 0 (0 < 0 < 360°) cosec 0 + 2 = 0 are

sati sfyi ng

(a) 210°, 300°

(b) 240°, 300°

(c) 210°, 240°

(d) 210°, 330°

6

K

2

( c ) COS X > — X 71

38.The

mi nim um

and

maxi mum

value

( C ) 2C 2 + 3C 6 = 0

39. If

0

cos

32. If P is a poin t on the altitud e AD of the triangle ABC such that ZCBP = B/3, then AP is equa l to (a) 2asin (C /3 ) (b) 2b sin (A/3) (c) 2c sin (B/3) (d) 2c sin (C /3 ) 33. For what and only what value s of a lying be tw ee n 0 and 7t is th e in eq uality

and

co s x = tan y, cos y = tan = tan x then sin x equ als

(a) sin y (c) 2 sin 18°

of

+ c

cos x ab sin x + - a ) (I a I < 1, b > 0) respectively are (a){b-c,b + c] (b ){b + c,b-c} ( c ) { c - b , b + c] (d) Non e of these

(b) q + c 3 + c 5 = 6 =

(b) cos x < 1 - — x

(d) cos x < — 7t x

(a) Co + c 2 + c 4 + c 5 = 0

C4 + 2C6

K

cm co s x = 0

where c 0 , q , c 2 , ...., c 6 are constants, then

(d)

2 (a) cos x > i1 - — x

m

31. If sin * sin3x = ^ m

sin(a-P)

37 . If 0 < x < 71/2, then

30. The

3

(c) AA' - BB' = (A'B - AB') (d) None of these

(b) sin (d) sin ( y +

40 . The value of the express ion

cos sin

7

cos

2k

371

5K.

14

14

cos

107t

-

sin

7t

14

is

sin a cos 3 a > s in 3 a cos a valid ? (a) a e

(0, TI / 4 )

(b) a e

(0,

TC/2)

(a)0

[

K

—, — (d) No ne of these 34. Whic h of the fol lowi ng is correct (a) sin 1° > sin 1 (b) sin 1 ° < sin 1 K (c) sin 1 ° = sin 1 (d) sin 1° = y j ^ sin 1 35. If a , P , y do not d iff er by a multiple of n cos ( a + 0) = — co s( P + 0) and if f —i r sin (p + y) si n( Y+ a) _ cos (y + 0) k Then k equals sin ( a + P) (a) ± 2 (b)± 1/2 (c)0 (d)±l If 36 .

expression the A cos (0 + a) + B sin (0 + P) retain the same A' sin (0 + a) + B' cos (0 + P) value for all '0' then (a) (AA' - BB') sin (a - p) = (A'B - AB') (b) AA' + BB' = (A'B + AB') sin ( a - P)

(d ) 4 41. The val ue of

18

X cos (5r)°

r= 1

is, where x°

denotes the degrees (a) 0 (c) 17/2

(b) 7/2 (d) 25/ 2

An a = K then the numerical value of tan a tan 2 a tan 3 a tan (2n - 1) a = (a) -1 (b )0 (c) 1 (d)2

42. If

43. If tan a is an in teg ral

sol ut ion

of

the

2

equation 4 x - 1 6 x + 1 5 < 0 an d cosP is the slope of the bisector of the angle in the first qua dr ant between th ex and y axes then the value of sin ( a + P) : sin ( a - P) = (a)-l (b)0 (c) 1

(d) 2

236


44. If 2 cos 0 + sin 0 = 1 then 7 cos 0 + 6 sin 0 equals (a) 1or 2 (b) 2 or 3 (c) 2 or 4 (d) 2 or 6 45. Th e valu e of 2 ver sin A — ver sin 2 = 2 . s •2 (a) cos A (b) sin A (c) cos 2A (d) sin 2A 46. The ratio of the greatest value

(C)

(a) x - V3~(l - a) x + a = Q (b) V 3~ X 2 -(1-a)x of

(a) 1/4 (c) 13/4

(b) 9/4 (d) None of these K 71 47. If cos x+ sin x = a — 2
a4l-a

(c) a V2 + a

00?

51. If t a n x t a n y = a and x + y = —, then tan x 6 and tan y satisfy the equation

2 - cos x + sin x to its least value is

(a) a

2tc

+

(c)x2 + y!3(\+a)x-a

a^3=0 =0

(d) V3~x2 + (1 + a) x - a yl3= 0 1 3 cos 0 52. If sin" 0 — co s 0 sin 0 - c o s 0 V( 1 +cot 2 0)

- 2 tan 0 cot 0 = - 1, 0 e [0, 271], then (a) 0 6 (0,7 1/2) - {71/4} ( b ) 0 e f | , 711-{371/4} (c) 0 e | 7t,

y

{571/4}

sm 0

is minimum when 48. Expression 2 + 2 0= and its mini mum value is

, , , ~ _ , 7rc i - 1/V2 (b) 2nn + —, n e 1,2 7I7I ±

TI / 4 ,

n e /, 2'~

lW2

(d) None of these 49. If in a tria ngle ABC, cos 3A + cos 3B + cos 3 C = 1, then one angle must be exactly equal to 71 (a)

3

(C) 7t

*>T (d)

4 71

50. In a trian gle ABC, angle A is greater than B. If the measures of angle A and B satisfy the equation 3 sin x - 4 sin 3.v - k = 0, 0 < k < 1, then the meas ure of angle C is M-f

336 and 450° < a < 540°, 625 sin a / 4 is equal t o 1 1_ (a) (b) 5 V2 25 4 3

53. If

(a) 2/i7t + ~ , n e /,

(c)

(d) 0 e (0,7i)- {Tt/4,71/2}

§

sm a =

then

(c) 5 (d) 5 54. If sin (0 + a ) = a and sin (0 + (3) = b, then cos 2 ( a - p) - 4 ab cos (a — P) is equal to (a) 1 - a

2

-b2

(b) 1 - 2a 2 - 2b 2

(c) 2 + a l - b 2 55. The expression ( 37t I

(d) 2 - a - b ^

2

. 4,, +sin (371 + a)

• (1/7I > . fi , sin — + a + sin ( 5tt - a ) is equal to (a) 0 (c) 1

(b)- 1 (d) 3

MULTIPLE CHOICE -II

Each question in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c. d corresponding to the correct answcr(s). 56

- If 0 < x < | and sin" x + cos" x > 1 then (a) 11 e (2, 00)

(b)/ie(-°°. 2] (c) n e [— 1,1] (d) none of these


237

57. If tan x _ tan v _ tan z 0) and x + y+ z = 7t 1 ~ 2 ~~ 3 then (a) maximum value of tan x + tan y + tan z is 6 (b) minimum value of tan x + tan y + tan z is -6 (c) tan x = ± 1, tan y = ± 2, tan z = ± 3 (d) tan x + tan y + tan z = 0 V x, y, z e R 58. If 3 sin |3 = sin ( 2 a + P) then (a) [cot a + cot (a + P)] [cot P - 3 cot ( 2 a + P)] = 6 (b) sin P = cos (a + p) sin a (c) 2 sin p = sin (a + P) cos a (d) tan ( a + P) = 2 tan a 59. Let Pn («) be a polynomial in u of degree n. Then, for every positive integer rc, sin 2 nx is expressible is (a)P

2n

(c) cos x P2n- i (sin x)

60. If sin

^ k=

a cos a =

C, + C 3 =

0

(a) a

G

33 65

(c) a e 63. If

4 — a—b 2 2 a+b, 1,

-w

)=

2 , 1.2 a+b •

sin 6 x + cos 6 x = a2 has

2

-n

2

4

71 64

=1

set of

(d)/

values

of

3

32

=1

K 128

= 1

Xe R

such

5

that

tan 2 0 + sec 0 = X, holds for some 0 is (a) (-•», 1] (b) ( - - , - 1 ] (c) <> |

(d) [ - 1, °o)

68. If the map pin g/( x) = ax + b, a < 0 maps [- 1, 1] onto [0, 2] then f or all values of 0, real

A = cos 2 0 + sin 4 0 is (a) /^j
(b) a e

(b)/(0) (d)«e

tan a and tan p are the roo ts of 2 equation x + px + q = 0 {p * 0), then

2

/„ (0) = tan - (1 + sec 0) (1 + sec 20)

67. The

65

] I 2 ' 2

(d) cos (0 -

(c)/

63 65

( - 1, 1)

+ b2)

2

(1 + sec 2" 0). The n / 7t N = 1 (a ) / 2 (b)/ 16

a

62. The equation solutions if

(c) tan

e —
2

(1 +sec 40) ...

(b) sin ( a + P ) = | |

(d)cos(a-P) =

e — 4>

66. Let

(b) 0, 0, 0, (d) 0 , - 1 , 2 3 5 If cos a = — and cos P = — , then

(c) sin

(b) cos

2

c0+c2 =

(a) cos (a + P) =

e-<[> 2

(c) ( m + n ) cosec 0 = m (d) sin 0 = 2-375

Ck cos 2ka. then

(a) 0 , 1, 2 (c) 0, 2, 3 61.

(a) cos

(b) (a 2 + b2) cos 0 = 2 ab

C] + C 2 + C 3 + C0 =

and

64. If sin 6 + sin = a and cos 0 + cos (J) = b, then

(a) 4 sin 2 0 = 5

(d) sin x P2n _ , (cos x) 2

+ q co s 2 (a + p) = q (b)tan(a+p) = p/{q - 1) (c) cos (a + P) = 1 - q (d) sin ( a + P) = - p

65. Which of the follow ing statements are possible, a, b, m and n being non-zero real numbers ?

(sinx)

(b)P 2 „ (cos x)

4

(a) sin ( a + P) + p sin (a + P) cos (a + p)

(c)/f jl
(d)/(-l)

238


69. For 0 < <

x=

z =

L cos n=0

(c)

if 2

" (b , y =

In

E cos " d>sin n=0

In

X sin 2 " (!), n=0 (b, then

(a)

70. If tan

a-bN

sin <()/2

(d) None of these 75. The value of Jog tan 1 • + log . + log10tan 89* ]0 tan 2" + !og ]0 tan 3" e 10 IS (a)0 (c) \/e

(d) None of these tan A + tan tan C = 6 and • 2 fiD : • 2 sin tan A tan fi = 2 then sin A sin C is (a) 8 : 9 5: (b) 8 : 5 : 9 (c) 5 : 9 : 8

* c)

i i cos x + 2 sin x co s x + c sin x 2 2 z = a sin x - 2b sin x cos x + c cos x , then (a) y = z (b) y + z =

+c

(c)y-z =

- cf + 4 (d) y - z = / yi ^ sin + sin cos A + cos fi sin sin cos A - cos y / even or odd ) = ^ A- f i ^

(a) 2 tan" (c)0 3+cot 76° cot 16° 72. cot 76° + cot 16° (a) tan 16° (c) tan 46° 73. In tan tan (a) (c)

r

76. If in

(b)xyz = (c) xyz = x + y + z (d) xyz = yz + x

71.

V

(b) 2 cot" (d) None of these

(b ) co t 76° (d) cot 44°

a tria ngle tan A + tan + tan C = 6 and A tan , then the value s of A , tan and tan are 1,2, 3 (b )2 , 1, 3 1,2,0 (d) None of these

a C0 S m* Tr 74. If cos e =

+ b , J - * — - , then tan 9 / 2 = co s
(d) 5 : 8 : 5

77. If cot 9 + tan 9 = x and sec 9 - cos 9 = y, then (a) sin 9 cos 9 = — (b) sin 9 tan 9 = y , 2.2/3 , 2.2/3 , (c)w (x y) ,J S, 2 ,1/3 , , 2x1/3 . (d) 78. If — = C ° S ^ whe re A * fi then y cos fi A + fi x tan A + y tan fi (a) tan 2 x+y M-fi x tan A — y tan fi (b) tan x+y (c) sin (A + fi) sin (A - fi)

y sin A + x sin fi y sin A - x sin fi

(d) x cos A + y cos fi = 0 sin a - cos a 79 . If tan 9 = , then sin a + cos a (a) (b) (c) (d)

sin a - cos a = ± <2 sin 9 sin a + cos a = ± V I cos 9 cos 29 = sin 2a sin 29 + cos 2 a = 0

80. Let 0 < 9 < and x = Xc os 9 + Ksin 9 , y = X sin 9 - Y cos 9 such that 2 x 2 + 4xy + y 2 = aX2 + , where are constants. Then ( a ) a = - 1, 6 = 3

(a)A/

(b) 9 = 71/4

(b)Vf^ ' a—

(c) = 3, (d) 9 = 71/3

Trigonometrical, Ratios and

Identities

239


Time : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct ansuier(s). [10 x 2 = 20] 1 2 2 • Minimum value of Ax - 4x | si n 9 | - cos 9 6. If t an 9 = n t a n (j),then maximum value of is 2,

(a)-2 (b)-l (c) -1 /2 (d) 0 2. For any rea l 8, th e maxi mum value o f co s 2 (cos 9) + sin 2 (sin 9) is (a) 1

(b) 1 + si n 1

(c) 1 + cos 1 (d) does no t ex is ts 3. If in a triangle ABC, CD is the angular bi se ct or of the an gl e ACB then CD is equal to (a)y^cos (C/2)

t an (9 - <|>) is ( a ) ^ An

(b)

(n - If An

2 (2re - l) n+1 (d) An An If 1 tan A | < 1, an d | A is acute then Vl + sin 2A + Vl - sin 2A is equal to V1 + sin 2A - Vl - sin 2A (a) t a n A (b) - t a n A (c) cot A (d) - cot A 8. The maxim um value o f th e e xpression

(c)

I V(sin 2 x + la ) - V(2 a - 1 - c os 2 *) I,

(b) — —c o s (C/2) ab

where a and x are real numbers is (b)V2 (a)V3 (c)l (d)V5

(c)

cos (C /2 ) a+b b sin A (d) si n (B + C/2)

9. If

a

n+l

1 a , — a n a sin 29 cosec a ar e in 4. Tr it cos2* 9R. sec

=

J : y7

(1 + an)

then

A

A. P. th en 8 6 1 j • 8 6 cos n9 sec a , — an d si n n9 cosec a ar e in (a) A. P. (b) (c) H. P. (d) 5. Given that (1 + V(1+ x)) t a nx = Then sin Ax is equal to (a) Ax (b)

(c) x

G. P None of them l + V( l -x )

2x


cos

ai

Vi-qp is equal to <23 ...to 0°

(a) 1


(&-1

(d)A a 0 10. If in A ABC, ZA = 90° a nd c, sin B, cos B are rational numbers then (a) a is the rational (b) a is irrational (c) b is rational (d) b is irrational (c) a 0

must be 100%


240

Answer 1. (a)

2. (a)

3. (a)

7. (a) 13. (a) 19. (d)

8. (a) 14. (d)

9. (a) 15. (c)

20. (c)

5. (b)

6. (a) 12. (b)

22. (a)

11. (c) 17. (b) 23. (b)

24. (d) 30. (d)

4. (c) 10. (c) 16. (b)

18. (a)

25. (b)

26. (b)

21. (b) 27. (c)

28. (b)

29. (b)

31. (a)

32. (c)

33. (a)

34. (b)

35. (d)

36. (c)

37. (a)

38. (d)

39. (c)

40. (c)

41. (c)

42. (c)

43. (c)

44. (d)

45. (b)

46. (c)

47. (d)

48. (b)

49. (b)

50. (c)

51. (b)

52. (d)

53. (c)

54. (b)

56. (a),

57. (a), (b), (c)

58. (a), (b), (c)

59. (c), (d)

60. (b)

61. (b), (c), (d)

62. (b), (d)

63. (a), (b)

64. (a), (c), (d)

65. (b), (d)

66. (a), (b), (c), (d)

67. (d)

68. (a)

69. (b), (c)

70. (b), (c)

71. (b), (c)

72. (c), (d)

73. (a), (b)

74. (a)

75. (d)

76. (b), (d)

77. (a), (b), (c)

78. (a), (b) (c)

55. (c)

79. (a), (b), (c), (d)

80. (b), (c).

1. (b)

2. (b)

3. (c), (d)

7. (b)

8. (b)

9. (c)

4. (a) 10. (a), (c).

5. (c)

6. (b)

TRIGONOMETRIC EQUATIONS § 28.1. Reduce any trigonometric equation to one of the following forms

2rm 0 = rm + a,

rm. 2 ' e

...(1)

(0-a)

= 2rm

242


MULTIPLE CHOICE-I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. The num ber of val ues sin 2x + cos 4x = 2 is (a) 0 (c )2

(d) None of these

of x for which

8. The smallest positive root of the equation

tan x — x = 0, lies in

(b) 1 (d) infinite

(a) (0, n/2)

2. The numb er of soluti ons of the equati on x3 + x 2 + 4x + 2 sin x = 0 in 0 < x < 2n is (a) zero (c )t wo

(b) one (d) four

J 1 271 J 9. The numbe r of solutions of the equati on . 5 sin x-

x for which 2 cos x, I cos x I and 1 - 3 cos x are in G.P. The minimum value of I a - (3 I is (b) TT /4 (d) Non e of these

4. The number of solutions of the equation tan x + sec x = 2 cos x lying in the interval [0, 27t] is

(a )0 (c)2 5. If 2 tan

(b) 1 (d)3 2

- (sin x * c os x) is sin x

cos x

(b) 1 (d) none of these

2 10. The equation (cos p - 1) xz + (cos p) x + sin p =0, where x is a variable , has real roots. Then the interval of p may be any one of the followings (a) (0, 2n) (b) ( - 71, 0)

n

(c)

, ne N, then

cos x =

(a) 0 (c) infinite

n

(d) (0,7i)

2 ' 2

x - 5 sec x is equal to 1 for exactly 7

distinct v alues o f x e 0, nn'

1

(O IK .?

3. Let a , (3 be any two pos itive values of

(a) 71/3 (c) 7t/ 2

11. The numbe r of solut ions of the equati on 2 (sin 4 2x + cos 4 2x) + 3 sin' x cos 2 x = 0 is

(b) 1 (d) 3

the greatest value of n is (a) 6 (b) 12

(a) 0 (c) 2

(c) 13

cos 2x + a sin x =• 2a-1 possesses a solution for

(d) 15

6. The general solution of the trigonometrical equation sin x + cos x = 1 for n = 0, + 1, + 2, ... is given by (a) x = 2nn (b) x = 2nn + n/2

(a) all a

(b) a > 6

(b) a < 2

(d) a e [2, 6]

13.The

c omple te

Solution

of

the

equati on

7- cos 2 x + sin x cos x - 3 = 0 is given by (a) nn +n/2 (n e / ) (b) nn - n/4 (n e /)

(c) x = nn + (- n " — - — ' 4 4 (d) None of these The solution

(c) nn + tan"

(2 cos x-

(d) nn + ~~ , kn + tan" 1 (4/3) (k, n e /)

of set 1) (3 + 2 cos x) =0 in the inter val

0 < x < 2x is

1

(4/3) (n e I)

14. If 0 < x < ft and 81

sln

f

+ 8 1 co:

= 30 then x is

equal to (b) (c)

If if

5TC 3 571 -1 cos 3

(a) 7t/6 (c) 7t 15.If

(b) 71/2 (d) 71/4

1+ si n0 + si n 0 < 6 < 7i, 6 * n/2 then

2

9+.

: 4 + 2 <3,

Trigonometric Equations

243

(a) 6 = 71/6 (b) 6 = 71/3 (c) e = 71/3 or 7C/6 (d) 0 = 71/3 or 271/3 16. If tan (7i cos 0) = cot (nsin 0), then the value(s) of cos (0 - 7t/4) is (are) (a) | (C)

cos (71 Vx- 4 ) COS (71 Vx) = 1 is (a) None (b) One (c) Tw o (d) More than two 24. The number of solutions of the equation

sin [YJ3 )

(d) None of these

2^2*

17.The

23. The number of solutions of the equation

a sin x ' + b co s .v = c

equation

where

=

*2-2^3x +4

(a) Forms an empty set

2

I c I > y
sin .v + cos x = min

1,<3~-4(3 + 6}

aeR

(a) 2/m (b) 2nn + j

(c ) 2/171 +

value

of

0

such

that

solutions of/(x)

In

(d) None of these

27. Values of x and y satis fying the equation

(d) None of these 19. If / ( x) = sin x + cos x. Then the most general :

71

/

10

are (where [x] is

the greatest integer less than or equal to .v.) (a) 2/171 +-,116/

(b)/17t. /! 6 /

, 11 e I (d) None of these

20. If x e [0,2t i], y e [0, 2n]

and sin x + sin y = 2 then the value of x + y is (b)

(a) 71

7t

(c) 37t (d) Non e of these 21. The number of roots of the equation x + 2 tan x = n/2 in the inte rval [0, 2n\ is (a) 1 (b) 2 (c) 3 (d) infinit e 22. If .v = X cos 0 — K sin 0, y = X sin 0 + Y cos 0 and

general

6

m + (- l ) ' ' J - f

(c) 2/m ± ~

26.The

sin 20 = V 3/ 2 and tan 0 = - j = is given by VT In / \ (b) nn ± (a)/i7t+ —

are given by

( c ),

(b) is is only only two one (c) (d) is greater than 2 25. The solution of the equation log cos x sin x + log sin x cos x = 2 is given by (a) x = 2/i7i + 7t/4 (b) x = nn + n/2 (c) x = /m + 7C/8 (d) None of these

2

A 2 + 4.vv + y =

2

AX + BY' ,

0 < 0 < 71/2 then (a) 0 = n/6 (b) 0 = 7t /4 (c) A = - 3 (d) B = 1

sin 7 y = I x 3 - x 2 - 9x + 9 I + I x 3 - 4x - x 2 + 4 I + sec 2 2y + cos 4 y are (a) x = 1, y = nn (b) x = 1, y = 2 / m + 7t/2 (c) x = 1, y = 2/m (d) None of these 28.Numbe r of real roots of the equation sec 0 + cose c 0 = Vl5~ lyin g betwe en 0 and 2n is

(a) 8 (b )4 (c)2 (d)0 29. The soluti on of the equat ion

4- . • 10 10 29 sin x + cos x = 7 7 cos 2x is 16

nn A

(b) x = nn + —

(c) x = 2/m + —

(d) None of thes e

(a) x

30. Solutions of the equation I cos x I = 2 [x] are (where [ . ] denotes the greatest integer function)

(a) Ni ll

(b) x = ± 1

244

Objective Mathematics (c) x= 71/3

(d) No ne of thes e

(c) nn-n

31. The general solution of the equation .1 00 100 . . sin x - cos x = 1 is (a ) 2nn + j , n e / (b) nn + ^ ,ne (c) mt + ^ , « e /

I

35. If max {5 sin 9 + 3 sin (9 - a)} = 7, then the set of poss ible val ues of a is 9 e R

(d) 2wJt - - j , « e /

32. The num ber of solutions of the

(d) None of these

34. If x e (0, 1) the greatest root o f the equa tion sin 271 x = V I cosTtcis (a) 1/4 (b) 1/2 3/4 (c) (d) None of these

equation

(a) { x : x = 2nn ± — , n e I

cosx

2 = I sin x I in (a) 1 (c )3 33. The general solution

[-27 1,27t ] is (b)2 (d) 4 of the equation

(b)\x:x n

(c)

cos2* , , . «-mi. + 1 =3.2 IS (b) nn + n

(a) nn

>ne I

2tc

3 ' 3 .

. 2

2

= 2nn±Y

(d) None of these

MULTIPLE CHOICE -II

Each question in this part has one or more than one correct answers). a, b,c,d corresponding to the correct answer (s).

40 . The equation sin JC = [1 + sin JC] + [1 - cos JC ] has (where [JC] is the greatest integer less than or equal to J C)

36. 2 sin x co s 2x = sin x if ( a X=MI )

+ n / 6 (NE

(b) x = rat- n/6 (ne (c) x


I)

I)

= nn (n e I)

n (d) x =nn + n/2 (ne 37. The equation

I)

(a) no solution in

X 2 X 2 2 2 2 sin — co s x - 2 sin — sin x = cos x - sin

2

2

(b) no solution in

x

has a root for w hich (a) sin 2x = 1 (c) co s* = 1 / 2

371]

(c) no solution in

(b) sin 2x = - 1 (d ) cos 2x = — 1/ 2

(d) no solution for JC e R 41. The set of all JC in ( - 7t, 7t) satis fying I 4 sin x - 1 I < VF is given by

38. sin x + co s x = 1 + sin x cos x, if (a) sin (x + n/4) =

( a ) x e |— ( b ) * e

(b) sin (x — JC /4 ) =

(c)

(c) cos (x + Jt/4) =

^

(d ) cos( .r- 71/4 ) =

^

39 . sin 0 + V Ic os 0 = 6 x - x 2 -ll,O <0<47 holds for (a) no value of x and 0 (b) one value of x and two values of 0 (c) two values of x and two values of 0 (d) two pairs of values of (JC, 0)

n

' 2 ' 2

t,xe

R,

JC

e

-

371 JC,

(-

K

10' 10

^

(d)xe (-7t,7t)

42. The solution of the inequality log 1 / 2 sinjc > log 1 / 2 cosjc in [0, 2n] is (a) JC e (0,71/ 2) (b) JC e (0, ic/ 8) (c) JC e (0, Jt /4) (d) None of these

Solutions 43 . of the sin 7JC + cos 2JC = - 2 are 2*7t 3JC . , (a)x = — + —,n,ke I (b) x = nn + ^ , n e I

equation

Trigonometric Equations

245

(c) x = nn + n/2, ne I (d) None of these 44. The solutions of the system of equations sin x sin y = V 3/ 4, cos xcos y = V3/4 are

(a) 7 (c) 21

48. The number of solution(s) of the equation •3 , - 2 2 3 , sin x cos x + sin x cos x + sin x cos JC = 1 in the interval [0, 27t] is/are

(a) No (b) One (c) Two (d) Three 49. The most general values o f JC for which

(a) x, = | + | ( 2 n + k) (b )y ,= | + |( *- 2n )

(c)x

2

(d)y

sin x - cos x = min {2, e 2, n, X2 - 4A. + 7 ] \eR

= | + | ( 2 ,n + t ) 2

2

(b) 14 (d) 28

are given by (a) 2nn

=| + |( * - 2n )

(b) 2nn + ~~

2

45. 2 sin x + sin 2x = 2, - n
(d)«7t + (-1)

- - 4 3 50. The solution of the equation 103 103 cos'~ JC - sin x = 1 are (a)--

(b) 0

71 (, ON-

(d)7t

Practice Test Time: 30 Min

M.M. : 20

(A) There are 10 parts in this question. Each part has one or more than one correct answerfs). [10 x 2 = 20] .2 3 . 1 3.The solution se t of th e inequ alit y sin x - — sinx + — 2 then 2 1. 1. If I cos x I 2=1, cos 8a < — is z pos sible va lu es of x (a) nn or nn + (- l) n 7t/6, n e I (a) | 8/(8/1 + 1) | < 8 6 < (8/1 + 3) (b) nn or 2 nn + J or i

nn + (- l) n § , ne O

(c) nn + (— l) n ~, n e I b (d)nn , n e I 2. tan | x | = | t a n x | if (a)x e ( - it (2/fe+ l) /2 , - nk] (b) x e [nk , JC (2k + l)/2) (c ) x e (-nk , - 7i (2k- l)/2) (d) x e (7C(2 k - l)/2, nk), k e N

I (b) | 8 /( 8n - 3) | < 8 < (8n - 1) | , (c) 8/(4j» + 1) T < 8 < (4n + 3) 4 (d) None of these 4. If [y] = [si nx] and y= cos x ar e

e /1

two

given equations, then the number of solutions, is : ([•] denotes the greatest integer function) (a) 2 (b) 3 (c)4


246

(d) Infinite ly ma ny solutions (e) None of these 1 then x must lie in the 5. cos (sin x ) ~ interval n n (a) 4 ' 2

(a) zero (c) 2

8. The numb er of soluti ons o f the equ ati on 1 (0 < x < 271) is cot X I = cot X + si n x 0 (a)

it,

(c)

(b) 1

(d)

2

3

9. The real roots of th e e quat ion

)

cos 7 x + sin 4 x = 1 in th e in te rv al ( - 7t, 7t) are

6. A solutio n of th e e qua tio n (1 - t a n 6) (1 + t a n 9) sec 0 + 2 .tan 9 = 0 7t 71 I .18 where 0 lies in the interval 2 ' 2

given by (a) 0 = 0 (b) 0 = (c) 0 = - 71/3 (d) 0 = 7. The number of solutions of 1 + sin x sin 2x / 2 = 0 i n [-

(b) 1 (d) 3

J

(a) (C)

TI/2

7i/2,

0

71/2,

(b) ( d )0,

, 0

7

0 , TI / 2

I/ 4, 7I /2

10. Nu mb er of solutions o f the equ ati ons y = -j [sin x + [sin x + [sin x]]] and

TI/3

denotes [v + [)']] = 2 cos x, whe re [.] greatest integer function is (a )0 (b) 1 (c) 2 (d) in fin ite

TI / 6

the equation rt, 7t] is

Max. Marks 1. First attempt 2. Second attempt 3. Third atte mpt

must be 100%

Answers l. (a )

2. (b)

3. (d)

4. (c)

5. (d)

6. (c)

7. (b)

8. (c)

9. (a)

10. (d)

11. (a)

12. (d)

13. (d)

14. (a) 20. (a)

15. (d)

16. (c)

17. (c)

18. (c)

19. (d)

21. (c)

22. (b)

23. (b)

24. (b) 30. (a)

25. (a)

26. (d)

27. (b)

28. (b)

29. (a)

31. (b)

32. (d)

33. (a)

34. (c)

35. (a)

36. (a), (b), (c)

37. (a), (b), (c), ( dj

38. (a), (c), (d)

39. (b), (d)

40. (a), (b), (c), (d)

41 . (b)

42. (c)

43. (a), (c)

44. (a), (b), (c), (d)

45. (b), (c)

46. (d)

47. (b)

48. (a)

1. (c), (d) 7. (a)

49. (b)

*

50. (a), (b)

2. (a), (b)

3. (c)

4. (d)

8. (c)

9. (b)

10. (a)

5. (a), (d)

6. (b), (c)

the

INVERSE CIRCULAR FUNCTIONS

(i) sin"

1

1

(- x) = - sin " 1

(ii) COS" (- X) = (iii) tan"

1

(iv) cot"

1

( - x) =

(v) sec"

1

(- x) =

1

(- x) =-tan"

(vi) co se c"

1

JI

x,

- cot"

7i - se c"

(viii) tan" 1 x + cot"

(x) sec"

1 1

(xi) cosec"

1

x = cos"

1

x = sin " tan

(xii) cot

1

x.

- 1 < x < 1

R

x e 1

x. 1

x e

x.

x =

TI/2,

R

x < 1

1

1 or x > 1

x. - 1 < x < 1

R

x = i t/ 2, x e

x + cosec"

1

1

(- x) = - co se c"

(vii) sin" 1 x + co s" ' x =

(ix) sec"

x, - 1 < x < 1

71 - c os"

x = k/2, x < - 1 or x > 1

'i—

I, x

< - 1 or x > i

1

I— >. x < - 1 or x > 1 x I 1(11 , x > 0 I I l x ;

7t + tan

| — . x < 0 l x (xiii) If x > 0. y > 0, xy < 1, then tan"

1

x + tan"

1

y = t an "

1

! K

y I


248

[xVi - y

2

+ yV i - x -y2 +y

2

] -

2n -

_2x_]

1 f i ^ z [1+x r

2x

^

-1

2

1 -x2!

MULTI PLE CHO IC E -I

question in this part has four choices out of which just one is correct. Indicate your c hoice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate.

(

17

TI

xy + yz + zx

i_2E

2.

«f

16 ( d )

x+

n

y =

,f ( c )

6

m

(d)7t

V

-'V2

Inverse Circular Functions

249

(c) two

infinite

16. The sum of the infini te series

8. A solu tion of the equa tio n tan" 1 (1 + (a) = JC 1 (c) JC = 0

JC)

+ tan - 1 (1 - JC) (b) JC =

cot-1 = -

JC3

1

(a) 7t

cos 20 -

JC2

then X tan ;= l

„

- 1

JC

V ( 1 + JC

2)

- V ( L -

JC

2)

,

2

2

(b) sin 2 a

(c) tan 2a

(d ) cot 2a

-1

-1

f — 1= yr

-1

(tan cos

(a) x

-1

-1

s i ny j

| (b) sin (d) sin

1

sin|-( ( sin

( - 1/2) + si n

(a) TE/4

(b)

5N /1

2

(C)3TC/4

( d )

1371/

12

In

-1

(- 1/2)

2n

14. If ^T sin " Xj = nn then

JC,

is equal to

i=1 (b) 2n

i=1

(a) 1 3 (c) 11

-x2)

2

(cosec

JC,

(b ) l

(c)3

(d)

4

21. The numbe r of positive integral solutions of tan"

1

JC

+ cot

-1

y = t an

-1

22. The value of cos is (a) 0 (c) 87T - 24

-1

(cos 12) - sin

JC > 2 + V 9

-1

(b) 7 I (d) Non e of these

,

1

\ O<

JC

are

< 1

1 + JC

(b)

0,7 T/ 4

(c) - 7t/4,

2TC

(d) V9-27T

is

3

(a) one (b) two (c)three (d) four

tan

( 2 - V 9 - 2 7 T , 2 +

a

(JC, y) where

5) > JC - 4JC JC= 2 + V 9 -

has

y = co s - 1 (cos x), - 2n < x < In, is

23. The smallest and the largest values of

JC

(b)

3)

(d) - 1 < a < 1 20. The number of real solutions of

(d) None of these

15.

1

(b) 1 5 (d) No ne of thes e

2tc

is equal to

(a) n n (w + 1) (c)

JC)} is equal to

(b) V(1

(a) 2

13. The value of t a n - 1 (1) + cos

...

(d) No ne of these

I y i = sin

12. — — is the princi pal value o f

7TI

3 2 +

(b) a < 1

f ^ 1+ ta n xr

(b) 71 /2 (d) None of these

co s —

-1

19. The equation si n - 1 JC = 2 s i n - 1 a solution for (a) all real values of a

11. If JC2 +y 2 + z2 = r 2, then

(a) cos"

+ cot

18. The value of tan (sec ' 2 ) + cot is

(a) cos 2 a

(c)0

18

(d) None Of these

, = — r = = = a, then x = /(1 +x2) +V ( 1 -J C )

1+ ta n zr

-1

+ cot

(b) 7t /2

17. sin {cot

co s P - sin P = 0.

(b)7t/2-p (d) - p

t a n - 1f

8

X; =

(a)P (c) 7t p— 10. If tan

-1

(c) TI/4

JC

sin 2P +

+ cot

is equal to

9. If JC], JC 2, JC3, JC4 are roots of the equation JC 4 -

2

T I / 2 is

)

7T /4,7T

71/4 /2

24. If - 1 < JC < 0 the n sin (a)

- c os

-1

(Vl -

JC

2)

1

x equals

(sin 12)

250

Objective Mathematics (b) tan

-i

sin

J (sin J I /3)

(c) - cot

COS

1

(d) cosec

x 1

25. The value of sin

(b)

10 -3 T C

(d) None of these

(a) tan

a + b ,b = {

i b2 = ya2bi and so on then

(c) tan

xk-k2

2x — 2xk + 2k2 (d) None of these cos

-7C /2

is

(b)

cos" 1 (a/b)

(c) b x =

^

x' — 2.xk + k2 1 ^ - 1 ' x + 2xk - 2k

29. The value of tan

cos - 1 (a/b)

(b) b„ =

- kx)

x — 2xk + k 2 ^ x+2

(b) tan

-laj? ,

2x +xk-k2

-1

26. If a, b are positive quantities and if

(aK. -

T(: x +k

where - < j c < 2 £ , £ > 0 | i s

(c) 3TI - 10

al+bi a2 = —-—,

7T /6

COS

(sin 10) is

(a) 10

ai

-

V(JC2 + k2 — kx)

(d)

—

cos (b/a) (d) None of these 27. tan

( C\x-y ) + tan C\ y + x \

1

r c2-c, \ 1 + c 2 c, ^

c

+ tan-1

C3 ~ 2 1 + c3c2

1

(a) tan

(c) - tan"

(y/x) 1

f—

30. Sum infinit e term s of the series cot

l

i2

+

4

.. + tan - 1 + cot (b) tan"

22 +

+ cot

-1

32 +

+

1

y

(d) None of these

(a) TC/4 (c) tan"

(b) tan 1

3

1

2

(d) None of these

28. The value of

MULTIPLE CHOICE -II Each question, in this part, has one or more than one correct answer(s). For each question, write the letters a, b, c, d corresponding to the correct answer(s). 31. The x satisfying

sin" 1

JC

(a) 0 (c)l 32. If 2 tan"

(1 - x) = cos" 1 x are (b) 1/2 (d)2 2x 1 x + sin" 2 is independent of x 1 +x

+ sin"

1

1

then ( a ) x e [1, + °°) (b) x e [— 1, 1] ( c ) t € (— — 1 ] (d) None of these

33. If ^ cos 1 x holds for (a) all values of JC (b) JC e (0, 1 /V2) (c)jce (1 /V2, 1) (d) JC = 0-75

34. sin

35. 6 sin" 1 (x2 -6x + 8-5) = 71, if


251

(a)x=l (c)x=3

(b)x = 2 (d) x = 4

-1

36. If cot

n

(c) a = 71/4 42The .

x)A + (cot

1

5ft x)2 = ~ , then

(

x equals

(a) 7t/2 (c)

is

2r (b) n / 4 (d) 271

ft

(a) 7t /4

c (a + b + c) . is ab (b) 71/2

(c) 7t

(d) 0

44 . If au a2, a3,.., an is an A.P. with common

difference d, then

39 . The numb er of the positive integral solutions of

.tan -1 x + , cos (a) 1 (c )3 40 . Let/

VT+ y

tan tan

.-if 3 ^ = sin -i— [VToJ

E

(b)/[

' sin

C O S

871

+rt/3)

then

a, +a

d

-1

1 +a„-i
n

(b) n

(d)

(n-

1) d

1 +axan an-ax a„ + a

then the value

iiti/12

JIC + cos

1 + a1a-i

,

45 . If tan 0 + tan ^ — + 0 | + t a n | - ^ - + 0 |= it tan 30,

of k is

= e

(a) a = 0

1 + tan 2

V

nd 1 + axan

(c)/ [-t)=" 2

41 . If a < sin "

d

1 +a ,a

(n-l)d

871 13ir/18 T J - «

1

(

is equal to

5)1/18

J In \ (d )/ [" T j

_1

+ ... + tan

* (b) 2 _ (d )4 (JC) =

{a + b + c) ca

+ tan 1

tan " r= 1

of

d»f

a (a + b + c) i -\\b + tan be

(d) -3

38 . Th e valu e of ^

values

4

i

tan

(b)-l oo

least

x) 3 are

In

(c) 43 .

(a)0 (c)-2

1

32

(b)5 (d) None of these 1

JC)3 + ( cos

3

value of n is

37 . If (tan

and

, , ft -1

(sin

> | ~ ], n e N, then the maximum

(c)9

(d) p = 7t

greate st

- 1

JC + tan

- 1

x < p, then

(c)3

(b) P = 7t /2


Practice Test M.M: 20

Time : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct 1. Th e pri ncip al va lu e of cos

(- l

(a) It (c) 7C/3

2n) ^cos

Y

J

- - i f - 2ti V ^sinyjis + s i n (b) 7i/2 (d) 471/3

answer(s). [10 x 2 = 20] 2. The su m of th e inf init e ser ies . -i( 1 > . - i f V 2 - n . -i fV l- V2 ) sin [ ^ h ^ ^ J +

+ S ln

. -lfV^-Vn-l [
l+

'


252

x

(d)7t

(C);

100

100

+y

100

+z

x 3. The

soluti on

sin [2 cos

1

of

[cot (2 tan

the 1

equation

a = 2 tan"

are 1

101 ,

+y

+z

101

is

(b)l (d) 3

x)]] = 0 are

(a) ± 1 (b) 1 ± <2 (c) - 1 + <2 (d) None of these 4. a, (3 a nd y

(a) 0

101 ,

8. If - ^ < x < ^ , then the two curves

y = cos x

andy = sin 3x intersect at (a) the

(V2~- 1),

angles

given

by

+ sin" 1 (- 1/2) and y = cos" 1 (1/3) then (a) a > p (b) p > y (c) ya> (d) none of the se 5. The number of dist inct roots of the eq uation 3 3 A sin x + B cos x + C = 0 no t wo of which differ by 2n is (a)3 (b)4 (c) inf init e (d) 6 6. If fsin" 1 cos" 1 sin" 1 tan" x] = 1, where [•] denotes the greatest integer function, then x is given by the interval (a) [tan si n cos 1, tan si n cos sin 1] (b) (tan sin cos 1, tan sin cos sin 1) (c) [- 1, 1] (d) [sin cos tan 1, sin cos sin tan 1] 7. If sin" 1 x + sin" 1 y + si n the value of

(b)

P = 3 sin" 1 (1/V2)

1

z = 3K/2

K

1

N

and

4'72

/

_n

8'

and

C

S

V -8' ° 8

4'72

n _ 1

It

and

72

2

of -

71

8'"COS8

Wl-T. 4'721 9. The solution — 1

°S 8

\

1

4'

K C

the

inequality

1

(cot x) - 5 cot x + 6 > 0 is (a) (cot 3, cot 2) (b) ( cot 3) u (cot 2, « ) (c) (cot 2, <* >) (d) None of these 10. Indicate the relation which is true (a) tan I ta n -1 (b) cot | cot

-1

(c) tan

x I = |x \

X

=

X

ta nx | = I x

(d) sin I sin

then

1

1

x I = xI

Max Marks 1. First attempt 2. Second attempt 3. Third attempt

must be 100%

Answers 1. (d)

2. (c)

3. (b)

4. (b)

5. (c)

6. (a)

7. (c)

8. (c)

9. (b)

10. (b)

11. (b)

12. (b)

13. (c)

14. (b)

15. (c)

16. (c)

17. (a)

18. (c)

19. (c)

20. (c)

21. (b)

22. (c)

23. (b)

24. (b)

25. (c)

26. (b)

27. (b)

28. (c)

29. (a)

30. (b)


31. (a), (b)

32. (a)

33. (a), (b), (d)

34. (c), (d)

35. (b), (d)

36. (b)

37. (b)

38. (b)

39. (b)

40. (b), (c)

41. (a), (d)

42. (a), (c)

43. (c), (d)

44. (b)

45. (c)

1. (a)

2. (c)

3. (a), (c)

7.(c)

8. (a)

9. (b)

4. (b), (c) 10. (a), (b), (d)

5. (d)

PROPERTIES OF TRIANGLE b, c

A,

C

ABC be

A = ^ ca

B = ^ ab

a) (s - b) (s- c) B

b2

C

C

A

C.

a + b+ c

2

A

A

B

B

2

C

ABC, A

B

C

any A ABC, a2

A =

ABC, a = b

2 be

C + c

„2 , „2

c +a - b B = 2 ca

.2

n

B, b = c

A + a

_

„2 ,

C

a +

C, c = a

t?-c2 2 ab

B + b

ABC, B-C

b-c b + c

. .

4

(

C-/4

c- a c+ a

T

A- B 2 |

A/2

4

A/2 =

c) be

V

a-b a+ b

w

(S-c)

Is(s-b) be

C/2

B/2,

A.

Properties of Triangle

255

cosq/2

=

V^ST ' ab y s(s-b)

tan c / 2

= "

Ft, r,

'

SIS

-

C)

m.

B

a)

(s-b) b

B/2 C/2 cos A/2 r =4 R

B/2

i n = —

. >~2 _ =

c) C/2 cos B/2 C/2

A

r , r. s - b

3

c

= s-

B/2, r 3 T2 = b r3 =

4R R

ABC BE P.

P

DEF,

PA = 2R PD = 2R

PB

R PE = 2R

PC

B/2 C/2

R PF

256



257

(ii)

. n H2 . 2n Area = ——- sin — n 2 Radius = R = f cos ec — n

An l

(i) aa co c osBB++c ccot co sCC==24f(Rt sin (ii) cotsAA ++ bbcot + t)A sin B sin C. (iii) r-\ + r2 + rz = 4R+ r

Fig. 30.4.

= s2

(iv) rir2 + r 2 r 3 +

MULTIPLE CHOICE-I

question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. In a trinagle ABC, (a + b + c) (b + c - a) = k be if (a )A :<0 (h) /t > 6 (c) 0 < <&4 ( d) *> 4 2. If X be the perimeter of the A ABC then , 2C 2B . b cos — + c cos — is equal to (a) X (c)X/2

(b) G.P. (d) Non e of these

In a triangle ABC, ZB = n/3 and Zc = 71/4 let D divide BC internally in the ratio 1 : 3. BAD) . sin (Z Then . ) y ^ . J . eq uals sin (Z CAD)

(b) 2X (d) Non e of these

3. If the area of a triangle 2

(a) A.P. (c) H.P.

ABC is given by

2

A = a ' - (b - c) then tan A / 2 is equal to (a)-l (b)0 (c) 1/4 (d) 1/2 4. The perimeter of a triangle ABC is 6 times the arithmetic mean of the sines of its angles. If the side a is 1 then ZA is (a) 30° (b) 60° (c) 90° (d) 120°

5. a3 cos(B-Q + b 3 co s (C - A) + c3 cos (A-B)(a) 3 abc (b) 3 (a + b + c) (c) abc (a + b + c) (d)0 cos A cos B cos C 6. If and the side a = 2, a b c then area of triangle is (a) 1 (b)2 (c) V3 /2 (d) <3 7. If in a A AB C, cos A + 2 cos B + cos C = 2, then a, b, c are in

(c)

(d)

7T

9. If D is the mid point of side ABC and AD is perpendicular to (a) 3a 2 = b 2- 3c 2 2

2

(c) b = a - c

2

(b)3 b2 = 2

of a triangle then

a2-c2 2

(d) a + b = 5c 2

g, h are the internal bisectors of a A 10. Iff, . 1 A 1 fi 1 C then — cos — + — cos — + — cos — = 2 h / 2 g ., 1 1 1 a b c a b c , , 1 1 1 (d) none of these (c) - + T + a b c 11. If a, b, c, d be the sides of a quadrilateral and 8 (x)=f(f(f(x)))

where f(x) =

a +b2 + c 2

(a)>*(3) (c)>g(2)

(b) < g (3) (d) < g (4)

then

258

Objective Mathematics (b) isosceles (c) right angled (d) None of these

12. If in a AABC, sin 3 A + sin 3 B + sin 3 C = 3 sin A. sin B. sin C, then the value of the determinant

a b c

b c a

c a

20. Let A0AXA2A3A4A5 be a regular hexagon inscribed in a circle of unit radius. The produ ct of the length of the line segments A0 A j, Aq A2 and A 0 A4 is

is

b

(a)0

(a) 3/4

(b)(a + b + cf (c) (a + b + c) (ab + be + ca) (d) None of these 13. In a AABC, if r =

(c)3

r2 + r 3 - rx, and Z A >

j

21. If in a A ABC, r, r2 + r 2 r 3 + r 3 r, = r] r 2 , r 3 are the ex-radii and

j then the range of — is equal to (a) (c)

(a) s2

(d) (3,

14. If the bisector of angle A of triangle makes an angle 0 with BC. then sin 0

(a) cos

B—C

(c) sin LB -

(b) sin

(b) 2s2 2

22,

,3

(where 2s is the

per imeter)

(b) 1

(b) 3 <3 343 (d)

ABC

B-C

(d) sin

15. With usual notations, if in a triangle

(c) 3 s (d) As2 In a A ABC, the value of

a cos A + b c os B + c cos C a+b+c :R (a) £ r 2r r^ (c) (d) v_/ R R 23. In a A ABC, the sides a, b, c are the roots of

ABC,

b+c c +a a+b 11 " 12 ~ 13 then cos A : cos B : cos C = (a) 7 : 19: 25 (b) 1 9 : 7 : 2 5 (c) 12 : 14 : 20 (d) 19 : 25 : 20 B C 16. If b + c- 3a, then the value of cot — cot — = 2 2 (a) 1 (b) 2 (c) 43 (d) 41 If17. in a triangle ABC, cos A co s B + sin A sin B sin C= 1, then the sides are proportional to (a) 1:1:42 (b) 1:42:1 (0)42:1:1 (d) None of these 18. In an equilat eral triangle, R : r : r2 is equal to (a) :11 : 1 (b) 1 : 2 : 3 (c)2 : 1 : 3 (d) 3 : 2 : 4 19. If in a A ABC, a 2 + b2 + c2= 8 R 2 , where R = circumradius, then the triangle is (a) equilateral

the equation x 3 - ll x 2 + 38* - 40 = 0. Then cos A cos B cos C . + —: — + is equal to a b c (a) 1 (b) 3/4 (c) 9/16 (d) None of these 24. If the base angles of a triangle are 22^° and

112^°, then height of the triangle is equal to (a) half the base (c) twice the base

(b) the base (d) four times the base

25. In a triangle ABC, a = A,b = 3,ZA = 60°. Then c is the root of the equation (a) c 2 - 3c - 7 = 0 2

(c) c - 3c + 7 = 0

(b) c 2 + 3c + 7 = 0 (d) c 2 + 3c - 7 = 0

26. The area of the circle and the area of a regular polygon inscribed the circle of n sides and of perimeter equal to that of the circle are in the ratio of n> n .E (b) cos I-n n n >In n 71 n ' n n ' n


259

27. The ex-radii of a triangle r b r2 , r 3 are in H.P., then the sides a, b, c are (a) in H .P. (b) in A.P. (c) in G.P. (d) Non e of these 2

7K In t any triangle, ABC, sin A + 1 .is A D „ -z, sin A +:— sin A always greater than (a) 9 (b) 3 (c) 27 (d) Non e of these 29. If twice the square of the diamet er of a circle is equal to half the sum of the squares of the sides of inscribed triangle ABC, then 2

2

2

sin A + sin B + sin C is equal to (b)2 (a) 1 (d)8 (c)4 30. If in a „ cos A cos B „cos C 2 + —- — + 2

triangle ABC, a b , , = — + — then the c be ca

a b value of the angle A is (a) 71/3 (b) 7t/ 4 (c) JC /2 (d) n/6

31. If in a tria ngle

1 - ^

1 --

= 2, then the

triangle is (a) right angled (b) isosceles (d) equilateral (d) None of these 32. Angles A, B and C of a triangle ABC are in .b_<3 A.P. If — = -J— , then angle A is c H' (a) n / 6 (b) 71/4 (c) 571/12 (d) tc /2 33. In any A ABC, the distance of the orthocentre from the vertices A, B,C are in the ratio (a) sin A : sin B : sin C (b) cos A : cos B : cos C (c) tan A : tan B : tan C (d) None of these 34. In a A ABC, I is the incentre. The ratio IA : IB : IC (a) cosec A/2is: equal cosec toB/2 : cosec C/ 2 (b) sin A/2 : sin B/2 : sin C/2 (c) sec A / 2 : sec B/2 : sec C / 2 (d) None of these

35. If in a triangle, R and r are the circumradius and inradius respectively then the Hormonic mean of the exradii of the triangle is (a) 3r (c )R + r

(b) 2R (d) Non e of these

36. In a AABC, a = 2b and \A-B\ = n/3. the Z C is (a) 7t/ 4 (b) n/3 (c) 7t/ 6 (d) Non e of these

Then

37. In a A ABC, tan A tan B tan C = 9. For such triangles, if tan 2 A + tan 2 B + tan 2 C = X then (a)9- 3 V3~
(c)X<9-3<3

(d) A, 4 27

38. The two adja cen t sides of a cyclic quadrilateral are 2 and 5 and the angle between them is 60°. If the are a of the quadrilateral is 4 <3, then the remaining two sides are (a) 2, 3 (b) 1, 2 (c>3 ,4 (d) 2, 2 39. If in a A ABC, a 2 co s 2 A = b2+c2 then (a ) A < J

(b) f< A
(O A> f

(d)A=f

40. In a A ABC, the tangent o f half the diffe rence of two angles is one third the tangent of half the sum of the two angles. The ratio of the sides opposite the angles are (a) 2 : 3 (b) 1 : 3 (c) 2 : 1 (d) 3 : 4 41. If P\,P2,Pi are altitudes of a triangle

ABC

fr om the vertice s A, B, C and A, the area of the triangle, then p\ ! + p2

s—a (a) A s-c

(b)

~Pi is equal to

s-b

(c)

<

42. If the media n of AABC through perpendicular to AB, then (a) tan A + tan B = 0 (b) 2 tan A + tan B = 0 (c) tan A + 2 tan B = 0 (d) None of these

A is

260


ABC, co s A + co s B + cos 43. In a tria ngle C = 3/2, then the triangle is (a) isosceles (c) equilater al

(b) right angled (d) Non e of these

44. If A xA2A3 ...An be a regula r polygo n of 1

sides and -

At A2

(a) = 5n

(c) n 7= (d) Non e of thes e 45. If p is the product of the sines of angles of a triangle, and q the product of their cosines, the tangents of the angle are roots of the equation (a)qx -px

+ (1+ q) x-p 2

( b ) p x - qx + (l+p)x-q (c) (1 + q) x - px + qx - q (d) None of these

= 0 =0 = 0

a, a, b, a,

(d) (2 — 43) r

53. In a triangle ABC\ AD, BE and CF are the altitudes and R is the circum radius, then the radius of the circle DEF is (a) 2 R (c) R/2

(b) R (d) No ne of these

54. A right angled trapezium is circumscribed about a circle. The radius of the circle. If the lengths of the bases (i.e., parallel sides) are equal to a and b is

(c)

c, b are in A.P. b, c are in A.P. a, c are in A.P. b, c are in G.P.

(b) • a b a+b

a+b ab

{d)\a-b\

55. If a , b , c , d are the sides of a quadrilateral, 2 . . . . . fa then the min imum value of

47. In a triangle, the line join ing circumcentre to the incentre is parallel to then cos B + cos C = (a) 3/2 (b) 1 (c) 3/4 (d) 1/2

the BC,

48. If the ang les of a triang le ar e in the ratio 1 : 2 : 3, the corresponding sides are in the ratio (a) 2 : 3 : 1 (b) VT: 2 : 1 (c) 2 : V3~: 1 (d)l:V3~:2 49. In a AABC, a cot A + b co t B + c cot C = (a) r+R (b)r-R (c)2 (r + R)

< 0 ^ 3 ®

(a ) a + b

46. In a A ABC, tanA/2 = 5/6 and tan C/2 = 2/5 then (a) (b) (c) (d)

(US (2 + 43)

a,A4

n =6

(b)

(a)(2 + V3)r

n

1 • + — 1— , the n

AxA 3

52. Three equal circles each of radius r touch one another. The radius of the circle touching all the three given circles internalW is

(d ) 2 (r - R)

50. In a triangle, the lengths of the two larger sides are 10 and 9. If the angles are in A.P., then the length of the third side can be (a) 5 ± 4 6 (b) 3 4 3 (c) 5 (d) 45 ± 6 51. In a triangle , a2 + b2 + c2 = ca + ab4f. Then the triangle is (a) equilateral (b) right angled and isosceles (c) right angled with A = 90°, B = 60°, C = 30° (d) None of the above

(a) 1 (c) 1/3

+ b2 + c2 . = is d2

(b) 1/2 (d) 1/4

56. If r, , r2, r 3 are the radii of the escribed

ABC and if r is the radius incircle, then r r r i 2 3~ r (rir2 + r2r3 + r 3 r i) i s equal to (a )0 (b) 1 (c) 2 (d) 3 a c , h — the n 57. If in A ABC, 1be ab c a the trangle is circles of a triangle of

its

(a) right angled

(b) isoscele s

(c) equilatera l

(d) No ne of these

58. A triangle ABC exists such that (a){b + c + a)(b + c-a) = 5 be (b) the sides are of length V19", V38~, VTT6 2 2 2 ,2 2 a -bJ.2 b -c c -a = 0 + ;— + (c) a' b B+C B-C = (sin B + sin C) cos (d) cos


261

59. The ratio of the areas of two regular oc tagons which are respectively inscribed and circumscribed to a circle of radius r is , . n (a) cos / \

(c)CO S

(b)sin —

2 7t

(d) tan

—

2 71

60. In a triangle inradius is

ABC right angled at

B, the

(a)AB + BC-AC (b) AB+AC-BC , .AB + BC-AC <0 (d) None of these

MULTIPLE CHOICE - II Each question, in this part has one or more than one correct answer (sj. For each question, write the letter(s) a, b, c, d corresponding to the correct answer (s). (a) tan A = a/b

61. If in a trian gle ABC, ZB = 60 then (a) (a - b)~ = c 2

-ab

2 ' (e) sin A + sin B + sin" C = 2

(c) (c - a) = b - ac (d) a2 + b2 + c2 = 2b2 + ac 62. Given an isosceles triangle with equal sides of length b, base angle a < n/4. r the radii and O, I the centres of the circ umci rcle and incircle, respectively. Then

(a) R = ~ b cosec a (b) A = 2b 2 sin 2a b sin 2a (c ) r = 2 (1 +c os a) b cos (3a/2) (d) 01 2 sin a cos (a/2) 63. In AABC, A = 15°, b= 10 <2 cm the value of 'a' for which these will be a unique triangle meeting these requirement is (a) 10 >/2~cm (b)15cm (c)5(V3~+l)cm (d)5(V3~1) cm AABC;

7t

a

= 5, b = 4, A = ^ + B for 2 the value of angle C (a) can not be evaluated (b) tan

i

(d) 2 tan"

(40

(c) tan

ab

2

1.2

64. If

(d) tan A + tan B =

(c) cos C = 0

2 (b) (b- • c) = a" - be

B = b/a

(b) tan

66. There exists a triangle ABC satisfying (a) tan A + tan B + tan C = 0 .. .s in A si nB sin C (c) ( a + b)2 = c2 + ab and V^(sin A + cos A) = ,,, . , . _ <3 + 1 „ V3" (d) sin A + sin B = — - — , cos A co s B = — = sin A sin B 67. In a AABC, 2cos (A-C

2

JL 40

(a) B = 71/3 (b) B = C (c)A,B, Ca re in A.P. (d) B+ C = A 68. If in A ABC, a cos A + b cos B + c cos C _ a + b + c a sin B + b sin C+ c sin A 9R then the triangle ABC is (a) isoscele s (b) equila teral

Given b > c. ZC = 23° and AD = then ZB =

65. If tan A, tan B are the roots of the quadratic

a, b. c

(d) None of these

69. In a tria ngle ABC. AD is the altitude from A. abc

(e) None of these

abx2 — c^ x + ab =0. where sides of a triangle, then

a+c V( a 2 + c2 - ac)

Then

(c) right angled -1

J

are the

(a) 53°

(b) 113° (d) None of these

b'-c'

262


70. If a, b and c are the sides of a triangle such

b. c = X2, then X and A is that

(a) C>2X sin C / 2 (c) a>2X sin A / 2

the

relati on

a,

is

78. If side s of a tria ngle ABC are in A.P. and a is the least side, then cos A equals

3c-2b 2c 4a-3b (c) 2c

71. In a A ABC, tan C< 0. Then

2

. PT M

inequation 43 x2 -Ax + 4f<0

then

(a ) a + b 2 + ab > c 2 (b ) a2+b2-ab< 2

(c ) a +b

>c

2

c2

(d) Non e of these

74. If A, B, C are angles of a triangle such that the angle A is obtuse , then tan B tan C < (a) 0 (b)l ( c) 2 (d) 3 75. If the sines of the angles of a triangle are in the rati os 3 : 5 : 7 their cotan gent are in the ratio (a ) 2 : 3 : 7 (b) 33 : 65 : - 15 (c) 65 : 33 : - 15 (d) No ne of thes e 76. For a triangl e ABC, which of the foll owing is true? cos A cos B cos C (a)(b)

co s A

a , . sin A

2

cos B cos C + —:— + sin B

2

a +b" + c 2abc 3

sin C

cot A (b)-

(c) 999

B satisfy the

2

43 + 1

.

cot B + cot C

b)/c

(d) satisfies sin A + cos A = (a +

, 1

80. In a A AB C, b2 + c2= 1999 a 2 , then

c x -c(a + b)x + ab = 0, then the triangle (a) is acute-angled (b) is right-angled (c) is obtuse-angled tan

(d) None of these

(a) 42 : 2 : 43 + 1 (b) 2 : <2 : <3 + 1

(d) tan A + tan B + tan C > 0 72. If the sines of the angl es A and fi o f a triangle ABC satisfy the equation

73. In a A ABC tan A and

Ac-3b 2c

79. If the angle s of a triang le are in the rat io 2 : 3 : 7, then the sides opposite these angles are in the ratio

(a) tan A tan B < 1 (b) tan A tan B > 1 (c) tan A + tan B + tan C < 0

2

(b)

(a)

b > 21 sin A/2 (d) None of these

(b)

1

1999 (d) 1999

81. There exists a triangle ABCsat isfyi ng the conditions (a) b sin A = a, A < n/2 (b) b sin A > a, A > n/2 (c) b sin A > a, A < n/2 (d) b sin A < a, A a 82. If co s (0 - a ) , cos 9, cos (0 + a ) are in H .P.,

83

then cos 0 sec a / 2 is equal to (a) -1 ( b) -V 2 (c) 42 (d) 2 If sin (3 is the G.M . be twe en sin a and cos a , then cos 2P is equal to 2 71 (b)2cos 2 f J (a) 2 sin 4 ~ a n ^ + a (d) 2 sin sin"2 -y + a (c)2 cos

84. If I is the median from the vertex A to the side BC of a A ABC, then (a) AI2 = 2b2 + 2c - a (b) 4/ 2 = b2 + c2 + 2bc cos A (c) 4/ 2 = a2 + Abc cos A

(d)

sin2 A

a

1

sin2 B sin2 C

b

,L

c

t

77. Let f ( x + y)=f(x).f(y) for all x a n d y and / ( l ) = 2. If in a triangle ABC, a =/(3), b =/( l) +/(3 ), c =/(2) +/(3), then 2A^= (a) C (c) 3C

(b) 2 C (d) 4C

(d) 4/ 2 = {2s- a) 2 - Abc si n 2 A/2 85. If in a A ABC, /-,; : 2r 2 = 3r 3 , then (a) a/b = 4 / 5 ( b ) a / b = 5/4 (c) a/c = 3/ 5 (d) a/c = 5/3


263

sin B tan B • and 2 is a sin C 2 x - 9x + 8 = 0, then

86. In a AABC, 2 co s A =

solution of equation AABC is (a) equilate ral (b) isosceles (c) scale ne (d) right-an gled 2 2 2 sin A _ sin (A - B) then a , b ,c are in 87. If sinC sin (B-C) (a) A.P. (b) G.P. (c) H.P. 88. In a

(d) None of these A ABC, A : B : C = 3 : 5 : 4. a + b + c <2 is equal to (a) 2b (b) 2c (c) 3b

The n

89. If in an obtuse-angled triangle the obtuse angle is 371/4 and the other two angles are equal

to two values of a tan 9 + b sec 9 = c, when 2

9

satisfying

2

then a —c is equ al to (a ) ac (b)2 ac (d) None of these (c) a/c 90. In a AABC, A - : n / 3 and b : c= 2 : 3. <3 tan 9 = , 0 < 9 = : 71/2, then (a ) B = 60° + 9 (c) B = 60° - 9

If

(b) C = 60° + 9 (d) C = 60° - 9

(d) 3a


Time : 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct 1. IfA,A 1 , A 2 , A 3 are the areas of incircle and the ex-circles of a triangle, then 1

(a)(c)

1

1

2_

(b) TA

VA 1_

(d)

2VA

(a) a > b > c

( sin 2 A + sin A + 1 is always greater [ si n A than (a)9 (b)3 (c) 27 (d) No ne of th es e 3. If A is the a re a a nd 2s the su m of thre e sides of a triangle, then s

2

(b) a <

2

(c) A >

V3

abc , a +b+ c (d) AABC is rig ht angled if r + 2R = s where s is semi perimeter

5. In a tri ang le if r j > r

n

3 \3~

inradius (O B - >

(e) None of these

3

In any A ABC

(a) A <

answer(s). [10 x 2 = 20]

(d) None of these

4. In A ABC, which of the following statements are true (a) maximum value of sin 2A + sin 2B + sin 2C is sa me as the ma xi mu m val ue of sin A + sin B + sin C (b) R > 2r where R is circumradius and r is

2

> r 3 , then

(b)a b & b < c (d) a c funct ions 6. If th ere are only two linear f a nd g which map [1, 2]-on [4, 6] and in a AABC, c=f(l) +g(l) and a is the maximum value of r , wh er e r is the distance of a variable point on the curve 2

2

x +y - xy = 10 fro m th e srcin, th en sin A : sin C is (a) 1 : 2 (b) 2 : 1 (c) 1 : 1 (d) No ne of th es e 7. In a tria ngl e :si n A _ si n (A - B) sinC ~ sin (B-C) (a) cot A, cot B, cot C in A.P. (b) sin 2A, sin 2B, sin 2C in A.P. (c) cos 2A, cos 2B, cos 2C in A.P. (d) a sin A, b si n B, c sin C in A.P. 8. If in a AA BC si n C + cos C + si n (2 5 + C) - cos (2B + C) = 2 <2 , the n AABC is


264

(a) equ ila te ral (b) isosceles (c) ri ght angl ed (d) obtu se angl ed 9. The radiu s of the circle pass ing throu gh the centre of incircle cf AABC and through the end points of BC is given by (a) (a /2 ) cos A (b) (a /2 ) sec A / 2 (c) (a /2 ) sin A (d) a sec A / 2

10. In a tri ang le ABC, 4a + 4b -4c (a) always positive

is

(b) always negative (c) positive only when c is smallest (d) None of these

Record Your Score

must be

Answers 3. (c)

4. (a)

9. (b) 15. (a)

10. (c)

21. (a) 27. (b)

22. (c) 28. (a)

32. (c) 38. (a)

33. (b) 39. (c)

34. (a) 40. (c)

43. (c) 49. (c)

44. (c)

45. (a)

50. (a)

51. (c)

55. (c)

56. (a)

61. (c), (d) 66. (c), (d)

1- (c) 7. (a) 13. (a)

2. (c)

25. (a)

8. 14. 20. 26.

(a) (a) (c) (a)

31. (a) 37. (b)

19. (c)

16. (b)

5. (a) 11. (b)

6. (d) 12. (a)

17. (a) 23/(c) 29. (c)

24. (a) 30. (c)

18. (c)

35. (a)

36. (b) 42. (c)

46. (a) 52. (b)

41. (c) 47. (b) 53. (c)

57. (a)

58. (d)

59. (c)

60. (c)

62. (a), (c), (d)

63. (a), (d)

64. (b), (d)

65. (a), (b), (c), (d), (e)

67. (a), (c)

68. (b)

69. (b)

70. (c)

71. (a), (c)

72. (b), (d)

73. (a), (b)

74. (b)

75. (c)

76. (b), (c)

77. (a)

78. (b)

79. (a), (c), (d)

80. (a)

81. (a,) (d)

82. (b), (c)

83. (a), (c)

85. (b), (d)

86. (a)

87. (a)

88. (c)

84. (a), (b), (c), (d) 89. (b)

1. (b) 6. (c)

48. (d) 54. (b)

90. (b)

2. (c) 7. (a), (c), (d)

3. (a)

4. (a), (b), (c). (d)

8. (b), (c)

9. (b)

5. (a) 10. (a)

HEIGHTS AND DISTANCES

Horizontal line X

Horizontal line Fig. 31.1.

Fig. 31.3.

Fig. 31.4.

Fig. 31.5.

266


Fig. 31.6.

ABC, AD AB 2 + AC 2 = 2 (AD2 + 0D 2 )

BD: DC = m : n (m + n) (.m + n) n B - m

C.

AD

BAC BD AB DC ~ AC

V2 7t 2

4=1-15, f

rt

e=

1

= 0-87, *

Heights and Distances

267

MULTIPLE CHOICE -I Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. An isosceles triangle of wood is placed in a vertical plane, vertex upwards and faces the sun. If 2a be the base of the triangle, h its height and 30° the altitude of the sun, then the tangent of the angle at the apex of the shadow is (a)

2ah<3 3h —a

ah 43 (c) 12 2 h —a

(b)

2ah<3 2

3h + a

(d) None of these

2. As seen from A, due west of a hill HL itself leaning east, the angle of elevation of top H of the hill is 60°; and after walking a distance of one kilometer along an incline of 30° to a poi nt B, it was seen that the hill LH was pri nted at right angles to AB. The height LH of the hill is (a) - i - km \3

(b)V3 km

(c) 2 4 T km

(d)

km

3. A tower subt ends a ngles 0 , 20 and 30 at 3 poi nts A, B, C respectively, lying on a horizontal line through the foot of the tower then the ratio AB/BC equals to

sin 30 sin 0 cos 30 (c) cos 0 (a)

4. ABC is

sin 0 sin 30 tan 0 (d) tan 30 (b)

a triang ular park with AB = AC = 100 metres. A clock tower is situated at the mid point of BC. The angles of B elevation of the top of the tower at A and are cot - 13.2 and c os ec - 1 2.6 respectively. The height of the tower is (a) 16 mt (b) 25 mt (c) 50 mt (d) Non e of these 5. From a station A due west of a tower the angle of elevation of the top of the tower is seen to be 45°. From a station B, 10 metres from A and in the direction 45° south of east

the angle of elevation is 30°, the height of the tower is (a) 5 V2~(V5~+ 1) metres (b)

5

metres

, ), 5(V 5 + 1) metre s (c (d) None of these A tree is broken by wind, its upper part touches the ground at a point 10 metres from the foot of the tree and makes an angle of 45° with the ground. The entire length of the tree is (a) 15 metres (b) 20 metres (c) 10 (1 + V2) metres 43 (d) 10 1 + metres 7. A person towards a house observes that a flagstaff on the top subtends the greatest angle 0 when he is at a distance d from the

house. The length of the flagstaff is d 0 d cot 0 (a) | tan (b) (c) 2dtan 0

(d) None of these

8. A tower and a flag staff on its top subtend equal angles at the observer's eye. if the heights of flagstaff, tower and the eye of the a, b and h. then the observer are respectively distance of the observer's eye from the base of the tower is

(a) (b )b

a + b-2h i+b + b-2h\ a-b

•h \ a +b J (d) None of these (c)

9. The angle of elevation of the top of a tower

from a point A due sout h of it, is tan ' 6 and that from B due cast of it. is tan

1

7-5. If h is

268

Objective Mathematics the height of the tower, then where X" = (a) 21/700 (c) 41 /90 0

AB = Xh,


10. A ladder rests against a wall at an angle a to the horizontal. If foot is pulled away through a distance a, so that it slides a distance b down the wall, finally making an angle (3 with the horizontal. Then

tan (a) b/a (c) a-b

(b) a/b (d) a + b

11. Two rays are drawn through a point A at an angle of 30°. A point B is taken on one of

them at distance a from the point A. A perpendicular is drawn fr om the point B to the other ray, another perpendicular is drawn from its foot to AB, and so on, then the length of the resulting infinite polygonal line is (a) 2 <3 a (b) a (2 +-If) (c) a (2--If) (d) None of thes e 12. From the top of a cliff h metres above sea level an observer notices that the angles of depression of an object A and its image B are complementry. if the angle of depression at is 9, The height of A above sea level is h cos 9 (a)sinh 9 (b)

h 29 (c)sin

(d)

A

h cos 29

MULTIPLE CHOICE -II Each question, in this part, has one or more than one correct answer(s). letters(s) a, b, c, d corresponding to the correct answer(s) : 13. A person standing at the foot of a tower walks a distance 3a away from the tower and observes that the angle of elevation of the top of the tower is a . He then walks a distance 4a perpendicular to the previous direction and observes the angle of elevation to be p .

The height of the tower is (a) 3a tan a (b) 5a tan P a P la tan P (c) 4tan (d) 14. A tower subtends an angle of 30° at a point on the same level as the foot of the tower. At a second point, h meter above the first, the depression of the foot of the tower is 60°, horizontal distance of the tower from the point is

(a) h cot 60° (c)-

h cot 60°

(b) I

h cot 30°

(d) h cot 30°

15. The upper (3/4)th portion of a vertical pole subtends an angle tan" 1(3 /5 ) at a point in

horizontal plane through its foot and distant 40 m from it. The height of the pole is (a) 40 m (b) 160 m (c) 10 m

(d) 200 m

For each question, write the

16. The angles of elevation of the top of a tower from two points at a distance of 49 metre and 64 metre from the base and in the same straight line with it are complementry the distances of the points from the top of the tower are

(a) 74-41 (c) 85-04

(b) 74-28 (d) 84-927

17. The angle of elevation of the top P of a pole OP at a point A on the ground is 60°. There is a mark on the pole at Q and the angl e of elevation of this mark at A is 30°. Then if PQ = 400 cm (a) OA = 346-4 cm (b) OP = 600 cm (c) AQ = 400 cm (d) AP = 946-4 cm 18. ABCD is a square plot. The angle of elevation of the top of a pole standing at D from A or C is 30° and that from B is 9 then tan 9 is equal to

(a) <6 (c) V3~AT

(b) 1/V6 (d) -I f/- If

19. If a flagstaff subtends the same angle at the points A, B, C, D on the horizontal plane through the foot of the flagstaff then A, B, C, D are the vertices of a

Heights and Distances (a) square (c) rectangel

269 (b) cyc lic quadrilateral (d) None of these

foot of the tower are a, 2a, 3a respectively. If AB = a, the height of the tower is (a)tan a a (b) a sin a (c)sin a 2a (d) a sin 3a

20. The angle of elevation of the top of a T.V. tower from three points A, B, C in a straight line, (in the horizontal plane) through the


M.M.: 10

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] From the top of a building of height h, a , , 5 (b)| m (a) — m tower standing on the ground is observed to make an angle 0. If the horizontal distance between the building and the tower is h, the height of the tower is 2 h sin 0 (a) sin 0 + cos 0 2h tan 0 (b) 1 + tan 0 (c)

2h

1 + cot 0 2 h cos 0 (d) sin 0 + cos 0 2. From the top of a ligh t house, the ang le of depression of two stations on opposite sides of it at a distance a apart are a and p. The height of the light house is

, . a ta n a ta n B £ tan a + tan p

. a cot a cot P (b) cot a + cot P a (0 ° (d) cot a + cot p cot a cot P 3. A vertical lamp-post, 6m high, stands at a distance of 2m from 2m from a wall, 4m high. A 1-5m tall man starts to walk away from the wal l on the other side of the wall, in line with the lamp-post. The maximum distance to which the man can walk remaining in the shadow is (a)

(c) 4m (d) None of th ese 4. The angle of elev ation of the top of a tower standing on a horizontal plane, from two points on a line passing through its foot at distances a and b respectively, are complementary angles. If the line joining the two points subtends an angle 0 at the top of the tower, then

•b a+ b 2
5. A man standing between two vertical posts finds that the angle subtended at his eyes by the tops of the posts is a right angle. If the heights of the two posts are two times and four times the height of the man, and the distance between them is euqal to the length of the longer post, then the ratio of the distances of the man from the shorter and the longer post is (a) 13 : (b) 3 : 2 (c) 1 : 3 (d) 2 : 3


must be 100%

270


Answers 1. (a)

2. (d)

3. (a)

4. (b)

5. (d)

6. (c)

7. (c)

8. (b)

9. (c)

10. (b)

11. (b)

12. (d)

16. (a), (c)

17. (a), (b), (c)

18. (b)

13. (a), (b)

14. (a), (b)

19. (b)

20. (c)

1. (a), (b), (c)

2. (a), (c)

15. (a), (b)

3. (a)

4. (a), (d)

5. (a), (c).

VECTORS

X|ai

>

—>

0

*i ai + X2a2 +. ,. . + Xna n = 0.

X.

A, B, C

X

X.

wd*

w

I D I

r -* IV

iV

272

Objective Mat hem ati cs

A A

AA

A A

.

.A A

I.I=J.J=K.K

AA

I

. Ai. az1+

A A

J

K

K

bz j + bs a*, b* = ai£>i + azbz + azbz

b? + b£ + b£) a-\b\ + azbz + azbz +

+

^b?

+ bi + b£)

(i.e. a 2b 2

a* = ai f + az f + az

= £>-m +

to •

£>3 ft A A A I

bi

A

A

A

A

A

ix j = k , j x k

J

az bz

A

K

bz

A

A

= I , KX I = J.

A

_ I

Vectors

273 EMS I a * x b* I

(c) 1 i

= -

c

^

x

-f> i

*x *x

*x $

=

.

=

*x

ai bi C1

r-> —>

*x

*x

*

*x

x

*x

*x

*x

x

= =

Cz

C3

—» —>

*x

*x

274


—>

—^

—^

—>

—^

x

—>

—>

—>

—y

x

x

—>

—>

—>

—>

x

) =

—>

—>

^

^

—>

x

) x

x

ii*

a b c ]

—>

x

x

x

[ a b c ]

d c

[ a bc ]

1

7

'x c '

- >

, b* =

/

x

r tZ / x b

f -*• F*= A

+ _L_

P.V.

Vectors

275 A -»

-»

to E

—»

_

I (c*=

- >

—»

x

if-l =

I rj l

^

r—> —> r-> "V>,

p (F± 15 . (r*- bV = 0

276


MULTIPLE CHOICE -1

Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. The two vectors {cr= 2 i+j + 3 = 4 i - Xj + 6 are parallel if X = (a) 2 (c)3

h?

(b )- 3 (d)-2

2. If I a + bl = I a^- b l , then

If at bt c~are non-coplanar vectors, then a . ( b x ct

b! ( cx a '

b t (c~x a t (a) 0 (c) 2

ct ( a x bt

+

c^(bx

is equa l to a*(bxct ( b) l (d) None of these

(a) ats parallel to b*(b) atl b*

10. If at bt ctire three non-coplanar unit vectors,

(c) I a t = Ib l

then [a*b*ct is (a) ± 1 (b) 0 ( c) ±3 (d) 2 11. The value of c so that for all real x, the vectors cx f — 6 j + 3 ic,x f + 2 j + 2 cx £ make an obtuse angle are (a) c < 0 (b) 0 < c < 4 / 3 (c) - 4 / 3 < c < 0 (d) c > 0

(d) None of these

3. If a and b are unit vectors and 0 is the angle between them, then

., . e

IS

(a) sin -

(b) sin 0

(c) 2 sin 0

(d) sin 20

4. If

a + b + c~= 0,1 a t = 3,1 bl = 5, 1 c t = 7,

then the angle betwee n a and b is (a) n/6

(b) n/3

(c) 2 J I /3

If

(d) 5 n / 3

a t b t c * are

unit

vectors

such

th at

a + b + c = 0, then the value of .—.—» —> —>. a . b + b . c + c . a is (a) 1 (c) - 3/ 2


6. Vectors aa nd b* ar e inclin ed at an ang le 0 = 120°. If

I a t = 1,1 bt = 2,

then {(a + 3 bf x (3 a^- bf} (a) 225 (c) 325

2

is equal to

(b) 275 (d) 300

7. If a , b , c are vectors s uch that a . b = 0 and a + b = c then 2

2

(a) I a t +1 bt = I c t 2

2

(b) I at = I b l +1 ct

2 2

(c ) I b l 2 = I at 2 +1 c i 2 (d) None of these 8. If atbtcâre three non-coplaner vectors, then [ a xb , b x c , c x a j is equal to (a)0 (c) [at bt ct

(a) all are acute angles (b) all are right angles (c) at least one is obtuse angle (d) none of these 13. If f , j \ fc are unit orthonormal

(d) 2 [at bt c t

vectors and a^

is a vector if a x r = j , then a . r is (a)—1 (c) 1 14. If


I a t = 3,1 bt = 4 and I a + bt = 5,

l^-b1 = (a) 3 (c)5 15. If ( t x bt IbU (a) 16 (c) 3

then

(b) 4 (d)6 2

+ ( at bt

2

= 144 and I a t = 4, then

(b) 8 (d) 12

16. The projection of the vector 2 f + 3 f - 2 £ on the vector f + 2 jV 3 ic is (a)

(b)[itbt^1 2

12. If a , 6 and c are thre e unit ve ctors , such that a + 6 + c is also a unit vect ors 0], 0 2 and 0 3 are angles i^l es between the vectors a , t>; D , c and c , a respectively, then among 8,, 0 2 and 0 3

, ,

v k 3

(b)

Vl4

(d) None of these

282

Vectors 17. For

non-zero

vectors

a t b t c^

(a) a^b*= 0, b* ct r 0 (b) b^ (c)

0, c* a t 0

a t : 0, a t bt= 0 (d) a* b*= b *~z= ~t= 0

18. If

a* is

a

unit

vector

such

that

A

a x ( f + j + 1c) = f - £ , then at= 1

A

I X ( A * x t ) + j x( A* x j ) + ftx(A*x ft) is equal to (a)0*

(b) 2 A*

(c)-2A>

(d) None of these

of vectors, then a x p + b x q + c x r equals

(b)j A A

(d) None of these

26. If a , b , c and p , q , r are reciprocal system

A

(a) — J (2 f + . f + 2 ft )

(c)y(f+2j

(c) 2

25. For any vector A*, the value of

I ( a x b5 . c t = l a t lb 1l c t holds iff

/ s r-hr*—* (a) [a b c j

+2k)

(d)f 19. Let a an d b~*be two unit vector s and a be the angle betwe en them, then a + b is a unit vector if (a) a = n/4 (b) a = 7t/3 (c) a = 2n/3 (d) a = T I / 2 If a + 2 b + 3c = 0 and a x b + b x c + c x a i s

(c )( f (d)a + b + c* 27. If at band ctare unit coplanar vectors, then the scalar triple product (a) 0 (c)-VT

21. If


oA. = f + 3 _f - 2 (c and ofe = 3 ?+.f— 2 ic,

then

o b which bisects the angle AOB is given by (a) 2i-2 j-2 fc (b) 2 t + 2 j + 2 ft (c )- 2t +2 j- 2 ft (d)2 t+2? -2ft

22. If

att+j +

c = ,i + a j + p f t

ft,bt4 are

i + 3 j + 4f t linea rly

and

depende nt

vectors and I c t = 4T, then (a) a = 1, P = - 1 (b) a = 1, P = ± 1 ( c ) a = - 1, P = ± 1 (d) a = ± 1 , P = 1 23. If l^xtl

a^ 2

(a) ( I f

by

+ li1<|l

any 2

+ l^x ftl

b , r^ b+ + cc ax (b xc ) = - ^ - ,

2

(b) -1

then the angle between aan d b i s (a) (b) 7t/ 2 (c)

(d) 3tc/ 4

29. Let at(jc) = (sin JC) f+ (cos x) f and (cos 2x) i + (sin 2x) j be two variable vectors (JC E R), then a (JC) and b (JC) are (a) collinear for unique value of x (b) perpendicular for infinitely many values ofJC (c) zero vectors for unique yalues of JC (d) none of these 30. Let the vectors a t bt ca nd d*be such that ( ax bt x ( c x d t = ot

Let

PlnndP2

be

planes determined by the pairs of vectors a , b and c , d respe ctive ly. Then the angle

then

=

(b) 2 (

( c ) 3 ( at 2 A (d)0 24. If the vectors a ?+ j + ft , t + j + c ft (a * b, c jt 1) are the value of1r 1 —a +1 -1-b — - +1 (a) 1

vect or,

(b )l (d)V3~

28. If at b t (Tare non-coplanar unit vectors such that

equal to X ( bt< c^ then X = (a)3 (c) 5

... —» —> (b) p + q + r

between P\ and P2

31 . Let

t + b j + lc and cop lanar , then . 1 — isc

(b) Jt / 4 (d)

(a) 0 (c) ji /3

I a t = l , l b 1 = 2, Ic 1= 3

atl. ( b + ct , btL ( c + at

and

Then I at- b + ct is (a) V6 (c) VTT

and

ctl . ( a + bt .


278

Objecti ve Mathemati cs

32. If a an d Ft arc two vectors of magnitu de 2 inclined at an angle 60° then the angle bet wee n a and a + b is (a) 30° (b) 60° (c) 45°

(d) Non e of these

is equal to (a) 3 7*

(b)F*

(d) None of these (OO* 34. a , b , c are non-coplan ar vector s and p , q , r are defined as

40 . The position vectors of the points A, B and C A A A A -A A A _A _ A are l + j + k, i + 5 j - k and 2 I + 3 j + 5 k respectively. The greatest angle of the triangle ABC is (a) 90° (b) 135° -1 (c) cos (d) cos

r-» 41. Let a and b are two vectors making angle 0 with each other, then unit vectors along bisec tor of a and b is

—>

—» c x a a x b P= q = . ) ). i > r = [b'c'aV [ c *a 'b b l^ [ a *ct bxc

then

(c) a and b are collinear (d) None of these

(a + ^ . p + (b + c t . q + (c + a t . r t

A

A

(c)±

is equal to

„,

(a) 0 (b)l (c)2 (d)3 , —» - A A „ A , .—> A A such that a t c = Ic t , I

Tr—

a t = 2 >fl

and the ang le betw een a x b and c is 30° then

n umbe r

of

vectors

unit

length

perpendicular to the vec tor s a = (1, 1, 0) and

a and b

(b) 2 (d) infinite are

two

vect ors,

such

that

a* b^< 0 and I a — t >b l =-> I a^x b l , then angl e betwe en vec tors a & b is 7 T T /4

(a ) 7i

(b )

( c ) T I /4

(d ) 3 T C/4

38. if

X ( a~x b t + M- ( b x c t + v ( c~x a t and

(a) d* ( a + b + c t b

a ,b ,c

"

(b)

2

d*.( b

+ ^

+ c t (d ) 8 dt ( t + b + c t

39. If a t b a n d c ^ a r e any thre e vectors, t hen —> —>—> —> —> — i a x ( b x c ) = ( a x b) xc if and only if (a) b amd c are collinear (b) aand care collinear

(a + b)

|S+ £ I

be

thre e ,

vect ors

,

such

that

-»

a x b = c and b x c = a then (a) a , b , c are orthogo nal in pairs (b) I a t = I b l = i c i = 1 1

(d) I a i * I b l * I c i 4 44. A vector a has components 2p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the srcin in the clockwise sense. If with respect to new system, a has compo nents p + 1 and 1, then (a) p = 0 (b)p = 1 or

[ a V c t = 1/8, then X + p. + v =

(c) 4 ^ (

v(d)± y

2 cos 0/ 2

(c) I a i = I b l = c1 i

b*= (0, 1,1) is (a)l (c)3 If

A

a + b

T->

of

A

the tetrahedro n whos e with position vectors i-6j+\0t -1- 3j +1%, 5 t—j + }Jc and 7 (' - 4 j + 7 k is 11 cubic units if X equals (a) - 3 (b) 3 (c) 7 (d) - 1

43. If

l ( a x b t x c t is equal to (a) 2/3 (b) 3/2 (c)2 (d)3 36. The

A

a+ b

(b) ± 2 cos e

42.The vo lume of vertices are

35. Let a = 2 i + j - 2 k and b = I + j . If c is a vector

A

, . , a +b

(a) ± — — 2

p =

(c) p = - 1 or p = (d) p = 1 or p = - 1 45. If a \ n d b are parallel then the value of

( at x bt x ( ct x dt + ( a~x c t x ( ET^x dt equal to (a) { ( a x c t . bt d* (b) {(b* x c t . at d*

is

Vectors

279

(c) {( a~x t t • c t d 4 (d) None of these

52. If

= a x b ^ ( a % 0, b*± 0), (b)r

(c)

(d) None of these

0*

( a ) a*. I

2 a *x c t

posi tio n

0

I a t = Ic t = 1,

Ibt = 4 and Ibt x c t = 4 l 5 , b - 2 c = A, a . Then A. equals

if

nce fr om A to

(b)-l (d) — 4

48. Let AD be the angle bisector of the angle of A ABC, then At. = a A§ + p (a) a =

(b)a =

„ , P = !A B + AC I ' AB I + I AC

I AC I AB + AC I AB l+IAC I

P = I AB I

I AC I

I AC I

I AB I AB l+IAC

1AB l+IAC IAB I 0 I AC I (d) a = — , p = AC I I IA B I

three

BC is

a . b

(c) I b*- Si (d) None of these 53. If the non-ze ro vect ors and T? are perpen dicula r to each other the n th e sol ution .

T+

of the equati on r x a = b is . . 1 .—J (a) r = x a + (a x a •a

(b) r =

, where

IAB I

(c)a =

A

of

respectively,

b*- c t a*

-¥

(a) 1 (c)2

vec tor s

A ,B ,C

(b)

47. Le t a t bt, c^be three vecto rs such that ^ and a*x

, b, c are

the shortest dista

then

a x b

(a) r = a x b

a

non-collinear points

46. If f x (r "x f) + f x ( F x j) + £ x (r~ x ic)

JC

1

V

x

.(a

(c) r = x (a X b] (d) None of these

dX

54. Let

= S t 0% = 10 a + 2 b*

and

wher e A and C ar e non-col linea r poi nts. Let p denote the area of the quadrilateral OABC, and let q denote the area OA and OC as of the parallelogram with adjacent sides. If p = Icq, then k = (a )2 (b) 4 (c) 6 (d) Non e of thes e 55. Let rtatb*and c*be four non-zero vectors

49 . p f + 3 _f + and ~iq f + 4 (c are two vect ors, where p,q > 0 are two scalars, then the length of the vectors is equal to

3

- If

( f x £) = ( a )- l (c) 2

I r x b*\= \ T*\\b =

(d )2

ABCDAXBXCXDX with lower

bas e ABCD, upper base

A\BXCXDX and the

lateral edges AA, , BBy , CC, and and M, are the centres of the faces an

A,B,C,D, respectively.

DDX ; M ABCD and

O is a point on the

line MM, such that

acute angle for

51

(a) - 1

r ' a * = 0,

=r It I c t then (b )0

56. Given a cube

vj A+ a C and

q* = - 3 f + a lo g 3 x j + lo g 3 x£ include (a)= a0 (c) a >0

that

I r x c (c) 1

(a) All values of (p, q) (b) Only finite number of values of ( p , q) (c) Infinite number of values of ( p , q) (d) No value of ( p , q) 50. The vectors p* = 2 f + log

such

a < 0 (b) (d) no real value of

a

f x ( i t x i ) + | x (o^x 3 +

..•{(?•?)?+(?•?)}+(?•£)*) (b) 0 (d) None of these

OA

then

+ OB OM

(a) 1/16 (c) 1/4

+ OC + OO = AOM, if A =

(b) 1/8 (d) 1/2

= OM, ,

280

Objective Mathematics —> —>

^

57. Let a, fc and c be three non-zero and non-coplanar vectors and p*, q* and r* be three vectors given by p = a + b - 2 c, q If

= 3a -2 b + c

r = a - 4 b +2 c

and

the

volume

of

the

parallelopiped

determined by a , F*and c*is V, and that of the parallelpiped determined by p*, ^ a n d r* is V2 then V2 : V, = (a) 3 : 1 (b) 7 : 1 (c) 1 1 : 1 (d) 15 : 1 58. A line Lx passes through the point 3 i and parallel to the vector - i + j + £ and another line L2 passes through the point i + j and

parallel to the vector ?+£, then point of intersection of the lines is (a) 2 i + | + f t (b) 2 t - 2 j + ft (c) t + 2 j ft (d) Non e of these The line joi ning the points 6 a* - 4 b*- 5 c*, - 4 c 4 and the li ne joining the points —> „ r * „ —»—> „ r-> _ ->. - a - 2b — 3c , a + 2 b - 5 c intersect at (a) 2 c*

(b) - 4 c*

(c) 8 C*

(d) Non e of these

60. Let A*, Bând ( f be unit vectors. Suppose that A*. B* = A*. C? = 0 and that the angle between

B

and

C

is

then

A* = k ( i f x (?) and k = (a) ± 2

(b )± 4

(c) ± 6

(d) 0

MULTIPLE CHOICE -II

Each question, in this part, has one or more than one correct answer (s). For each question write the letters a, b, c, d corresponding to the correct answers) 61.If

a

vector

r

sati sfie s

the

equation

65. If 3 a*- 5 b*and 2 a*+ b*are perpendicular to

F x ( f + 2 j V it) = f - ic, then r~is equal to

each other and a + 4 b , - a + b

(a) f+3 jj + £ A (b) 3 1+7j+3£ (c) j A+1 (f+ 2 J+ ic) where t is any scalar (d) f+( f + 3 ) M where t is any scalar

mutually perpendicular then the cosine of the

62. If d A = a*, A& = b* and c S = k a*, where k > 0 and X, Y are the mid points of

DB and —»AC respectively such that I a*l = 17 and I XY I = 4, then k is equal to (a) 8/17 (c) 25/17

(b) 9/17 (d) 4/17

63. a*and c*are unit vectors and I b*l = 4 with a x b = 2 a x c . The angle between a —> -1 —* —> —> and c is cos (1 /4 ). Then b - 2 c = Xa , if A. is (a) 3 (c) - 4

(b) 1/4 (d) - 1/ 4

64. If / be the incentre of the triangle ABC and

a, b, c be the lengths of the sides then the force IB + cI C = ( a) -l (c) 0


are also

angle between a*and b*is / ^

17

(c)

^^

(a) ?w

™

19

(b)yw (d) None of these

66. A vector a* = (x, y, z) makes an obtuse angle with"

y-a xis,

equal

angles

with

b = (y, - 2z, 3x) and c = (2z, 3x, - y) and a

is perpendicular to d = ( 1 , - 1 , 2 )

if

I a I = 2 <3 , then vector a is (a) (1,2, 3) (c) ( - 1, 2 , 4)

(b )( 2.- 2,- 2) (d) None of these

67. The vector ^directed along the bisectors of the

angle

between

the

vecotrs

a = 7 * — 4 j - 4 ft and b t = - 2 i - j + 2ft if

I c t = 3 V(Tis given by (a) 1 - 7 j + 2 ft . . . A _ % _ A (b) * + 7 j - 2 ft (a)-f +7j-2f e

Vectors

281 (a) all values of m (c) A> - 1/ 2

68. Let a t n d t t be no n collinear vector s of which a is a unit vector. The angles of the triangle whose two sides are represented by

(b) X < - 2 (d) A,e [- 2, -1/ 2]

76. The vect ors a t ; c ' i - 2 j + 5 ft

V3"(a~x bt) and b*- (at bi atare

and b"t *+ y j - z ft are collinear if

(a) 7t /2, 7t /3, 71 /6

(b) 7t/2, ti/4, 7i/4

(c) 7t/3, 7t/3,7 C/3

(d) data insufficient

( a ) x = 1, y = - 2, z = - 5 (b) x = 1/2, y = —4, z = - 10 (c) x = - 1/ 2, y = 4, z = 10 (d) x = - 1, y = 2, z = 5

6"- Let a and bi ,e two non-collinear If iT= at- ( a* bt band

unit vector s.

I vl = a~x b t then I v t

77. Let

is u l (a) I

(b)

I ul + I u

(c) I u l + u* I bi 70. If A, B, C

(b)

4

at

A

A

A

,

A

A

(c)

a - b +3 c

(b) (d)

2

has the value 2,0.2 2 a +3 b - c a + 3 tf + c 2

72- If | a t = 4, Ib 1 = 2 and the angle aa nd bi s the n ( a x b t 2 is (b) (^t (d) 32

r

is

any

vector

(b) 2 ?+ 3 j + 3 ft

(c) — 2 i - j + 5 ft (d) 2 t + j + 5 ft 78. The vectors (x, x + 1, x + 2), (x + 3, x + 4, x + 5) and (x + 6, x + 7, x + 8) are coplanar for

79

(a) all val ues of x

(b) x < 0

(c) x > 0

(d) No ne of thes e

If a t bt ca re three non-copla nar vectors such that r] = a - b + c , r

betw een

2

= b-) -c -a ,

r = A,] r, + X2 r 2 + A3 r 3 , then (a) A, = 7 / 2

(b) A, + A, = 3

(c) A, + A2 + A3 = 4 (d) A 2 + A3 = 2

2

80. A parallelogra m is construct ed on the vectors

73. If at bt (Tare non-coplanar non-zero vectors and

(a)2t+3j-3ft

r 3 = c + a+ b , r = 2 a - 3 b + 4 ci f

6

(a) 48 (c) 16

in

space

then

a t 3 a - jf, b t o + 3 j fif

I otl = I pi = 2 and

angle between otand (its 7t/3 then the length

[ b c r ] a + [c a r ] b + [a b r ] c is e qual

of a diagonal of the parallelogram is

to

(a) 4 V5~ (c) 4 4 T

(a) 3 [ a V c t r*

(b) [ a V c t r^

(c) [ b* c V j r*

(d) [ c V b t

74. If the unit vectors aand {Tare inclined at an angle 20 such that

I a^- b1 < 1 and 0 < 0 < 71,

then 0 lies in the interval (a) [0,71 /6) (c) [7t/6, 71/2]

and

of magnitude V(2/3) is A

p i+ + vectors 1 + j , 1- j and respectively, then the points are collinear if (a)p = q = r=l (b)p = q = r = 0 (c) p = q, r=0 (d)p=\,q = 2,r = 0 71. In a parall elogram ABCD, I c. n and I I = The o 2 , .2 2 3a + b - c (a)

ft

the plane of btind ctvhose projection on ats

I ul + u^ ( a + b t

are three points with position A

a~=2 1- j + ft, b = t + 2 j -

c t ' i + j - 2 f t b e three vect or s. A ve ct or in

(b) (5 tc /6, 7t] (d) [7t/2, 57t/6]

75. Th e vec tor s 2 i - A j + 3A ft and (1 + X) t - 2Xj + ft include an acute angl e for

(b) 4 V3~ (d) None of these

81. The position vectors a t b t cti nd dôf four poi nts A, B, C and D on a plane are such that ( I t dt . ( b 4 - ^ t = ( b t t ) then the point D is (a) Centroid of A ABC (b) Orthocentre of A ABC (c) Circumccnlre of AABC (d) None of these

• ( c*-a} = 0,

282


82. The vector b* bisects the ang le between the vectors a and 6 if

makes an obtuse angle with the z-axis, then the value of a is (a) a = (4n + 1) n - tan

(b) angle between aand bis zero

(b) a = (4n + 2) n - tan

(c )l at +l b1 = 0

(c) a = (4n+ 1) n + tan

(d) None of these

(d) a = (4n + 2) n + tan"

83. If a* , b t c^ ar e three non-zero vectors

such

2

1

2

-1

2

1

2

88. The resolved part of the vector aâlong the

that b*is not perpendicular to both a and c

vecto r b i s iCand that perpendicular

and ( a x b ) x c = a x ( b x c ) t hen (a) atind câre always non-collinear

(I^Then

(c) a and c are always perpendicular

_ (a*. b t b*

(d) a , b and c a re always non-c opla nar 84. Image of the point P with position vector 7 f - j V 2 ft in th e line whose vect or equation is T*= 9 i+ 5j + 5 ft + X (f + 3 ) + 5 ft) has the pos ition vector (a) — 9 f + 5 jV 2 ft (b) 9 f + 5 (c) 9 f - 5 j - 2 ft (d)9f+5j A +2ft 85. If a x b = c x d and a x c = b x d , then (a) ( at. dt =

to t t i s

(a*, bt a*

(b) a and c are always collinear

(b*. b t i * -

b t b*

89. Let the unit vectors aand bt?e perpendicular

and the unit vector e inclined at an angle 9 to both a an d bÎf c"t= a a t- P b1- y ( a t - b l then

X (bt-

(a) a = P

(b) at- d = X ( b + c t

(b) y 2 = 1 - 2 a 2

(c) (at- bt = X ( c + d t (d) None of these 86.

1

(a) I a t = I b l

(c) y 22= - cos 20 (d)P= ^ p

If (I xb tx ^= Ix (b x^ t , then (a) bt< ( ct< a t = 0 (b) ( c x i t x tt= 0

(c) cNone x ( aof x bthese t= 0 (d) 87. If the vectors tt = (tan a , - 1, 2 Vsin a / 2 ) 3 and c = j tan a , tan a , are Vsin a / 2 orth ogona l and

a vect or a = (1, 3, sin 2a )

90. Consider

a

tetrahedran

F2, FLet 3, F4.

with

faces

be the

vectors whose magnitudes are respectively equal to areas of F ,, F 2 , F 3 , F 4 and whose directions are perpendicular to their faces in outward direction. Then

I \ t + V^ + v t + V^ I

equals (a) 1 (c) 0


Vectors

283


Time 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] 1. Let the unit vectors a and b be perpendicular to each other and the unit vector c"*be inclined at an angle 6 to both a* and b*. Ifc * = x a*+y b*+z (a x b), then (a) x = cos 0, y = sin 0 , z = cos 20 (b) x = sin 0, y = cos 0 , 2 = - cos 20 2

(c) x = y = cos 0, z = cos 20 (d)x = y = cos 0, z 2 = A =

a a•

a j* o

a • c

then

- >

o • c

passing through the points 3 i + f - 2k is

, ^

,

*

„ „ - i - llj

i - j + 2k and A

^

A

+ lk

(c) - 2 i - 2/ + ^ (d) Non e of these 4. Let O be the circumcentre, G be the centroid and O ' be t he orthoce ntre of a AABC . Three vectors are taken through O and are represented by a and c

then a + b + c is

(a) 0&

(b) 2

6&

(c) 6 6 ' (d) None of then 5. If the

vectors

c, a = xi+yj+z

fi

and

b = j are such that a , c and b form a right handed system, then c is (a )zi-x% (c)yj

(b)o* )-zi+xk (d

6. The equation of the plane containing the line r = a + k b and perpendicular to the planer

n = q is

(b) (r (c)

a)

x a)

0

—>

[n* x (a* x b ) ) = 0 (n x b] = 0

(d) (r*—b) • [n x (a ' x 6')} = 0 Find the val ue of A so that the points P, Q, R, S on the side OA, OB, OC and AB of a regular tetrahedron are coplanar. You

0$ _1 ot = 2

(a) A = 0 (b) A = 1 (c) A = any non-zero value (d) None of these A A /S 3. The image of a point 2i + 2\ j- k in the lin e

(a) 3i + 11/ + Ik(b)

b)

,, , 0? 1 1 are given that =—; 3 OA

cos 29 —» c a •— c>

a 2.

(a)

' ofc

1 3

, OS

oS (a)=A1/ 2 (b) A. = - 1 (c) A =0 (d) for no val ue of A x and y are two mutu ally perpendicular unit vectors, if the vectors ax + ay + c (x x y), A

A

A

J

A

A

A

A

.

x + (x + y) and cx + cy + b (x x y), lie in a plane then c is (a) A.M. of a and b (b) G.M. of a and b (c) H.M. of a and b (d) equal to zero 9. If a*

i+j + fi and b = i - j ,

vectors (a

1) 1 + (a J)J + (a •

then

the

%,

(fc • ^ ? + (b*-j)j + (F 4 £) £ ,and f + y -•2% (a) are mutually perpendicular (b) are coplanar (c) Form a parallelopiped of volume 6 unit s (d) Form a parallelopiped of volume 3 un its 10. If unit vectors 1 and j are at right angle s to other and p* = 3 1 + 4 j, , ,„ _.. , - * -> -» g = 5 i, 4 r = p + g, then 2 s = p - q —> —> —> —> (a) I r + k s I = Ir - k s I for all real k each

(b) r is perpendicular to s —>

(c) r* + sîs perpen dicular to r (d) | ^ f

= |

=

= |


284

Answer

CO-ORDINATE GEOMETRY-3D yi, PQ

yi,

m

;y=-

i + m2

]Z

mi + rri2

mi + m 2

m 1 yi - rri2 yi mi - mz

mi X2- rri2 xi mi - m2

Z1

mi - rm X

Xxz ±xi x= A+1

Xy2 ± yi . _ Xz2 ± zi X+1 ' X±1 x±-\

17

zi 2

'

2

+Z2

'

2

ABC C

(X3,

yz

(X1+X2

A

B (X2,

+ XZ

(xz, yz, zz ( X1

a,

a,

A

Z3 )

+ X

2

yz

zi

D (X4,

+ X3 + X 4

y i + N + y3 + y4

ZL + Z2 + Z3 + Z4

y m, n. a, p5, y BA y.

AB -y )

(n - P)

a) ,

i.e.,

i.e., OP

I, m, n

OP= r,

Ir, mr, ni). a

p

y=

P-

B

286

Obje ctive Mathe matic s

x2-xi,y2-yi,z2-zi.

mi, m)

(tz, mz, nz)

h k + mi mz + m nz = 0 h _ rm _ m tz mz nz bz, ci aia2 + bi b2 + ci C2 C22) &

o; .. o

i

S n

V(a-p + b f + ci

'

Vx(f ri C2 -b 2C i)

+ b\bz +

2

2

+b$ + c£)

) V(a| ai

bi

ci

az

bz

cz

(X2, =

(X2

/1X1

a(x-xi)

+ b(y-yi)

+ c(z-zi)

* y z i - + JL + - - 1 a b c

=0

Co-ordinate Geometry-3D

287

/1

mi

m

=0

bzy+

I axi + byi + czi + d I

d= +

+

r

+ d — ^d >

0 o r

——u

^< 0 —

dz =

288

Objective Mathematics I d - di

^/(a2~H b 2 + c 2 ) : Find the co-ordinates of any point on one of the given planes, preferably putting x = 0, y = 0 or y = 0, z = 0 or z = 0, x = 0. Then the perpendicular distance of this point from the other plane is the requir ed dista nce betwe en the planes. ^

Any

pla ne

pass ing

through

the

line of intersec tion

of the

plan es

0

and

be rep resen ted by the equation

(aix + £>iy+ ciz + di) = 0

Equations of the bisectors of the planes Pi 0 & P 2 = aix+ biy (where >0) are (ax+

+ C1Z+ di = 0

(aix+ biy

V(a 2 +

+ ciz+ di)

c 2)

c2)

Conditions

Acute angle Bisector

aai + bbi + cci > 0

-

+

aai +

+

-

+ cci < 0

Obtuse angle Bisector

The imag e of A (xi, yi, zi) wit h r espe ct to the p lane mirror

0 be

(x2, y 2 , z 2 ) is given

by

Y2 - yi _ z 2 -

X 2 - X1

Z1

- 2 (axi + byi + czi + 2

c ) The f eet of perpendic ular from a point A (xi, y , Zi) on the pla ne given by X2 - xi a

= _

yz - yi b

=

~

z2-bi c

=

- (axi + byi + czi + d)

~

(a 2 + b 2 + c2)

The reflection of the plane ax+

0 be

(x2, y2, z 2 ) is

0 on the place aix+ biy+ ciz+ di = 0 is

2 (aai + bbi + cci) (aix+ biy+ ciz+ di) = (ai + b? + cf' (ax+ by+

If A y z , Azx, AXy be the projecti ons of an a re a A on the co-ordinate pla nes yz, zx and xy respectivel y, then A = V(A^ + A| f +A x ^) If vertices of a triangle are (xi, yi, zi), (x 2, yi 1 y2 V = 2 y3

y 2, z 2 and (X3, y3, z3 ) then zi z2 Z3

Z1

X1

z2

.x 2

Z3

X3

Co-ordinate Geometry -3D

289

&

AXy=± I X3 ys

1 1

Corollary PAIR OF PLANES 1. Homogeneous Equation of Second degree:

degree.

= 0

i.e., 2. Angle betwee n two P la ne s:

e = tan

- i f 2 Vf

2

Corollary THE STRAIGHT LINE

1. Equation of a straight line (General form):

Corollary

0

z=0

0 = y.

2. Equation of a line Passing through a Point and Parallel to a Specified Direction: d

a,

3. Equation of line Passing through two Points:

(X2, yz, zz)

xz -

yz -

zz - z\

4. Symmetric Form:

I, m, n

Objective Mathematics X-X1 I

y-yi m

z-zi n

: Put x = 0 (or y= 0 or z = 0) in the given equations and solve for y and z. The values of x, y and z are the co-ordinates of a point lying on the line. : Since line is perpendicular to the normals to the given planes then find direction cosines. Then write down the equation of line with the help of a point & direction cosines.

Z-Z1 If angl e bet wee n t he line ——— a - - —b— the angle between normal and the line

and the plane aix+biy+ ciz+ d= 0 is 0 then 90° - 9 is

iaai + bb] + cci) co s (90 - 9) = j P p o i * \ = r "V(a + tf + c f) V(af + bf + cf ) (aai + bb\ + ccQ sin 9 = - 2 Vta + b 2 + (?) V(af + b? + c?) : If line is parallel to the plane then aai + bbi + cci = 0

i.e., or

is where al+bm + cn = 0

x- xi y- yi Z-Z1 I m n a (x- xi) + b (y-yi) + c(z-

zi)

If the two lines are

z-z\ x - -x— x-xi y-yi 2 : y-y2 &— h m m2 ni /2 X2-X1 yi - yi Z2 - zi Coplanar then h rm m = 0 Iz mz nz the equation of plane containing the line is x- xi y- yi z -z i mi = 0 m /1 mz nz Iz Let

z-z2

n2

are

X - X1 y-yi Z-Z1 and I m n aix+ biy+ ciz+ di =0= azx+bzy+czz+dz The condition for coplanarity is aixi + biyi + C1Z1+ di a-|/+ bi m + c m a2Xi + b2yi + C2Z1 + dz ~ azl+ bzm + czn Let lines are aix+ biy+ ciz+ di = 0 =

azx + bzy +czz+ dz

The condition that this pair of lines is coplanar is


291 ai a2 a3 a4

bi bz bs b4

C1 C2 C3 C4

dt dz da ck

Two straight lines in space are called skew lines, when they are not coplanar. Thus skew lines are neither parallel, nor intersect at any point.

(S Let and are two skew lines and a line which is perpendicular to both of the line is called the shortest distance between and Let equations of the given lines are x-xi = y - y i = z-zi z-zz a n d x-x2 = y-ys =

h

/T71

m

fe

/TJ2

and

Then the length

fJ2

Let S.D. lie along the line x-a

y~p

z-

y

I

and x - X1

h I

y-yi m m

~ m ~ n S.D. = 11 (x 2 - xi) + m (yz - y, } + n (z 2 - z\) I is Z- Z1 X-X2 y-yz z-zz = 0& = 0 mz nz k m n m n I

If vertices of tetrahedron are (xi, yi, zi), (X2, yz, z 2), (xa, yz, Z3) and ( M , y4, ZA ) is xi yi zi 1

1

x2

y2

z2

6 X3 yB Z3 X4 y4 Z4

MULTIPLE CHOICE Each question in this part has four choices out of which just one is correct. Indicate your choice of correct answer for each question by writing one of the letters a, b, c, d whichever is appropriate. 1. The four lines drawn fro m the vertices of a ny tetrahedron to the centroid of the opposite faces meet in a point whose distance from each vertex is k times the distance from each vertex to the opposite face, where k is

(d) none of these The pair of lines whose direction cosines are given by the equat ions 3/ + m + 5n = 0, 6mn - 2nl + 5 Im = 0 are (a) parallel (b) perpendicul ar (c) inclined at cos

(a)} (d) * 4 2. Which of the stat ement is true ? The coordinate planes divide the line joining the points (4, 7, - 2 ) and (-5, 8, 3) (a) all externally (b) two externally and one internally (c) two internally and one externally

-

(d) none of these 4. The distance of the point A ( - 2 , 3, 1) from the line PQ through P(-3,5,2) which make equal angles with the axes is (a)

^3" , , 16

(b)

(d)i

292


5. The equation of the plane through the point (2, 5, -3) perpendicular to the planes x + 2y + 2z = 1 and x - 2y + 3z = 4 is

12. The point equidistant from the four points (a, 0, 0), (0, b, 0), (0, 0, c) and (0, 0, 0) is

(a) 3JC — 4y + 2z - 20 = 0 (b) 7;r - y + 5z = 3 0 (c)x-2y + z= 11 (d) 10r —y — 4z = 27 6. The equation of the plane through the points (0, —4, -6) and (-2, 9, 3) and perpendicular to

(C)

(a )- 2 (c) 2

7. The equation of the plane passing through the poin ts (3, 2, -1 ), (3, 4, 2) and (7, 0, 6) is 5x + 3y -2z = X where X is (b) 21 (d) 27

8. A variabl e plane whic h remains at a constant distance p from the srcin cuts the coordinate axes in A, B, C. The locus of the centroid of the tetrahedron

OABC is y~z + Z V +

xy1

2 2 2

= kxy z where k is equal to (a) 9p 2

I®- The point on the line

^

=^

=

Z +

(b) - 1 (d )l

(c) cos 0 = j

(d) cos 0 =

^

15. The acute angle between two lines whose direction cosines are given by the relation 2

2

2

betwe en I + m + n = 0 and I +m -n = 0 is (a) n / 2 (b) Jt /3 (c) 7t/ 4 (d) Non e of these lin lines es

J

=

2

=^

>

2

=

1Z

(a) parallel lines (b) intersecting lines (c) perpendicular skew lines (d) None of these 17. The direction consine s of the line drawn fr om P(- 5, 3, 1) t o g (1 ,5 ,- 2) is (a) (6, 2, - 3) (b) (2, - 4, 1)

(b) (3 ,- 2, 1) (d) (3, 2, 1) X

theSC

14. The angle between any two diagonals of a cube is <3 1 (a) cos 0 = — (b) cos 0 =

The me

(b)4 P

( o ^ (d )4 p p 9. The line joining the points (1, 1, 2) and (3, - 2, 1) meets the plane 3x + 2y + z = 6 at the point (a) (1 ,1 ,2 ) (c) (2, - 3 , 1 )

° n e °f

13. P, Q, R, S are four coplanar points on the sides AB, BC, CD, DA of a skew quadrilateral. ^ BQ CR DS J AP The pr od uc— t- • equals

the plane x - 4y - 2z = 8 is (a) 3JC + 3y - 2z = 0 (b)x-2y +z=2 (c)2x+y-z =2 (d) 5x-3y + 2z = 0

(a) 23 (c) 19

(d ) N

( 2 ' 2 ' 2 )

^ at (c) (- 4, 8,- 1)

a distance of 6 from the point (2, -3, -5) is (a) ( 3, -5 ,- 3) (b) (4, -7 , -9 ) (c) (0,2 -S 1 ) (d) (- 3, 5, 3) 11. The plane passing through the point (5, 1, 2) perp en dicu la r to the lin e 2 ( j t - 2 ) = y - 4 = z - 5 will m eet the line in the point (a) (1 ,2 , 3) (b) (2, 3, 1) (c) (1 ,3 , 2) (d) (3, 2, 1)

(d)[^f.f.-f

18. The coordinates of the centroid of triangle ABC where A,B,C are the points of interse ction of the plane 6x + 3 y - 2 z = 18 with the coordinate axes are (a) (1,2,-3) (b) (- 1, 2, 3) (c) (- 1, -2 ,- 3) (d) (1, -2, 3) 19. The intercepts made on the axes by the plane which bisects the line joining the poin ts (1 ,2 , 3) and (-3,4, 5) at right angles are


(0

9.-f,9

293 (b)( 2.9,9

i \

(d )| 9 , | , 9

(c)

2

20. A line mak es angle s a , P, y, 8 with the four diagonals of a cube. Then cos 2

2

a + cos 2 P

2

+ cos (a) 4/3 (c) 3

cos 8 is

x

.

2 1

y

2 1

2 P 4

2 ~~

Z 1

X y Z P 23. The distance between two points P and Q is d and the length of their projections of PQ on the coordinate planes are d\,d2,d3. Then

d\ + (a) 1 (c)3

d] = Kd2 where K is (b)5 (d) 2

24.The

line | = = f is vertical. The 2 3 1 direction cosines of the line of greatest slope in the plane 3x - 2y + z = 5 are Proportional

to (a> (16, 11 ,- 1) (b) (- 11 , 16, 1) (c) (16, 11,1 ) (d) (11, 16 ,- 1) 25. The symmetric form of the equations of the line JC + y - z = 1, 2x - 3V + z = 2 is

=

JC-

1 3

z ( b )

5

£ = 2= in i 2 3 5

1

3

w

5

3

2

5

26. The equati on of the plane whi ch passe s through the jc-axis and perpendicular to the line


21. A variable plane passes through the fixed point (a, b, c) and meets the axes at A, B, C. The locus of the point of intersection of the pla nes throu gh A, B, C and parallel to the coordinate planes is , . a b c a b c (a)- + - + - = 2 (b) —h — H— = 1 x y z x y z . . a b c . . . . a b c ( c ) - + - + - = - 2 (d) - + - + - = - 1 x y z x y z 22. A plane moves such that its distance fro m the srcin is a constant p. If it intersects the coordinate axes at A, B,C then the locus of the centroid of the triangle ABC is 1 1 1_ 1 (a) - J + ~2 + ~2 - ~2 x y z p ,, 1 1 1 9 (b) -xj + -yj + ~2 Z = P— (w

x

( J C -1)

(y + 2)

(z-3)

cos 0 sin 9 (a) jctan e + y sec 0 = 0

0

is

(b) x sec tan 00 == 00 (c) JC cos 00 + + yy sin (d) JC sin 0 - y cos 0 = 0 27. The edge of a cube is of length of a. The shortest distance between the diagonal of a cube and an edge skew to it is (a) a 42 (b) a (c) 42/a (d) a/42 28. The equation of the plane passing through the intersection of the planes 2x — 5y + z = 3 and JC + y + 4z =5 and parallel to the plane JC + 3y + 6z = 1 is x + 3y + 6z = k, where k is (a) 5 (b) 3

(c )7

(d) 2

29. The lines which intersect the skew lines y = mx, z = c; y = - mx, z = - c and the x-axis

lie on the surface (a) cz = mxy (c) xy = cmz

(b) cy = mxz (d) Non e of these

30. The equation of the line passing through the poin t (1, 1, - 1 ) and per pendi cul ar to the plane x - 2y - 3z = 7 is y - 1 z+ 1 :x - 1 (a ) - 1 2 2 3 '

x- 1 y- 1 z+ 1 (b) : ' - 1 ~ - 2 3 y - 1 z+1 :x - 1 (c ) 1 ~ -2 ~ - 3 (d) none of these 31. The plan e 4x + 7y + 4z + 81 = 0 is rotated through a right angle about its line of intersection with the plane 5x + 3y + 10z = 25. The equation of the plane in its new position is x - 4y + 6 z = k, where k is (a) 106 (b) - 8 9 (c) 73

(d) 37


294

32. A plane meets the coordina te axes in A, B, C such that the centroid of the triangle ABC is the point (a, a, a). Then the equation of the plane i s x + y + z = p where p is (a) a (b) 3 /a a/3(c) . (d) 3 a 33. If from the point P (a, b, c) perpendiculars PL, PM be drawn to YOZ and ZOX planes, then the equation of the plane OLM is (a)- + £ + - = 0 a b c

(a) 1 : 1 (c) 3 : 1

( b) 2 : 1 (d) 1 : 3

39. A plane makes in tercepts OA, OB, OC whose measurements are a, b, c on the axes OX, OY, OZ. The area of the triangle ABC is

(a) ^ {ab + bc + ca) 1

(

2 . , 2 2 . 2 2,. 1/2

(b) —(a b + b c +c

(b) — - £ + - = 0 a b c

a)

(c) ^ abc (a + b + c)

a b c

a b c

34. A variable plane makes with the coordinate plane s, a tetrah edron of constant vol ume 64 k 3. Then the locus of the centroid of tetrahedron is the surface (a) xyz = 6k2 (b)xy + yz + zx = 6 k 2 (c) x 2 + y 2 + z 2 = &k 2 (d) none of these

—+ + - = k, meets the a b c co-ordinate axes at A, B, C such that the

35. The

plane

centroid of thek is triangle ABC is the point (a , b, c). Then (a) 3 (b )2 (c) 1 (d)5 36. The perpendicul ar distance of t he origin from the plane which makes intercepts 12, 3 and 4 on x, y, z axes respectively, is (a) 13 (b)ll (c) 17 (d) none of thes e 37. A plane meets the coordinate axes at A, B, C and the foot of the perpendicular from the srcin O to the plane is P, OA = a, OB = b, OC = c. If P is the centroid of the triangle ABC, then (a) a + b + c = 0 (b) I a I = I b I = I c I (c) — + 7 + —= 0 (d) none of these a b c 38 . A, B, C,D is a tetra hedron. A,, Bt, Ch D, are respectively the centroids of the triangles

BCD, ACD, ABD and ABC\ AA,, BB h DDj divide one another in the ratio

CCh

(d )±(a

+ b + c)2

40. The project ions of a line on the axes are 9, 12 and 8. The length of the line is (a) 7 (b) 17 (c) 21 (d) 25 41 . If P, Q, R, S are the points (4, 5, 3), (6, 3, 4), (2, 4, -1), (0, 5, 1), the length of projection of RS on PQ is (b)3 (c )4 (d) 6 42 . The distance of the point P (- 2, 3, 1) fro m the line QR, through Q ( - 3, 6, 2) which

makes equal angles with the axes is (a)3 (b)8 (c) <2 (d) 2 <2 43 . The direction ratios of the bisector of the angle between the lines whose direction l\, l cosines are 2, m2, n2 are

(a) /| + l2, nj] + m2, «] + n2 (b) /|ffj

2

—

h mb

m n

i 2 ~ w 2 ni> n\h ~

n

ih

(c) l]m2 + l2mx, mxn2 + m2nx, nxl2 + n 2 /] (d) none of these 44. The poin ts (8, - 5 , 6), (11, 1, 8), (9, 4, 2) and (6, -2, 0) are the vertices of a

(a) rhombus (b) square (c) rectangle (d) parallelogram 45. The straight lines whose direction cosines are al + bm + cn = 0, given by fmn + gnl + him = 0 are perpendicular if


295

(a)^ + f + ^ = 0 a b c 2 , 2

...a

2

b

c

g

h

(b) — + — + — = 0 /

(c) a\g + h) + b 2 (h +f) + c 2 (f + g) = 0 (d) none of these 46. The three plane s 4y + 6z = 5; 2x + 3)- + 5z = 5; 6x + 5y + 9z = 10 (a) meet in a point (b) have a line in commo n (c) form a triangular prism (d) none of these 47. The

line

x + 1 y + 1

z + 1

2 - 3 ~ 4 plane x + ly + 3z = 14, in the (a) (3 ,- 2, 5) (b) (3, 2, (c )( 2, 0, 4) (d) (1, 2, 48. The foot of the perpendicu lar x +1 —y - 2 z+ to the line - 2

meets

the

point -5) 3) fro m P (1, 0, 2) 1 i. is the point

(a) (1,2,-3)

0»|

(c) (2, 4, - 6 )

(d) (2, 3, 6)

49. The length o f the perpend icular fr om (1, 0, 2) x+ 1 y- 2 z+ 1. on the line is 3 ~ -2 ~ -1 3 46 (a) 2 • 5 (c) 3 4l (d) 2 Vf 50. The plane containing the two x-3 v-2 z —1 , x-2 = and ~r=4 — z+l is 11 x + my + nz = 28 where 5 (a) m = - 1, n = 3 (b) 7M = 1, n = — 3 (c) m = - 1, n = - 3 (d )m =l ,n = 3

lines y+3

- 1

Practice Test MM: 20

Time: 30 Min.

(A) There are 10 parts in this question. Each part has one or more than one correct answer(s). [10 x 2 = 20] 3. The distan ce of th e point (2, 1 , - 2) from th e 1- The projection of the line X + ^ = « = Z Q ^ 1 y+1 2 -3 line — 1 z u measured parallel 2 1 - 3 on the plane x - 2y + z= 6 is th e line of to the plane x + 2y + z = 4 is intersection of this plane with the plane (a) <10 (b) V20 (a) 2x +y +z =0 (c) V5 (d) V30" (b) 3x+y -z = 2 4. The shortest distance betwe en th e lines (c) 2x - 3y + 8z = 3 x - 3 y +15 z - 9 , x + 1 y— 1 (d) none of the se 2. A variable plane passes through a fixed 2 -9 . po int (1, - 2 , 3) an d me et s the co-ordinate = — I S axes in A, B, C. The locus of the point of (a) 2 V3 (b) 4 V3 inter sectio n of th e pl anes thro ugh A, fi, C (c) 3 V6 (d) 5 V6 pa ra ll el to the co-ordinate pl an es is th e 5. The area of the triangle whose vertices are surface , , 1 1 at the points (2, 1, 1), (3,1, 2), (-4, 0,1) is + -zx = 6 (a)xy--yz (a)Vl9" (b)|Vl9 (b) yz - 2zx + 3ry = xyz (c) xy - 2yz + 3zx = 3xy2 (c)|V38 (d)|V57 (d) none of these


296 6. The equation to the plane through the points (2, -1, 0), (3, -4, 5) parallel to a line with direction cosines proportional to 2, 3, 4 is 9x - 2y - 3z = k where k is (a) 20

(b) -2 0

(c) 10 (d) - 1 0 7. Through a point P ( f , g, h) a plane is drawn at right angles to OP, to meet the axes in A, B, C. If OP=r, the centroid of the triangle ABC is (a) ' f_ JL A 3r ' 3r ' 3r (b)

3f 2

3g 2

r r r (c) {3f'3g'3h

(a) ±VF + m cos a m

(b)

si n a

+ m tan a (d) none of these 9. If a stra ight line makes an angle of 60° with each of the X and Y axes, the angle which it makes with the Z axis is (b);

'i

(d)-3n 4 10. The condition for the lin es x = az + b, y = cz+d and x = a tz + b\,y = c\z + di to be perpendicular is (a) aci + 0^0 = 1 (Of

3h 2

(b) aa1 + cc1 + 1 = 0

(d) none of t hes e The plane Ix + my =0 is rotated about its line of intersection with the xOy plane through an angle a. Then the equation of the plane is Ix + my +nz = 0 where n is

(c) bci + b]C + 1 = 0 (d) none of th ese

Record Your Score Max. Marks 1. First attempt 2. Second attempt 3. Third attempt

must be 100%

Answers 4. (b) 10. (b) 16. (c)

5. (d)

6. (c)

9. (b) 15. (b)

11. (a) 17. (d)

12. (c) 18. (a)

(c) (c) (d) (c)

21. (b) 27. (d)

22. (b) 28. (c)

23. (d) 29. (b)

24. (d) 30. (c)

33. (c) 39. (b)

34. (a) 40. (b)

35. (a)

36. (d) 42. (d)

44. (b) 50. (c)

45. (a)

46. (b)

l.(c) 7. (a) 13. (d)

2. (c) 8.(d) 14.(d)

19. (a) 25. (c)

20. 26. 32. 38.

31. (a) 37. (b) 43. (a),(c)

3. (c)

41. (a) 47. (d)

48. (b)

5. (c)

6. (&)

49. (a) 1. (a)

2. (b)

3. ((d)

7. (c)

8. (c)

9. (b),(d)

4. (b) 10. (b)

Sk Goyal maths book

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