ALGEBRA (Practice Sheet).pdf

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Practice Question Question based on

Q.1

LEVEL –1

Roots of Quadratic Equation

The roots of the equation (x+2) 2 = 4 (x+1) – 1

Question based on

Q.8

are-

Q.2

Q.4

Nature of roots

If roots of the equation ax 2 + 2 (a+b)x + (a + 2b + c)= 0 are imaginary, then roots of the

(A) ±1

(B) ± i

equation ax2 + 2bx + c = 0 are -

(C) 1,2

(D) – 1, –2

(A) rational

(B) irrational

(C) equal

(D) complex

The roots of quadratic equation x 2 + 14x + 45 = 0 Q.9

are -

Q.3

Quadratic - Equation

If a and b are the odd integers, then the roots of

(A) – 9, 5

(B) 5, 9

the equation 2ax2 + (2a + b) x + b = 0, a

(C) – 5, 9

(D) – 5, – 9

be(A) rational (C) non-real

The roots of the equation x 4 – 8x2 – 9 = 0 are(A) ±3, ±1

(B) ±3, ±i

(C) ±2, ±i

(D) None of these

Q.10

Which of the following equations has 1 and –2 as

 0, will

(B) irrational (D) equal

If the roots of the equation 6x 2 – 7x + k = 0 are rational then k is equal to (A) – 1 (B) –1, –2 (C) – 2

(4) 1, 2

the roots Q.11

(A) x2 – x – 2 = 0

(a2 + b2) x2 – 2(bc+ ad) x + (c2 + d2) = 0 are equal, if (A) ab = cd (B) ac = bd

(B) x2 + x – 2 = 0 (C) x 2 – x + 2 = 0 (D) x2 + x + 2 = 0 Q.5

Q.6

(C) ad+ bc = 0

Roots of 3x + 3 – x = 10/3 are(A) 0, 1

(B) 1, – 1

(C) 0, – 1

(D) None of these

Q.12

For what value of m, the roots of the equation

(A) ]

If f(x) = 2x 3 + mx2 – 13x + n and 2 and 3 are and n are-

(4) None of these

x2 –x + m = 0 are are not real1 4

,[

1 1 (C) ] – , [ 4 4

roots of the equations f(x) = 0, then values of m

Q.7

The roots of the equation

Q.13

(B) ] – ,

1 4

[

(4) None of these

Roots of the equation

(A) 5, 30

(B) – 5, 30

(a + b – c)x 2 – 2ax + (a – b + c) = 0,

(C) – 5, – 30

(D) 5, –30

(a,b,c  Q) are – (A) rational

(B) irrational

(C) complex

(D) none of these

The number of roots of the quadratic equation 8 sec2  – 6 sec  + 1 = 0 is (A) Infinite

(B) 1

(C) 2

(D) 0

Q.14

The roots of the equation x 2 – x – 3 = 0 are(A) Imaginary Imaginar y (B) Rational (C) Irrational (D) None of these

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1

Q.15

The roots of the equation x2 + 2 3 x + 3 = 0 are are-(A) Real and equal (B) Rational and equal (C) Irrational and equal

Question based on

Q.23

2

If the roots of the equation ax + x + b = 0 be real, then the roots of the equation Q.24

x2 – 4 ab x + 1 = 0 will ill be be (A) Rational (C) Real Q.17

(B) Irrational Irrationa l (D) Imaginary Imaginar y

If one root of equation x2 + px + 12 = 0 is 4, while the equation x 2 + px + q = 0 has equal roots then the value of q is-

Q.18

(A) 49/4

(B) 4/49

(C) 4

(D) None of these

(A) A.P.

(B) H.P.

(C) G.P.

(D) None of these

(C) a

 

1 4 1 16

(B) a



(A) 3, 3/2

(B) 3, 1

(C) 1, 3/2

(D) None of these

  are roots of the equation x 2 + px – q = 0 and   are roots of x 2 + px + r = 0, then the value of (  – ) ( – ) is-

If

(A) p + r

(B) p – r

(C) q – r

(D) q + r

If ,  are roots of the equation 2x 2 – 35x + 2 = 0, then the value of (2  – 35)3. (2 – 35)3 is equal to(A) 1 (C) 64

Q.26

(B) 8 (D) None of these

If ,  are roots of the equation px 2 + qx – r = 0, then the value of (A) –

If the roots of x 2 – 4x – log 2a = 0 are ar e real, then(A) a

Q.20

Q.25

If roots of the equation (a – b)x 2 + (c – a)x + (b – c) = 0 are equal, then a, b, c are in -

Q.19

For what value of a, the the difference difference of roots of the equation (a –2)x 2 – (a – 4)x – 2 = 0 is equal to 3

(D) Irrational and un equal Q.16

Sum and product of roots

1

p 2

qr

 2

(3pr + q 2)

q (C) – 2 (3pr –q2) pr

8

+

 2

is equal to-

(B) –

(D)

q pr 2

p pr 2

(3pr + q 2)

(3pr + q)

(D) None of these Q.27

If product of roots of the equation

If the roots of both the equations

mx2 + 6x + (2m – 1) = 0 is – 1, then m equals(A) – 1 (B) 1

px2 + 2qx + r = 0 and qx 2 – 2 pr x + q = 0 are

(C) 1/3

(D) – 1/3

real, then Q.28

(A) p = q , r  0

For what value of a the sum sum of roots of the

(B) 2q = ± pq

equation x2+ 2 (2 – a – a 2)x – a2 = 0 is zero (A) 1, 2 (B) 1, – 2

(C) p/q = q/r

(C) – 1, 2

(D) – 1, – 2

(D) None of these Q.29 Q.21


x2 – 7x – 9 = 0 is -

(p – 2)x2+ 2(p – 2)x + 2 = 0 are not real when-

(A) 7

(B) (B)

(C) 9

(D) 2 85

(A) p  [1, 2] (C) p  (2, 4) Q.22

The difference between the roots of the equation

 [2, 3] (D) p  [3, 4] (B) p

If the roots of the equation x 2 – 10x + 21 = m are equal then m is(A) 4

(B) 25

(C) – 4

(D) 0

Q.30

85

The HM of the roots of the equation x2 – 8x + 4 = 0 is (A) 1 (C) 3



2

Q.31

Q.32

If the sum of the roots of the equation

Q.39

ax2 + 4x + c = 0 is half of their difference, then

are

the value of ac is(A) 4 (C) 12

(p, q) is equal to -

If the

(B) 8 (D) – 12

sum of the

roots roots

of the equation

Q.40

2

(a + 1)x + (2a + 3) x + (3a + 4) = 0 is –1, then the product of the roots is (A) 0 (C) 2

(B) 1 (D) 3

Sum of roots is – 1 and sum of their r eciprocals is 1 6

(B) x – x + 6 = 0

2

(D) x – 6x + 1 = 0

If   are roots of the equation 2

2x – 5x + 3 = 0, then

 +   is equal to2

2

(A) 125, 216

(B) 125, – 216

(C) Only 125

(D) Only – 216

  are roots of the equation x 2 – mx + n then value of (1 +  + 2) (1+  + 2) is If

= 0,


If the equation

a xa

b

+

x  b

= 1 has roots equal

(B) – 15/4

in magnitude but opposite in sign, then the value

(C) 15/4

(D) – 15/2

of a + b is -

If   be the roots of the equation 2

2 2

p(x + n ) + pnx + qn x = 0 then the value of p ( +  )+ p + q 2

2



2 2

is -

(A)  + 

(B) 0

(C) p + q

(D)

 and  are roots of ax – bx + ( + 1) ( + 1) is equal to a  b  c a  b  c

If

(C)

a a  b  c a

(B)

(D)

Q.43

 +  + p + q 2

(A)

c = 0, then

a b  a  c

Q.44

(C) (1 + 2q) 2

(D) 2q – 3

If



and



(D) None of these

 and  are the root of ax 2 + bx + c = 0, then  1 1  the value of    is a  b a  b  If

a bc

c ab

(B)

b ca

(D) None of these

If roots of the equations 2x 2 – 3x + 5 = 0 and other then (a, b) equals -

Q.45

(A) ( –5, 3)

(B) (5, 3)

(C) (5, –3)

(D) ( –5, –3)

If the sum of the roots of ax 2 + bx + c = 0 be equal to sum of the squares, then -

are the roots of the equation

 )x += 0 then the values of  and  are (A)  = 1,  = –2 (B)  = 2,  = –2 (C)  = 1,  = –1 (D)  = –1,  = 1 x2 +(

(C) 1

ax2 + bx + 2 = 0 are reciprocals of the roots of the

If difference of roots of the equation (B) (1 – 2q) 2

(B) 0

(C)

a

(A) 2q + 3

(A) – 1

(A)

x2 – px + q = 0 is 1, then p 2 + 4q2 equals-

Q.38

If one root of the equation x 2 – 30x + p = 0 is

(A) 15/2

2

Q.37

(D) (2, 1)

(C) 1 – (m– n) + (m 2 + mn + n 2)

2

(C) 6x + x + 1 = 0

Q.36

(C) (– 2, 1)

(B) 1 + (m + n) + (m 2 + mn + n 2) 2

(A) x + x – 6 = 0

Q.35

(B) (– 1, 1)

(A) 1 + (m + n) + (m 2 – mn + n 2)

, then equation is – 2

Q.34

(A) (1, 1)

square of the other, then p is equal to-

Q.41 Q.33

 and  of the equation x 2 + px + q = 0 such that 3  + 4 = 7 and 5 –  = 4, then

If roots

Q.46

(A) 2 ac = ab + b2

(B) 2 ab = bc + c2

(C) 2bc = ac + c 2

(D) None of these

If one root of ax 2 + bx + c = 0 be square of the other, then the value of b 3 + ac2 + a2c is(A) 3 abc

(B) – 3abc

(C) 0

(D) None of these


3

Question based on

Q.47

Q.48

Formation of Quadratic Equation with given roots

Q.53

The quadratic equation with one root

The quadratic equation with one root 2i is-

(A) 2x2 + 2x + 1 = 0

(A) x2 + 4 = 0

(B) x 2 – 4 = 0

(B) 2x2 – 2x + 1 = 0

(C) x2 + 2 = 0

(D) x 2 – 2 = 0

(C) 2x2 + 2x – 1 = 0

Q.54

If

(A) x + 2x + 15 = 0 equation whose roots are

(B) x2 + 15x + 2 = 0 (C) 2x2 – 2x + 15 = 0

If  and  are roots of 2x 2 – 3x – 6 = 0, then the equation whose roots are

2 +

2 and

2 + 2 will Q.55

be -

(B) 3x 2 + 2x + 1 = 0

(C) 3x2 – 2x – 1 = 0

(D) x 2 – 3x + 1 = 0

If  and  be the roots of the equation 2x2 + 2(a + b)x + a 2 + b2 = 0, then the equation

(B) 4x2 – 49x – 118 = 0

whose roots are (  + )2 and ( – )2 is-

(C) 4x2 – 49x + 118 = 0

(A) x2 – 2abx – (a 2 – b2)2 = 0

(D) 4x + 49x + 118 = 0

(B) x2 – 4abx – (a 2 –b2)2 = 0

If  and  are roots of 2x2 – 7x + 6 = 0, then the quadratic equation whose roots are –

2



(C) x2 – 4abx + (a 2 – b2)2 = 0 (D) None of these

2 , – is-



(A) 3x + 7x + 4 = 0

2 = 5 – 3, equation whose roots are / and  / is-

(B) 3x2 – 7x + 4 = 0

(A) x2 – 5x – 3 = 0

(C) 6x2 + 7x + 2 = 0

(B) 3x2 + 12 x + 3 = 0 (C) 3x2 – 19 x + 3 = 0 (D) None of these

Q.56

2

(D) 6x2 – 7x + 2 = 0 If roots of quadratic equation ax 2 + bx + c = 0 are

 and  then symmetric expression of its roots is (A)

 2 +  

(C) 2 +22

Q.52

(A) 3x2 – 2x + 1 = 0

(A) 4x2 + 49x – 118 = 0

2

Q.51

 1  1 and will  1  1

be -

(D) x2 – 2x – 15 = 0

Q.50

is-

 and  are roots of x 2 – 2x + 3 = 0, then the

2

Q.49

1 i

(D) 2x2 – 2x – 1 = 0

The sum of the roots of a equation is 2 and sum of their cubes is 98, then the equation is -

1

(B) 2 –2+  –2  2

(D)

Question based on

Q.57

2

(1 

   1     1          

2

(B) x + x – 1 = 0 2

(C) x + x + 1 = 0 (D) x2 – x + 1 = 0

= 5 – 3,

then the

Roots under particular cases

For the roots of the equation a – bx – x 2 = 0

(B) negative and same sign (C) greater root in magnitude is negative and opposite in signs (D) greater root is positive in magnitude and opposite in signs

 3 ) is-

(A) x2 – x – 1 = 0

   but 2

(a > 0, b > 0) which statement is true (A) positive and same sign

The quadratic equation with one root 1

If

Q.58

If p and q are positive then the roots of the equation x2 – px– q = 0 are(A) imaginary (B) real & of opposite sign (C) real & both negative (D) real & both positive

IIT - ian’s PACE ; ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com

4

Q.59

If a > 0, b > 0, c > 0, then both the roots of the

Q.67

2

Q.60

equation ax + bx + c = 0 -

the equation x – 3x + 2 = 0 and x 2 + 2x – 3 = 0

(A) Are real and n egative (B) Have negative real parts (C) are rational n umbers (D) None of these

then the value of f( ) is (A) 3 (B) 2 (C) 1 Q.68

The roots of the equation ax + bx + c = 0 will be imaginary if -

=2

(C) m = 2

(D) – ac

If both the roots of the equations k(6x 2 + 3) + rx + 2x2 – 1= 0 & 6k(2x2 + 1) + px + 4x 2 – 2 = 0

(B)

(A) 1

2

x

+ mx – 2 = 0 are



=–2

(D) m = –2

Question based on

Q.70

(B) – 1

Q.71 Condition for common roots

(D) 0

For all real values of x, the maximum value of the x x

2

 5x  9

(A) 1 (C) 90

(B) –1 (D) 2

(C) 2

Quadratic Expression

expression

, then a will tend to-

Question based on

(B) a + c (C) ac

are common, then 2r – p is equal to -

If one of the roots of x(x + 2) = 4 – (1– ax 2) tends (A) 0 (C) 1

If the two equations x 2 – cx + d = 0 and

(A) 0 Q.69

If roots of the equation

(A)

(D) 0

x2 – ax + b = 0 have one common root and the second has equal roots, then 2(b + d) =

2

reciprocal of each other, then-

Q.62

 is a common root of

2

(A) a > 0, b = 0, c < 0 (B) a > 0, b = 0, c > 0 (C) a = 0, b > 0, c > 0 (D) a > 0, b > 0, c = 0 Q.61

If f(x) = 4x 2 + 3x – 7 and

is(B) 45 (D) None of these

If x is real, then the value of the expression 2

 34x  71 does not exist betweenx  2x  7

x

2

Q.63

If the equation x2 – ax + b = 0 and x 2 + bx – a = 0 have a common root, then(A) a = b (C) a – b = 1

Q.64

2

Q.73

If one of the factors of ax 2 + bx + c and bx2 + cx + a is common, then -

Q.74

x2 + k(2x + 3) + 4(x + 2) + 3k – 5 is a perfect square, if k equals (A) 2 (B) – 2 (C) 1 (D) – 1

Q.75

If  – x is a factor of x2 – ax + b, then (a – equal to-

The equation ax2 + bx + a = 0 & x 3 – 2x2 + 2x – 1 = 0 have two root in common, then (a + b) is equal (B) 0 (D) 2

(B) p = – 8 (D) p = – 6

(A) a = 0 (B) a 3 + b3 + c3 = 3 abc (C) a = 0 or a 3 + b3 + c3 = 3abc (D) None of these

If one of the roots of x 2 + ax + bc = 0 and

to (A) 1 (C) –1

(B) 5 and –9 (D) 5 and 9

The factors of 2x 2 – x + p are rational if (A) p = 3 (C) p = 6

(D) 1, 24

x2 + bx + ca = 0 is common, then th eir other roots are (A) a, b (B) b, a (C) b, c (D) c, a Q.66

Q.72

2

If x – 11x + a = 0 and x – 14x + 2a = 0 ha ve one common root then a is equal to(A) 0, – 24 (B) 0, 1 (C) 0, 24

Q.65

(B) a + b = 0 (D) a – b + 1 = 0

(A) –5 and 9 (C) –5 and –9

(A) –b

(B) b

(C) a

)

is

(D) –a


5

Q.76

If x + 1 is a factor of the expression 4

3

2

x + (p – 3)x – (3p – 5) x + (2p – 9) x + 6 then the value of p is(A) 1 (C) 3

Question based on

Q.84

(B) 2 (D) 4

Sign of Quadratic Expression

The diagram shows the graph of y = ax2 + bx + c. Theny

Q.77

If x be real then the minimum value of 40 – 12x + x2 is (A) 28 (C) –4

Q.78

(B) 4 (D) 0

If x be real then the value of

x

x 2  2x  1 x 1

will not

lie between(A) 0 and 8 (C) – 8 and 0

(B) – 8 and 8 (D) None of these

Q.85

Q.79

(B) b2 – 4ac < 0

(C) c > 0

(D) b2 – 4ac = 0

The

Inequality

(A) If x be real then 2x2 + 5x – 3 > 0 if (A) x < –2 (B) x > 0 (C) x > 1

Q.80

Q.81

x (x1, 0)

(A) a > 0

y= Question based on

(x2, 0) 0

(C)

maximum 1 4x 4 3

13 4

2

 2x  1

value

of

the

function

is(B)

5 2

(D) None of these

(D) –3 < x < 1/2

The solution of the equation 2x 2 + 3x – 9  0 is given by(A) 3/2  x  3

(B) – 3  x  3/2

(C) –3  x  3

(D) 3/2

x2

If for real values of x, x 2 – 3x + 2 > 0 and x2 – 3x – 4

 0, then-

(A) –1  x < 1 (B) –1  x < 4 (C) –1  x < 1 and 2 < x  4 (D) 2 < x  4 Question based on

Q.82

Quadratic Expression in two variables

If x2 + 2xy + 2x + my – 3 have two rational factors then m is equal to (A) 6, 2 (B) – 6, 2 (C) 6, –2

Q.83

(D) –6, –2

If 2x2 + mxy + 3y2 – 5y – 2 have two rational factors then m is equal to(A) ± 7 (C) ± 5

(B) ± 6 (D) None of these


6

LEVEL- 2 Q.1

If roots the equation

Q.8

x2 (1 + m 2)+ 2 mcx + c 2 – a2 = 0 are equal, then value of c is-

+ (a2 – 3a + 2) = 0 are of opposite signs, then a lies in the interval (A) (– , 1) (C) (1, 2)

(A) a

(1  m 2 )

(B) a

(1  m 2 )

(C) m

(1  a 2 )

(D) m

(1  a 2 ) Q.9

Q.2

If the roots of the equation

xa ax  1

=

x  b bx  1

are

Q.3

The equation x –

x 1

= 1 –

2 x 1

(A) p = R – (B) p

has -

Q.10

(B) x2 –

(A) only one real n umber (B) real and sum = 1

Q.11

The roots of the equation 2

2

x – 2px + p + q + 2qr + r = 0 (p, q, r (A) rational and different (B) rational and equal (C) irrational (D) imaginary

 Z) are

Q.7

p  q p  q

is-

 q  p    x – q p   

pq ( q  p ) 2

= 0


The imaginary roots of the equation (x2 + 2)2 + 8x 2 = 6x (x2 + 2) are (A) 1± i (C) – 1 ± i

Q.13

2 (A) 7

x – 3kx + 2e – 1 = 0 is 7, then(A) roots are integers and positive Q.14

(B) 2 ± i (D) None of these

If one root of the equation 2x 2 – 6x + c = 0 is 3  5i

log k

(B) roots are integers and n egative (C) roots are rational not integers (D) roots are irrational

If one root of the equations ax 2 + bx + c = 0 and

(C) a = b or b = c or c = a (D) None of these

If product of roots of the equation 2

– p

(A) a + b + c = 0 (B) a = b = c

If a, b, c are positive real numbers, then the number of real roots of the equations ax2 + b |x| + c = 0 is(A) 0 (C) 2

,

x2 + x + 1 = 0 is common, then-

Q.12 Q.6

q

(C) (p + q) x 2 + (p2 – q2) x – pq = 0 (D) None of these

(C) real and sum = 0 (D) real and product = 0

2

The equation whose roots are

(A) (p + q) 2 x2 + (p2 – q2) x + pq = 0

The roots of the equation |x| 2 + |x| – 6 = 0 are-

2

1     1,   2  

(D) p = – 1

(D) infinitely many roots

Q.5

1   ( – , – 1)    ,   2 

(C) p 

(A) no root (B) one root (C) two equal root

Q.4

For what values of p, the roots of the equation imaginary-

(B) b = 2 (D) b = 0 2

(B) (– , 0) (D) (3/2, 2)

12(p + 2)x2 – 12 (2p –1)x– 38p – 11= 0 are

reciprocal to each other, then (A) a = 1 (C) a = 2b

If roots of the equation 3x 2 + 2(a2 + 1)x

, then the value of c will be (B) – 7

(C) 17

(D) – 17

If ,  are roots of the equation ax 2 + bx + c = 0 and  –  =  then – (A) b2 – 4ac = c 2

(B) b2 – 4ac = a 2

(C) a (b2 + 4ac) = 2c

(D) b2 + 4ac = a


7

Q.15

Q.16

If x – 2 is a common factor of x 2 + ax + b and

Q.22

x2 + cx + d, then (A) d – b = 2 (c– a) (B) b – d = (c– a)

ax2 + bx + c = 0 is equal to the sum of the square

(C) 4 + 2c + b = 0 (D) b – d = 2 (c – a)

(B) c2 b, a2c, b2a are in G.P.

of their reciprocal, then(A) c2 b,a2c,b2a are in A.P.

(C)

If x = 6  6  6  ... , then (A) – 2 < x < 3 (C) x = 3

(D)

(B) 2 < x < 3 (D) x > 3 Q.23

Q.17

If the sum of the roots of the equation

b

,

a

c b b

,

a

c b

, ,

c a c a

are in H.P. are in G.P.

If the quadratic equations 3x2 + ax +1 = 0 and 2x2 + bx + 1 = 0 have a common root, then the

If x2/3 + x 1/3 – 2 = 0 then x(A) –2, 1 (B) –8, –2 (C) –8, 1 (D) None of these

value of the expression 5ab –2a 2 –3b2 is(A) 0 (B) 1 (C) –1

Q.18

If 8, 2 are roots of the equation x 2 + ax + 2

and 3, 3 are roots of x +



=0

x + b = 0 then roots of

Q.24

statement is true(A) reciprocal of roots of one another (B) reciprocal of roots of one another and opposite signs

If the difference of the roots is equal to the product of the roots of the equation

(C) roots are of opposite signs of each other (D) equal in product

2x2 – (a + 1)x + (a – 1) = 0 then the value of a is(A) 2 (B) 3 (C) 4

Q.25

(D) 5

For the roots of the equations 2x 2 – 5x + 1 = 0 and x2 + 5x + 2= 0 , which of the following

the equation x2 + ax + b = 0 are (A) 1, 9 (B) –1, 8 (C) 2, – 9 (D) –2, 8 Q.19

If x is real, then the values of the expression ( x  m)

Q.20

Q.21

If one root of the equation x – x – k = 0 is square of the other, then k equals to (B) 3 ± 2

(C) 2 ± 3

(D) 5 ± 2

The roots of a1x2 + b1x + c1 = 0 are reciprocal of

(D) between m and m + n Q.26

(B) (C)

a2 b1 b 2 a1 a2

= =

(D) a 1 =

b1 b 2 c1 a2

b1 c2 1 a2

= = =

 4x  1 is x 2  4x  2

c1

(A) any number (B) only positive number (C) only negative number (D) only 1

c2 a1 c2 c1

Q.27

b 2

, b1 =

If x is the real, then the value of the expression 2x 2

a2x + b2x + c2 = 0, if=

are not -

(B) greater than (m + 2n) (C) between 2m and 2n

2

a1

 4mn

(A) greater than (m + n)

the roots of the equation

(A)

2

2( x  n )

2

(A) 2 ± 5

(D) None of these

1 b 2

, c1 =

1 c2

If one root of the equations ax 2 + bx + c = 0 is equal to n th power of the other root, then (acn)1/ (n+1) +(anc)1/(n+1) equals (A) – b (B) b (C) (– b) 1/(n+1)

(D) (b)

1/(n+1)


8

Q.28

Q.29

|x2 + 4x + 3| + 2x + 5 = 0 is-

For what value of a the curve y = x 2 + ax + 25 touches the x-axis-

(A) 2 (C) 4

(A) 0 (C) ±10

The number of real roots of the equation

Q.37

(B) 3 (D) 1

If product of roots of the equation

Q.38

If roots of the equation 2x 2 – (a2 + 8a + 1) x + a 2 – 4a = 0 are in opposite sign, then (A) 0 < a < 4 (B) a > 0 (C) a < 8 (D) – 4 < a < 0

Q.39

If the roots of the equation

x2 – 4mx + 3e 2 log m – 4 = 0 is 8, then its roots are real, when m equals(A) 1 (B) 2 (C) 2 or –2 Q.30

(D) –2

For what value of c, the root of (c–2)x2 + 2 (c–2) x + 2 = 0 are n ot real (A) ]1,2[ (B) ]2,3[ (C) ]3,4[ (D) ]2,4[

(C) x Q.32

(B) x  0

 –1

(C)

(D) – 1  x  1 2

If roots of the equation ax + bx + c = 0 are

Q.40

  1 and , then (a + b + c) 2 equals 1  (A) 2b2 – ac

(B) b2 – ac

(C) b2 – 4ac

(D) 4b2 – 2ac

(A) | k |  2 (C) | k | > 2

Q.35

Q.36

2 1 2

7/9 7/9

(B) | k |  2

If

x

(C)

(D) None of these

If 7

log 7 ( x

2

 4 x  5)

Q.42

(B) 7

(C) – 2, –3

(D) 2, – 3

1 c

are

ab

(D)

1 2

1 2

(a2 + b2) ab

(B) – 4 < m  –3 (D) None of these

2

 2x  7 2x  3

< 6, x

 R, then -

3

3 2

< x < –1

3 2

If roots of the equation x 2 – bx + c = 0 are two successive integers, then b 2 – 4c equals (A) 1 (C) 3

= x – 1, x may have values -

(A) 2, 3

(B)

(D) –1< x < 11 or x <

If x > 1, then the minimum value of the expression 2 log10 x – logx (0.01) is (C) 1

x  b

=

2 (B) x > 11 or x < –1

(D) Never

(B) 4

(a2 + b2)

(A) x > 11 or x <

7/9

(A) 2

1

If both roots of the equation x 2 – (m + 1)x + (m + 4) = 0 are negative, then m equals -

If the product of the roots of the equation x2 – 3 kx + 2e sin k – 1 = 0 is 7 then its roots will be real if -

Q.34

1

(A) – 7 < m < – 5 (C) 2 < m < 5

Q.41 Q.33

xa

+

product is -

For x3 + 1  x2 + x(A) x  0

1

equal in magnitude but opposite in sign, then their

(A) Q.31

(B) ±5 (D) None of these

Q.43

2 The numbers of real roots of 32 x  7 x  7 = 9 is-

(A) 0 (C) 1

If ,  are roots of the equation

(B) 2 (D) 4

(B) 2 (D) 4

(3x + 2)2 + p(3x + 2) + q = 0, then roots of Q.44

x2 + px + q = 0 are (A) ,  (C)

1 3

( – 2),

(B) 3 + 2, 3 + 2 1 3

( – 2) (D)  – 2,  – 2

If a(p + q) 2 + 2apq + c = 0 and a(p + r) 2 + 2apr + c = 0, then qr equals (A) p2 + c/a

(B) p 2 + a/c

(C) p2 + a/b

(D) p 2 + b/a


9

Q.45

If a, b are roots of the equation x 2 + qx + 1 = 0

Q.52

and c, d are roots of x 2 + px + 1 = 0, then the value of (a – c) (b – c) (a + d) (b + d) will be2

(A) q – p

2

2

(C) – p – q

2

(B) p – q 2

 [7, 9] (C) a  [9, 7] (A) a

2

(D) p2 + q2 Q.53

Q.46

If one root of equation Ax 2 + Bx + C = 0 is AB

i(a – b) then

Q.47

x2 + 5 |x| + 4 = 0 are(B) 1, 4 (D) None of these

(B) 0

( a  b ) 2 1

(D) None of these

(a  b )

Two

 [9, 10) (D) a  [9, 12] (B) a

The real roots of the equation (A) –1, –4 (C) – 4, 4

equals-

1

(A)

(C)

C

If roots of x 2 – (a – 3)x + a = 0 a re such that both of them is greater than 2, then-

students

solve

a

quadratic

equation

2

x + bx + c = 0. One student solves the equation by taking wrong value of b and gets the roots as 2 and 5, while second student solves it by taking wrong value of c and gets the roots as – 3 and – 4. The correct roots of the equation are (A) – 2, – 5 (C) 2, 10 Q.48

(B) 2, – 5 (D) None of these

If in the equation ax 2 + bx + c = 0, the sum of roots is equal to sum of squares of their reciprocals, then

Q.49

b 2

bc

(A) 1

equals a2 (B) –1

(C) 2

(D) –2

ac

+

If ratio of roots of the equations x 2 + ax + b= 0 and x2 + px + q = 0 are equal, then -

Q.50

(A) aq = bp

(B) a 2q = bp2

(C) a 2 p = b2q

(D) aq2 = bp2

Let ,  be the roots of the equation ax2 + 2bx + c = 0 and 2

,  be

equation px + 2qx + r = 0. If G.P., then (A) q2 ac = b2 pr 2

2

(C) c pq = r ab Q.51

the roots of the

    are

in

(B) qac = bpr (D) p2 ab = a 2 qr

If real value of x and y satisfies the equation x2 + 4y2 – 8x + 12 = 0, then (A) 0 < y < 1 (B) 2 < y < 6 (C) – 1  y  1

(D) – 2 < y < 6


10

LEVEL- 3 Q.1

The

adjoining figure shows the

graph of

Q.6

The diagram shows the graph of y = ax2 + bx + c. Then -

y = ax2 + bx + c. Then -

y

y

x

O

(x1, 0)

(x2, 0)

O



(x2, 0)

x

(A) a < 0

x (x1, 0)

(A) a > 0

(B) b < 0

(C) c > 0

(D) b2 – 4ac = 0

(B) b2 < 4ac Q.7

(C) c > 0

If the roots of the equation a(b – c) x 2 + b(c – a) x + c(a – b) = 0

(D) a and b are of opposite signs

are equal, then a, b, c are in Q.2

The expression y =

ax2 +

bx + c has always the

same sign as c if (A) 4ac < b2 Q.8

(B) 4ac > b2 (D) ac > b2 If the roots of the equation (x – a) (x – b) – k = 0 be c & d then find the equation whose roots are a & b(A) (x – c) (x – d) + k = 0

(C) AP

(D) None of these

If (2 +  –2)x2 + (  + 2)x < 1 for all x

 R, then

Q.9

(A)

  2, 2     5 

(B) (–2, 1)

(C)

 2   , 1  5 

(D) None of these


(B) (x + c) (x – a) + k = 0

log2 (x2 – 4x + 5) = (x – 2) are -

(C) (x – c) + (x – a) = 0

(A) 4, 5


(B) GP

 belong to interval.

(C) ac < b2

Q.3

(A) HP

Q.10

(C) c < 0

(D) None of these

If f(x) = ax 2 + bx + c, g(x) = –ax2 + bx + c, where (A) At least three real r oots

+ b + c < 0, then (B) c > 0

(D) 3, 5

ac  0, then f (x) g (x) = 0 has -

Given that ax2 + bx + c = 0 has no real roots and a (A) c = 0

(B) 2, –3 (C) 2, 3

(B) No real roots (C) At least two real roots (D) Two real roots an d two imaginary roots

Q.5

The quadratic equation whose roots are reciprocal of the roots of the equation ax2 + bx + c = 0 is (A) cx2 + bx + a = 0 (B) bx2 + cx + a = 0 (C) cx 2 + ax + b = 0 (D) bx2 + ax + c = 0

Q.11

The equation

 x  sin 2 x  x 2  1 , 0  x   has  2 x2  2 

2 cos 2 

(A) No real solution (B) One real solution (C) More than one r eal solution (D) None of these


11

Q.12

The number of solutions of the equation (ex)

2 sin

Q.13

=

5x + 5 –x is

(B) 1

(C) 2

(D) Infinitely

Q.16

 (5  2

2 6 ) x 3

 10 is -

(A) 2

(B) 4

(C) 6

(D) None of these

If the equation ax2 + 2bx – 3c = 0 has no real

 3 c     a  b , then 4 



 a p     2  b q  c  R, a  0 1

(D) None of these

Q.20

a, b, and the quadratic equation 2 ax + bx + c = 0 has no real roots, then (A) a + b + c > 0 (B) a (a + b + c) > 0 (C) b (a + b + c) > 0 (D) c (a + b + c) > 0

Q.21

If the product of the roots of the equation

The number of real solutions of the equation

roots and

Q.15

-

(A) 0

2 (5  2 6 ) x 3

Q.14

(C)

x2 – 2 2 kx + 2 e 2 log k – 1 = 0 is 31, then the roots of the equation are real for k equal to (A) 1 (B) 2 (C) 3 (D) 4 Q.22

The number of real solutions of the equation x

(A) c < 0

(B) c > 0

(C) c  0

(D) c = 0

The product of all the solutions of the equation (x –2)2 –3|x –2| + 2 = 0 is (A) 0 (B) 2 (C) –4 (D) None of these

 9     3  x  x 2 is  10  (A) 0 (C) 2 Q.23


If the roots of the equation ax2 + bx + c = 0 are real and distinct, then (A) Both roots are greater than

If a, b, c are all positive and in H.P., then the roots of ax2 + 2 bx + c = 0 are (A) Real (B) Imaginary (C) Rational (D) Equal

(B) Both roots are less than

 b 2a

 b

(C) One of the roots exceeds

2a

 b 2a


The number of real roots of the equation (x – 1)2 + (x – 2) 2 + (x – 3) 2 = 0 is (A) 1 (B) 2 (C) 3 (D) None of these

Q.18

If ,  are the roots of ax 2 + bx + c = 0;  + h,  + h are the roots of px 2 + qx + r = 0, and D1, D2 the respective discriminants of these equations, then D1 : D2 (A)

(C)

Q.19

a2 p 2

c2 r 2

(B)

b 2

 b q      a p 

(D) None of these

(B)

The value of m for which one of the roots of x2 – 3x + 2m = 0 is double of one of the roots of x2 – x + m = 0 (A) 0 (B) –2 (C) 2 (D) None of these

Q.25

If ax2 + bx + 6 = 0 does not have two distinct real roots, where a  R, b  R, then the least value of 3a + b is(A) –2 (B) –1 (C) 4 (D) 1

q2

If ,  are the roots of ax 2 + bx + c = 0 and  + h,  + h are the roots of px 2 + qx + r = 0, then h= (A)

Q.24

1  b 2

q      a p 

Passage Based Questions (Q. 26-28)

Consider the expression y = ax 2 + bx + c, a  0 and a, b, c  R then the graph between x, y is always a parabola. If a > 0 then the shape of the parabola is concave upward and if a < 0 then the shape of the parabola is concave down ward. If y > 0 or y < 0 then discriminant D < 0.


12

Q.26

Q.27

Q.28

Let x2 + 2ax + 10 – 3a > 0 for every real value of x, then – (A) a > 5 (B) a < –5 (C) –5 < a < 2 (D) 2 < a < 5

Q.32

Statement I : x2 + bx + c = 0 has distinct roots

and both greater than 2 if b 2 – 4c > 0, b < –4 and 2b + c + 4 > 0. Statement II : x2 + 2x + c = 0 has distinct roots

The value of x2 + 2bx + c is positive if – (A) b2 – 4c > 0 (B) b2 – 4c < 0 (C) c2 < b (D) b2 < c

and both less than 1 iff c Q.33

The diagram show the graph of y = ax 2 + bx + c then –

 (–3, 1).

Statement I : We can get the equation whose

roots are 2 more than the roots of equation ax2 + bx + c = 0 by replacing x by (x + 2). Statement II : x2 + |x| + 5 = 0 has no real roots.

•

(x2, 0)

(A) a < 0 (C) b2 – 4ac < 0

•

(x1, 0)

(B) c < 0 (D) b2 – 4ac = 0

Questions based on statements (Q. 29 - 33) Each of the questions given below consist of Statement – I and Statement – II. Use the following Key to choose the appropriate answer. (A) If both Statement- I and Statement- II are true, and Statement - II is the correct explanation of Statement– I. (B) If both Statement - I and Statement - II are true but Statement - II is not the correct explanation of Statement – I. (C) If Statement - I is true but Statement - II is false. (D) If Statement - I is false but Statement - II is true. Q.29

Statement I : x2 + 4x + 7 > 0 Statement II : ax2 + bx +

 x  R c > 0 x R

if

b2 – 4ac < 0 and a > 0. Q.30

Statement I : The remainder obtained on

dividing the polynomial P(x) by (x – 3) is equal to P(3). Statement II : f(x) : (x – 8) 3 (x + 4)

 f '(x) may

not be divisible by (x 2 – 16x + 64). Q.31

,

Statement I: f(x) = ax2 + bx + c, then f(x) = 0

has integral roots when a = 1, b, c

I

and

b2 – 4ac is a perfect square of integer. Statement II : x3 + 1 = 0 has only one integral

root.


13

ANSWER KEY LEVEL- 1 Ques.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

B

D

B

B

B

B

D

D

A

D

B

A

A

C

C

D

A

A

C

C

Ques. 21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

C

C

A

D

C

B

C

B

B

A

D

C

A

C

B

C

C

A

C

B

Ques. 41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

Ans. D

B

B

C

A

A

A

D

C

A

B

D

B

A

B

C

C

B

B

B

Ques. 61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

B

C

C

C

B

B

D

C

D

A

D

D

C

C

B

D

B

C

C

B

Ques . 81

82

83

84

85

C

C

A

C

A

Ans.

Ans.

Ans.

LEVEL- 2 Ques.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

A

D

A

C

D

A

D

C

C

B

B

A

C

A

D

C

C

A

A

A

Ques.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Ans.

B

A

B

B

C

A

A

A

B

D

C

C

D

B

A

B

C

A

B

B

Ques.

41

42

43

44

45

46

47

48

49

50

51

52

53

Ans.

D

A

B

A

B

B

A

C

B

A

C

B

D

LEVEL- 3 Ques. Ans.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

A,D B

A

C

A B,C A

A

C

C

A

A

B

A

A

B

D

A

B B,D

24

Ques. 21

22

23

25

26

27

28

29

30

31

32

33

D

A

C A,B A

C

D

B

A

C

B

B

D

Ans.

20


14

IIT – ian’s

P A C E

216 - 217 ,2nd floor , Shopper’s point , S. V. Road. Andheri (West) Mumbai – 400058 . Tel: 26245223 / 09


Q.1

Q.2

Q.3

Q.4

LEVEL–1

Arithmetic Progression (A.P.)

10th term of the progression – 4 – 1+ 2 + 5 +.... is(A) – 23 (B) 23 (C) – 32 (D) 32 If 4 th term of an AP is 64 and its 54 th term is – 61, then its common difference is – (A) 5/2 (B) – 5/2 (C) 3/50 (D) – 3/50

Q.10

The 19 th term from the end of the series 2 + 6 + 10 + ....+ 86 is – (A) 6 (B) 18 (C) 14 (D) 10

Q.11

In the following two A.P.’s how many terms are identical? 2, 5, 8, 11.... to 60 terms, 3, 5, 7, ..... 50 terms (A) 15 (B) 16 (C) 17 (D) 18

Q.12

The first term of an A.P. is 2 and common difference is 4. The sum of its 40 terms will be – (A) 3200 (B) 1600 (C) 200 (D) 2800

Q.13

If nth term of an AP is 1/3 (2n + 1), then the sum of its 19 terms is(A) 131 (B) 132 (C) 133 (D) 134

Q.14

The sum of numbers lying between 10 and 200 which are divisible by 7 will be(A) 2800 (B) 2835 (C) 2870 (D) 2849

Q.15

If the sum of n terms of an AP is 2n 2 + 5n, then its nth term is(A) 4n-3 (B) 4n + 3 (C) 3n + 4 (D) 3n – 4

Q.16

If the ratio of sum of n terms of two A.P’s is (3n + 8) : (7n + 15), then the ratio of 12 th terms is(A) 16 : 7 (B) 7 :16 (C) 7 : 12 (D) 12 : 5

Q.17

If the ratio of the sum of n terms of two AP’s is 2n : (n + 1), then ratio of their 8 th terms is(A) 15 : 8 (B) 8 : 13 (C) n : (n– 1) (D) 5 : 17

Q.18

The sum of three consecutive terms of an increasing A.P. is 51. If the product of the first and third of these terms be 273, then third term is(A) 13 (B) 17 (C) 21 (D) 9

Which term of the series 3 + 8 + 13 + 18 + ... is 498(A) 95 th (B) 100 th (C) 102 th (D) 101th The number of terms 101 + 99 + 97 + .....+ 47 is(A) 25 (B) 28 (C) 30

in

the

Progression

series

(D) 20

Q.5

If (m + 2) th term of an A.P. is (m + 2)2 – m2, then its common difference is(A) 4 (B) – 4 (C) 2 (D) – 2

Q.6

If m terms of the series 63 + 65 + 67 + 69 + .... and 3 + 10 + 17 + 24 + ... be equal, then m = (A) 11 (B) 12 (C) 13 (D) 15

Q.7

If the 9th term of an A.P. be zero, then the ratio of its 29th and 19th term is(A) 1 : 2 (B) 2 : 1 (C) 1 : 3 (D) 3 : 1

Q.8

If fourth term of an A.P. is thrice its first term and seventh term – 2 (third term) = 1, then its common difference is(A) 1 (B) 2 (C) – 2 (D) 3

Q.9

If pth, q th and r th terms of an A.P. are a, b and c respectively, then a(q – r) + b (r – p) + c (p – q) is equal to (A) 0 (B) 1 (C) a + b + c (D) p + q + r

th


1

Q.19

If we divide 20 into four parts which are in A.P. such that product of the first and the fourth is to the product of the second and third is the same as 2 : 3, then the smallest part is(A) 1 (B) 2 (C) 3 (D) 4

Q.20

Three numbers are in A.P. The product of the extremes is 5 times the mean, also the sum of the two largest is 8 times the least, the numbers are(A) 3, 9, 15 (B) 6, 18, 30 (C) 3, 8, 13 (D) 6, 16, 26

Q.21

Q.22

Q.23

Q.24

If the angles of a quadrilateral are in A.P. whose common difference is 10º, then the angles of the quadrilateral are(A) 65º, 85º, 95º, 105º (B) 75º, 85º, 95º, 105º (C) 65º, 75º, 85º, 95º (D) 65º, 95º, 105º, 115º Three numbers are in A.P., If their sum is 33 and their product is 792, then the smallest of these numbers is – (A) 14 (B) 11 (C) 8 (D) 4 The sum of first four terms of an A.P. is 56 and the sum of its last four terms is 112. If its first term is 11, then number of its terms is(A) 10 (B) 11 (C) 12 (D) None of these If the numbers a, b, c, d, e form an A.P., then the value of a – 4b + 6c – 4d + e is(A) 1 (B) 2 (C) 0 (D) None of these

Q.25

If a2 (b + c), b 2 (c + a), c2 (a + b) are in A.P., then a, b, c, are in(A) A.P. (B) G.P. (C) H.P. (D) None of these

Q.26

If a, b, c are in A.P., then 1 1 1 , , are in b  c c a a  b (A) A.P. (B) G.P. (C) H.P. (D) None of these

 1 1   1 1   1 1  If a    , b    , c    are in A.P.  b c   c a   a b  then a, b, c are also(A) A.P. (B) G.P. (C) H.P. (D) None of these

Q.27

Q.28

If the roots of the equation (b – c) x2 + (c – a)x + (a – b) = 0 are equal , then a, b, c will be in(A) A.P. (B) G.P. (C) H.P. (D) None of these

Q.29

If

1 1 1 , , are in A.P. then p  q r  p q  r

(A) p2, q 2, r 2 are in A.P. (B) q 2, p2, r 2 are in A.P. (C) q 2, r 2, p2 are in A.P. (D) p, q, r are in A.P. The middle term of the progression 4, 9, 14,....104 is(A) 44 (B) 49 (C) 59 (D) 54

Q.30

Question based on

Q.31

Arithmetic Mean (A.M.)

If x, y, z are in A.P. and A.M. of x and y is a and that to y and z is b, then A.M. of a and b is (A) x (B) y (C) z (D) 1/2(x + y)

Q.32

If A1, A2 be two arithmetic means between 1/3 and 1/24, then their values are(A) 7/72, 5/36 (B) 17/72, 5/36 (C) 7/36, 5/72 (D) 5/72, 17/72

Q.33

The AM of 1, 3, 5, ...., (2n – 1) is – (A) n + 1 (B) n + 2 2 (C) n (D) n

Q.34

Given two numbers a and b, let A denotes the single A.M. and S denote the sum of n A.M.’s between a and b, then S/A depends on(A) n, a, b (B) n , b (C) n, a (D) n

Question based on

Q.35

Geometrical Progression (G.P.)

If the first term of a G.P. be 5 and common ratio be – 5, then which term is 3125 – (A) 6th (B) 5th (C) 7 th (D) 8th


2

Q.36

The fifth term of a GP is 81 and its 8th term is 2187, then its third term is(A) 3 (B) 9 (C) 27 (D) None of these

Q.46

Q.47 Q.37

Q.38

Q.39

Q.40

Q.41

Q.42

In any G.P. the first term is 2 and last term is 512 and common ratio is 2, then 5th term from end is(A) 16 (B) 32 (C) 64 (D) None of these Which term of the progression 512 18, –12, 8, .... is ? 729 (A) 9th (B) 10th (C) 8th (D) None of these If third term of a G.P is 4, then product of first 5 term is(A) 43 (B) 4 4 (C) 45 (D) None of these

(A) x = ± 2

(B) x = ± 2

(C) x = ± 3

(D) x = ± 3

The second; third and sixth terms of an A.P. are consecutive terms of a G.P. The common ratio of the G.P. is(A) 1 (B) 3 (C) – 1 (D) – 3

Q.44

Total number of terms in the progression 96 + 48 + 24 + 12 + .....+ 3/16 is(A) 9 (B) 10 (C) 15 (D) 20

Q.45

The sum of the first 10 terms of a certain G.P. is equal to 244 times the sum of the first 5 terms. Then the common ratio is(A) 3 (B) 4 (C) 5 (D) None

(D) – 4/3

2 4 8 + 2 + 3 + .... (upto ) x x x (B) x > 2 (D) x < 1/2

Q.48

If the sum to n terms of a series be 3(2 n –1), then it is(A) A.P. (B) G.P. (C) H.P. (D) None of these

Q.49

The value of 91/3. 91/9. 91/27... upto , is(A) 1 (B) 3 (C) 9 (D) None of these

Q.50

If 3 + 3  + 32 + ...  =

 equals(A) 15/23 (C) 7/15

45 ( > 0); then 8

(B) 15/7 (D) 23/15

Q.51

If the sum of an infinite GP be 3 and the sum of the squares of its term is also 3, then its first term and common ratio are – (A) 3/2, 1/2 (B) 1/2, 3/2 (C) 1, 1/2 (D) None of these

Q.52

Every term of an infinite GP is thrice the sum of all the successive terms. If the sum of first two terms is 15, then the sum of the GP is(A) 20 (B) 16 (C) 28 (D) 30

Q.53

A geometric progression consists of an even number of terms. The sum of all the terms is three times that of the odd terms, the common ratio of the progression will be(A) 1/2 (B) 2 (C) 3 (D) 1/3

Q.54

If first term of a decreasing infinite G.P. is 1 and sum is S, then sum of squares of its terms is(A) S2 (B) 1/S2 (C) S2/ (2S – 1) (D) S2/(2S + 1)

Q.55

If sum of three numbers of a G.P. is 19 and their product is 216, then its c.r. is(A) 1/2 (B) 1/3 (C) 3/2 (D) 3/4

Three numbers a, b, 12 are in G.P. and a, b, 9 are in A.P., then a and b are – (A) 3, 6 (B) – 3, 6 (C) 3, – 6 (D) – 3, – 6

Q.43

The sum 1 + is finite if – (A) x < 2 (C) x < 1

If third and seventh terms of a GP are 15 and 135 respectively, then its fifth term will be(A) 5 (B) 9 (C) 45 (D) 90 For which values of x do the numbers 1, x 2, 6 – x2 taken in that order form a geometric progression-

The sum of the infinite terms of 1 – 1/3 + 1/32 – 1/33 + ... is(A) 3/4 (B) 4/3 (C) – 3/4


3

Q.56

If the product of three numbers in GP is 3375 and their sum is 65, then the smallest of these numbers is (A) 3 (B) 5 (C) 4 (D) 6

Q.57

If the product of three terms of G.P. is 512. If 8 added to first and 6 added to second term, so that number may be in A.P., then the numbers are(A) 2, 4, 8 (B) 4, 8, 16 (C) 3, 6, 12 (D) None of these

Q.58

Q.59

Q.60

Q.61

Q.62

In the four numbers first three are in G.P. and last three are in A.P. whose common difference is 6. If the first and last numbers are same, then first will be(A) 2 (B) 4 (C) 6 (D) 8 Break the numbers 155 into three parts so that the obtained numbers form a G.P., the first term being less than the third one by 120(A) 5, 65, 125 (B) 10, 65, 120 (C) 5, 25, 125 (D) None of these

Q.65

If a, b, c, d are in G.P. then a + b, b + c, c + d are in(A) A.P. (B) G.P. (C) H.P. (D) None of these

Q.66

If a, b, c are in G.P. then (A) A.P. (C) H.P.

Question based on

If three geometric means be inserted between 2 and 32, then the third geometric mean will be(A) 8 (B) 4 (C) 16 (D) 12

Q.68

The product of three geometric means between 4 and 1/4 will be (A) 4 (B) 2 (C) – 1 (D) 1

Q.69

The ratio between the GM’s of the roots of the equations ax2 + bx + c = 0 and x2 + mx + n = 0 is-

Q.70

Q.63

The fractional value of 0.1 2 5 is(A) 125/999 (B) 23/990 (C) 61/550 (D) None of these

Q.64

If x, y, z are in G.P. then x 2 +y2, xy + yz, y2 + z2 are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

(A)

b an

(B)

c an

(C)

an c

(D)

cn a

If G be the geometric mean of x and y, then 1 2

G x

The sum of three positive numbers constituting an arithmetic progression is 15. If we add 1,4,19 to those numbers respectively. We get a geometric progression, then the numbers are(A) 2, 5, 8 (B) 8, 5, 2 (C) 5, 8, 2 (D) All of these 

Geometrical Mean (G.M.)

Q.67

Find three numbers in G.P. such that their sum is 14 and the sum of their squares is 84 (A) 3, 6, 12 (B) 2, 6, 18 (C) 1, 3, 9 (D) 2, 4, 8 Determine the first term and the common ratio of the geometric progression, the sum of whose first and third terms is 40 and the second and fourth term is 80 (A) 8, 3 (B) 8, 2 (C) 7, 3 (D) 7, 2

1 1 1 , , are in a b c (B) G.P. (D) None of these

(A) G2 (C)

2 G2

2

+

1 = G  y2 2

(B)

1 G2

(D) 3G2

Q.71

The A.M. of two numbers is 34 and GM is 16, the numbers are(A) 2 and 64 (B) 64 and 3 (C) 64 and 4 (D) None of these

Q.72

Two numbers are in the ratio 4 : 1. If their AM exceeds their GM by 2, then the numbers are-(A) 4, 1 (B) 16, 4 (C) 12, 3 (D) None of these


4

Q.73

Question based on

a, b, c are in A.P. If x is the GM between a and b and y is the GM between b and c, then the A.M. between x2 and y2 will be(A) a2 (B) b 2 (C) c2 (D) None of these

Q.80

Arithmetic-Geometrical Progression (A.G.P.)

Q.81 Q.74

Sum to infinite of the series 2 3 4 + 2 + 3 + .... is5 5 5 (A) 5/4 (B) 6/5 (C) 25/16 1+

Q.75

 1  x  (B)    1  x 

(C)

Q.76

1 x (1  x ) 2

Q.82

2

Q.83

Q.77

Q.78

1+2(1+1/n) + 3(1+1/n)2 + ...  terms, equals-

Q.84

If fourth term of an HP is 3/5 and its 8th term is 1/3, then its first term is– (A) 2/3 (B) 3/2 (C) 1/4 (D) None of these 1 1 The fifth term of the H.P. 2, 2 , 3 ,..... will be2 3 1 5

1 1 (B) 3 (C) 5 10

mn r  1

(C)

mn r

(D)

mn r  1

If b + c, c + a, a + b are in H.P., then a2, b2, c2 will be in(A) A.P. (B) G.P. (C) H.P. (D) None of these b b b If a, b, c be in H.P. then a – , , c – will be 2 2 2 in (A) A.P. (B) G.P.

Q.85

The HM between 1/21 and – 1/5 is (A)

1 8

(B) –

1 8

(C)

1 4

(D) –

1 4

If H is H.M. between two numbers a and b, then 1 1 + equals H  a H  b (A) a – b

(D) 10

If first and second terms of a HP are a and b, then its nth term will beab (A) a  ( n 1) ab

(C)

ab (B) b  ( n  1) (a  b)

ab (D) None of these b  ( n  1) (a  b)

(B) G.P. (D) None of these

Harmonic Mean (H.M.)

(C) Q.79

(D) None of these

If a, b, c are in A.P., then

(A) A.P. (C) H.P.

(B) n2 (D) None of these

Harmonic Progression (H.P.)

(A) 5

(B)

bc ca ab , , are inca  ab bc  ab bc  ca

Question based on Question based on

r mn

(C) H.P.

(D) None of these

(A) n (1+1/n) (C) n(1+1/n)2

(A)

(D) 16/9

The sum of infinite terms of the progression 1+ 3x + 5x2 + 7x3 + ......(x < 1) is1 x (A) 1 x

If the mth term of a H.P. be n and n th term be m, then the r th term will be-

Q.86

1 1 – a b

The HM between 2ab (A) a  b (C)

2ab 2 a  b 2

(B) a + b (D)

1 1 + a b

a b and is b a (B) (D)

2a 2 b 2 a 2  b 2 2a 2 b 2 a  b


5

Q.87

If 4 HM’s be inserted between 2/3 and 2/13, then the second HM is(A) 2/5 (C) 2/11

Question based on

Q.88

Q.89

(B) 2/7 (D) 2/17

Relation between A.M., G.M. & H.M.

If A,G & 4 are A.M, G.M & H.M of two numbers respectively and 2A + G2 = 27, then the numbers are(A) 8, 2 (B) 8, 6 (C) 6, 3 (D) 6, 4 If x, y, z are AM, GM and HM of two positive numbers respectively, then correct statement is (A) x < y < z (B) y < x < z (C) z < y < x (D) z < x < y

Q.90

Question based on

Sum of n terms of 1 + (1 + x) + (1 + x + x2) + (1 + x + x2 + x3) + .... is(A)

1 xn 1 x

(B)

x (1  x n ) 1 x

(C)

n (1  x )  x (1  x n ) (1  x ) 2

(D) None of these n

Q.95

 k 3 is equal tok 1 n

(A) 2  k 2

k 1

 n  (C)   k   k 1 

3

 n  (B)   k   k 1 

2

n

(D) 3  k 2 k 1

If sum of A.M. and H.M. between two positive numbers is 25 and their GM is 12, then sum of numbers is(A) 9 (C) 32

Q.91

Q.94

(B) 18 (D) 18 or 32

The A.M. between two positive numbers exceeds the GM by 5, and the GM exceeds the H.M. by 4. Then the numbers are(A) 10, 40 (B) 10, 20 (C) 20, 40 (D) 10, 50 Special Series

Q.92

Sum of the series 1+ 3+ 7 + 15 + 31+ .... to n terms is(A) 2n – 2 – n (B) 2n+1+2 + n (C) 2n+1 –2–n (D) None of these

Q.93

The number of terms in the sequence 1, 3, 6, 10, 15, 21, .... , 5050 is(A) 50 (C) 101

(B) 100 (D) 105


6

LEVEL- 2 Q.1

Find the sum of all the even positive integers less than 200 which are not divisible by 6(A) 6535 (C) 6534

Q.2

Q.7

2

a2 a3 + log 2 + ... is b b

 a  (A) n log    b  (B) n log (ab) (C)

n2 a n log + log (ab) 2 b 2

Q.4

Q.8

(A) 398 + 220

(B) 398 + 221

(C) 398 + 2 19

(D) None of these

Q.6

Certain numbers appear in both the arithmetic progressions 17, 21, 25.... and 16, 21, 26.... find the sum of the first two hundred terms appearing in both(A) 4022 (B) 402200 (C) 201100 (D) 398000

Q.9

 x18  1   (C)  2  x 1   

 x 11  1     x 9  + 20  

If 0 < x, y, a, b < 1, then the sum of the infinite terms of the series x ( ab + xy )

ax 1  b x 1  b

+

+

x 1 y

x 1 y

(B)

(D)

x 1  b ax 1  b

+

+

x 1 y x 1 y

If sum of 3 terms of a G.P. is S. product is P, and sum of reciprocal of its terms is R, then P2 R 3 equals to -

Q.10

(A) S

(B) S3

(C) 2S2

(D) S2/R

If A and G are respectively A.M. and G.M. of roots of a quadratic equation, then it is(A) x2 + 2Ax + G2 = 0 (B) x2 – 2Ax + G2 = 0 (C) x2 – Ax + G = 0 (D) None of these

1 1 1 + + + ...., 2 4 8

1 such that S – S n .< , then the least value of 1000 (B) 9 (D) 8

 x11  1    + 20  x9   

(C)

If S denotes the sum to infinity and Sn the sum of

n is(A) 11 (C) 10

 x18  1   (B)  2  x 1   

(A)

(D) None of these

n terms of the series 1+

 x 22  1    + 20  x 20   

+ x (b a + y x ) + ... is-

If first and (2n – 1) th terms of an A.P., G.P. and H.P. are equal and their nth terms are respectively a, b, c, then (A) a = b = c (B) a + c = b

Q.5

 x 20  1   (A)  2  x 1   

x( a+ x)+

The sum of 40 terms of the series 1+ 2 + 3 + 4 + 5 + 8 + 7+ 16 + 9 + ... is-

(C) ac – b2 = 0

2

(D) None of these

n2 a n (D) log – log (ab) 2 b 2 Q.3

2

1  1   1     x   +  x 2  2  +  x 3  3  + .... is x  x   x   

(B) 6539 (D) 6532

The sum of n terms of the series log a + log

The sum of 10 terms of the series

Q.11

If tn be the nth term of an A.P. and if t 7 = 9, then the value of the c.d. that would make t 1t2t7 least is(A) 33/40 (C) 33/10

(B) 33/20 (D) None of these


7

Q.12

Q.13

If mth terms of the series 63 + 65 + 67 + 69 + .... and 3 + 10 + 17 + 24 + ... be equal, then m = (A) 11 (B) 12 (C) 13 (D) 15

Q.18

(A)

A ball falls from a height of 100 mts. on a floor. If in each rebound it describes 4/5 height of the previous falling height, then the total distance travelled by the ball before coming to rest is-

(C)

Q.19

(A)  (B) 500 mts (C) 1000 mts (D) 900 mts Q.14

If sum of infinite G.P. is x and sum of square of its terms is y, then common ratio is-

the equation whose roots are a, b and c is given by(A) x3 – 3Ax2 + 3G3 x + G3 = 0

S be the sum of its all terms; then its common difference is-

 a2 2S    a 

2

a2 (C) 2S    a 

Q.17

(B)

Q.22

Q.23

2

(B)

( y  zx )

(C)

( z 2  xy )

(D) None of these

(C) 17

(D) 18

(A)

a ( p  q )q p  1

(B) –

(C)

a ( p  q ) p p  1

(D) None of these

a ( p  q )q p  1

Find sum of the series

If a, b, c are in A.P. and x = 1 + a + a 2 + ...,


The sum of 10 terms of the series. 0.7 + .77 + .777 + ... is(A)

1  7   89  10  9  10 

(B)

7  1   89  10  81  10 

(C)

7  1   89  9  81  10 

( x 2  yz) 2

(B) 16

y = 1 + b + b 2 + ... and z = 1 + c + c2 + ...., (where a, b, c < 1)), then x, y, z are in(A) A.P. (B) G.P. (C) H.P. (D) None of these

2

If x,y,z are in A.P. , then magnitude of its common difference is(A)

x 2  y2

1.32 + 2.5 2 + 3.72 + .... to 20 terms(A) 188090 (B) 94045 (C) 325178 (D) 812715

 a2 2S    a 

x2  y2

In an A.P. of which a is the first term, if the sum of the first p terms is zero, then the sum of the next q term is-

0

(B) G.M. (D) None of these

(D)

x2  y

Q.21

If a and  be the first and last term of an A.P. and

(A)

x 2  y2

x2  y

If 1 + r + r 2 + ....+ r n = (1+ r) (1+ r 2) (1+ r 4) (1+ r 8) , then the value of n is(A) 13 (B) 14 (C) 15 (D) 16

The G.M. of roots of the equation x2 – 2ax + b2 = 0 is equal to which type of mean of roots of x2 – 2bx + a2 = 0? (A) A.M. (C) H.M.

Q.16

x2  y2

(B)

In the following two A.P.’s how many terms are identical ? 2, 5, 8, 11.... to 60 terms, 3, 5, 7, ..... 50 terms

(C) x3 + 3Ax2 + 3(G3/H) x – G3 = 0 3(G3/H) x + G3 =

Q.20

(B) x3 – 3Ax2 + 3(G3/H) x – G3 = 0

Q.15

x2  y

(A) 15

If A,G and H are respectively A.M., G.M., and H.M. of three positive numbers a, b and c, then

(D) x3 – 3Ax2 –

x2  y

(D) None of these


8

Q.25

The

value

of

x alog b

where

a

=

0.2,

Q.31

1 1 1 + + + .... shall be1.2 2.3 3.4

 1 1 1  b = 5 , x =     .....  , is 4 8 16  (A) 1 (C) 1/2 Q.26

(A)  (C) 0

(B) 2 (D) 4

Find the sum of the series up to n term 1.3.5 + 3.5.7 + 5.7.9 + ..

Q.32

(B) n (8n3 + 11n2 – n – 3)

Q.27

Q.33

GM, then p : q is(A) 1 : 1 (B) 2 : 1

Q.34

3 ) : (2 – 3 )

(D) 3 :1 Q.28

The sum of the first ten terms of the geometric progression is S1 and the sum of the next ten

Q.35

terms (11th through 20th) is S2. then the common ratio will be-

Q.29

S1 S2

(B) ±

(C) ± 10

S2 S1

(D)

S2 S1

(C) x/ (1– x)3

(D) 1/(1– x)3

If the sum of four numbers in A.P. be 48 and that the product of the extremes is to the product of the means is 27 to 35 then the numbers are(A) 3, 9, 15, 21 (B) 9, 5, 7, 3 (C) 6, 10, 14, 18 (D) None of these 3 5 7 The sum of infinite series 1– + – + ... is2 4 8

bc ca (C) a b

(8) (1|cos x||cos

groups as follows ; (1), (2, 3), (4, 5, 6), (7, 8, 9, 10) and so on. Find the sum of the numbers in the nth group is-

2n ( n  1) (C) 3

n 2 (n  1) (D) 2

2 3

(C) –

2 9

(D)

9 2

2

x | |cos 3 x |.....)

= 43 in the interval

(– , ) are-

The series of natural numbers is divided into

n (n 2  1) (B) 4

(B)

The solution of the equation

(D) abc

1 (A) [n(n2+ 1)] 2

2 9

Q.37

z = c + (c/r 2) + (c/r 4) + ...., then (xy/z) is-

Q.30

(B) (1+ x)/(1– x)

If a, b, c are in G.P. and A.M. between a, b and b, c are respectively p and q, then (a/p) + (c/q) is equal to(A) 0 (B) 1 (C) 2 (D) 1/2

S1 S2

y = b – (b/r) + (b/r 2) – .... and

(B)

(A) (1+ x)/(1– x)3

Q.36

If x = a + (a/r) + (a/r 2) + ............,

ab c

The sum of the infinite series

(A)

(A) ± 10

(A)

The number of terms in the sequence

12 + 22 x + 32 x2 + ..... is-

If A.M. between p and q (p  q) is two times the

(C) (2 +


1, 3, 6, 10, 15, 21,...., 5050 is(A) 50 (B) 100 (C) 101 (D) 105

(A) 8n3 + 12n2 – 2n– 3 (C) n (2n3 + 8n2 + 7n – 2) (D) None of these

The sum to infinity of the following series

Q.38

(A) ±

  ,± 3 6

(B) ±

 ,± 3

(C) ±

 2 ,± 3 3

(D) None of these

If a, b, c, d are in G.P., then the value of (a – c)2 + (b – c)2 + (b – d) 2 – (a – d)2 is(A) 0 (B) 1 (C) a + d (D) a – d


9

Q.39

The third term of an A.P. is 9 and the difference of the seventh and the second term is 20. If the number 2001 is the nth term of the sequence then n is(A) equal to 499 (B) is equal to 500 (C) equal to 501 (D) can have no value

Q.40

Given the geometric progression 3, 6,12, 24,..... the term 12288 would occur as the(A) 11 th term (B) 12th term (C) 13 th term (D) 14 th term


10

LEVEL- 3 Q.1

Q.2

The maximum sum of the series

Q.7

1 2 20 + 19 +18 +……. is 3 3

then x is equal to (A) log43

(B) log34

(A) 310 (C) 320

(C) 1 – log34

(D) log3 0.25


Let a, b be the roots of x 2 – 3x + p = 0 and let c, d be the roots of x2 – 12 x + q = 0, where a, b, c, d form an increasing G.P. Then the ratio of q + p : q – p is equal to (A) 8 : 7 (B) 11 : 10 (C) 17 : 15 (D) None of these

Q.3

a  bx b  cx = = a  bx b  cx

If

a, b, c, d are in (A) A.P. (C) H.P. Q.4

Q.8

5

+

12

numbers, their squares, their cubes respectively, then

S3 (1  8 S1 ) S2 2

(A) 1

is equal to -

(B) 3

(C) 9

(D) 10

Q.9

The sum of three consecutive terms in a geometric progression is 14. If 1 is added to the first and the second terms and 1 is subtracted from the third, the resulting new terms are in arithmetic progression. Then the lowest of the original terms is (A) 1 (B) 2 (C) 4 (D) 8

Q.10

If S n denotes the sum of n terms of an A.P., then

c  dx (x  0), then c  dx


12  2 2

+

7 12  2 2  32

+……….. is -

(A)

6n n 1

(B)

9n n 1

(C)

12n n 1

(D)

15n n 1

n

If

 t r  2(3n  1) n  1 , then r 1

(A) 3

(B)

3 2

(C)

Sn+3 – 3S n+2 + 3Sn+1 – Sn is equal to (A) 0 (C) 1/2 Q.11

n

1 = r 1 t r

lim 

n 

then

a22 +

a32 –

a4

2 +

3 4

(D)

(A) 909 (C) 750

3 8

...... +

a 2n–12 –

n (a12 – a2n2) 2n  1

(B)

2n (a2n2 –a12) n 1

n (a12 + a2n2) n 1 (D) None of these (C)

(B) 75 (D) 900

The value of x + y + z is 15 if a, x, y, z, b are in A.P. while the value of

2

a2n is

1 1 1 5 + + is if X Y Z 3

a, X, Y, Z, b are in H.P., then a and b are-

equal to (A)

If a1, a2, a3, ......, a24 are in A.P. and + a3 + ...... + a 23 + a24 is equal to-

Let the sequence a1, a 2, a3, ......, an form an A.P., a12 –

(B) 1 (D) 2

a1 + a5 + a10 + a15 + a 20 + a24 = 225, then a1 + a 2

Q.12 Q.6

If S1, S2, S3 are the sums of first n natural

The sum of the first n terms of the series 3

Q.5

If 1, log9 (31–x + 2) and log3 (4.3x – 1) are in A.P.,

(A) 1, 9 (C) 7, 3 / 4

Q.13

If In =

 tan

(B) 3, 7 (D) 9, 1 n

x sec 2 x dx , then I1, I2, I3,... are in -

0

(A) A. P. (C) H.P.



11

Q.14

A G.P. consists of 2n terms. If the sum of the terms occupying the odd places is S1 and that of the terms at the even places is S2, then S 2/S1 is -

Then Sn–1 = t1 + t2 + t3 + ...... + t n–1, n > 1 (n  N)

(B) Independent of r (C) Independent of a and r (D) Dependent on r

subtracting, we get Sn – S n–1 = t n, n > 1 Further if we put n = 1 in the first sum then S 1 = t1 .Thus we can write

If x18 = y21 = z28, then 3, 3 log y x, 3 logz y, 7 logx z are in (A) A.P. (C) H.P.

Q.16

(B) G.P. (D) None

Q.19

find the terms of any kind of series, independent of its nature, provided the sum to first n terms is given. Q.21

Q.22

The sum of n terms of a series is a.2 n – b, where a and b are constants then the series is – (A) A.P. (B) G.P. (C) A.G.P. (D) G.P. from second term onwards

Q.23

If the sum to n terms of a series is given

64 . Then 5th term of the progression is 7

(A)

1 4

(B)

1 8

(C)

1 16

(D)

1 32

by

If the sum of the first 2n terms of the A.P. 2, 5, 8, ...... is equal to the sum of first n terms of the A.P. 57, 59, 61, ......, then n equals (A) 10 (B) 11 (C) 12 (D) 13 Let Sn =

1

+

1 2

+ …..+

1  2  .......  n

13 13  23 13  23  .....n 3 n = 1, 2, 3, ... Then S n is not greater than(A) 1/2 (C) 2

If sum of n terms of a series is of the form an2+ bn, where a and b are constants, then the fourth term of the series is– (A) a + b (B) 7a + b (C) 9 a + 3 b (D) 16a + 4b

(D) None of these

The sum of an infinitely decreasing G.P. is equal to 4 and the sum of the cubes of its terms is equal to

Q.18

tn = Sn – S n–1 and t1 = S 1. The above result can be used to

The sum of integers from 1 to 100 that are divisible by 2 or 3, is (A) 3300 (B) 3330 (C) 3000

Q.17

Suppose a series of n terms is given by S n = t1 + t 2 + t 3 + ...... + t n,

(A) Dependent on a

Q.15

Passage Based Questions (Q. 21 - 23)

;

(B) 1 (D) 4

n (n  1) ( n  2) then the nth term of the series 6

is – (A)  n2

(B) ( n)2

(C) n

(D) n + n

Questions based on Statements (Q. 24-28) Each of the questions given below consist of Statement – I and Statement – II. Use the following Key to choose the appropriate answer. (A) If both Statement- I and Statement- II are true, and Statement - II is the correct explanation of Statement– I.

Q.20

The number of common terms to the two sequences 17, 21, 25, ......, 417 and 16, 21, 26, ...... 466 is (A) 21 (B) 19 (C) 20 (D) 91

(B) If both Statement - I and Statement - II are true but Statement - II is not the correct explanation of Statement – I. (C) If Statement - I is true but Statement - II is false. (D) If Statement - I is false but Statement - II is true.


12

Q.24

Statement I : If A and G be the A.M and G.M.

between two positive real numbers a and b then a, b are given by A ± ( A  G ) (A  G ) . Statement II : Using x 2 – (a + b)x + ab = 0 ;

where a + b = 2A, ab = G2, we calculate x. Q.25

Statement I : The sum of all numbers of the

form n3 which lie between 100 and 10,000 is 53261. Statement II : If =

a  b a = then a, b, c are b  c c

in G.P. Q.26

Statement I : The number of terms of the A.P.

3, 7, 11, 15, ..... to be taken so that the sum is 465 is 15. Statement II : The sum of the integers from

1 to 100 which are not divisible by 3 or 5 is 2632. Q.27

Statement

I : If a, b, c, x are all real numbers

and (a2 + b2) x2 – 2b (a + c)x + (b2 + c2) = 0 then a, b, c are in G.P. and x is their common ratio. Statement II : If the ratio of the sum of m terms and n terms of an A.P. is m 2 : n 2 then the ratio of its mth and nth terms will be (2m – 1) : (2n–1) Q.28

Statement I : 1 + 3 + 7 + 13 + ....... up to n

terms =

n ( n 2  2) . 3

Statement II :

a n 1  b n 1 a n  b n

is Harmonic mean

1 of a and b if n = – . 2


13

ANSWER KEY LEVEL- 1 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

B

B

B

B

A

C

B

B

A

C

C

A

C

B

B

B

A

C

B

A

Q.No.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Ans.

B

D

B

C

A

A

A

A

A

D

B

B

D

D

B

B

B

A

C

C

Q.No.

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

Ans.

B

A

B

B

A

A

B

B

B

C

A

B

B

C

C

B

B

D

C

D

Q.No.

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

Ans.

B

A

A

B

B

B

C

D

B

B

C

B

B

C

C

D

B

D

C

C

Q.No.

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

Ans.

A

B

C

A

D

C

B

C

C

C

A

C

B

C

B

LEVEL- 2 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

C

C

B

C

B

A

A

D

B

B

B

C

D

B

A

B

B

B

C

C

Q.No.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Ans.

B

A

C

B

D

C

C

C

A

A

B

B

A

C

A

C

C

A

C

C

12

13

14

15

16

17

18

19

20

C

D

A

D

B

B

C

C

LEVEL- 3 Q.No.

1

2

3

4

5

6

7

8

9

10

11

Ans.

A

C

B

A

D

A

C

C

B

A

D A, D

Q.No.

21

22

23

24

25

26

27

28

Ans.

B

D

C

A

C

B

B

C


14

IIT – ian’s

P A C E



Q.1

LEVEL-1 Q.7

Fundamental Principle of counting

In how many ways 4 paintings can be hung on 4 walls of a room so that (i) one painting is hung on each wall and (ii) any number of paintings can be

The number of ways in which n distinct objects

hung on any wall?

can be put into two different boxes is-

Q.2

Permutation & Combinations

(A) n2

(B) 2n

(C) 2n

(D) None of these

The number of ways in which 3 persons can

Q.8

(B) 20

(C) 120

(D) 216

(B) 12, 3 4

(C) 4!, 4 4

(D) 4!, 43

In an examination, there are 3 multi-choice questions with one answer correct and each

occupy 6 rooms separately is(A) 2

(A) 4, 4!

question has 4 alternatives. If a student is declared pass only when he attempts all questions correctly, then number of ways in which he can

Q.3

Q.4

Find the total number of ways of answering

fail (if He attempts all the questions) is

5 objective type questions, each question having

(A) 1

(B) 12

4 choices.

(C) 27

(D) 63

(A) 46

(B) 54

(C) 6 3

(D) 45

Question based on

The number of ways in which first, second and third

prize

can

be

distributed

among

Q.9

5

Combinations nC + nC equalsr r–1

competitors is (no person can get more than a

(A) nCr+1

(B) n+1Cr

prize)-

(C) n+1Cr–1

(D) n+1Cr+1

(A) 10

(B) 15

(C) 60

(D) 125

Q.10

If 28C2r : 24C2 r–4 = 225:11 then the value of r is(A) 14

Q.5

10 different digits. The number of unsuccessful

Q.11

attempts to open the lock is-

Q.6

(B) 8

(C) 5

(D) 7

A lock consists of three rings each marked with If nCn–4= 5 then the value of n is (A) 5

(A) 310 – 1

(B) 93

(C) 10P3 –1

(D) 103 –1

Q.12

(B) 3

(C) 4

(D) 6

There are 13 players of cricket out of which 4 are bowlers. In how many ways a team of eleven be

5 Questions are asked in a question paper. Out of

selected from them so as to include at least two

which two questions can be solved by two- two

bowlers-

methods, two question by three-three and one

(A) 55

(B) 72

question can be solved by only one method then

(C) 78

(D) None of these

the number of possible attempts to solve the question paper are(A)

25

(C) 144

Q.13

How many cricket eleven can be formed from 15 persons if captain is included in every team?

(B)

22 .

(D) 288

32.1

(A) 364

(B) 1365

(C) 1001

(D) 1000


1

Q.14

Q.15

In how many ways a team of 11 be chosen from

Q.23

A bag contains 9 balls marked with digits

20 students of a class so that 2 particular students

1, 2, ...., 9. If two balls are drawn from the bag,

are always included and 5 are always excluded?

then number of ways of getting the sum of the

(A) 715

(B) 70

digits on balls as odd number is-

(C) 1365

(D) None of these

(A) 20

(B) 2 9

(C) 9C2

(D) 9P2

In how many ways can a committee of 6 be formed out of 6 men and 4 women so that

Q.24

6 candidates. A voter can cast any number of

committees include at the most 2 women? (A) 90

(B) 185

(C) 115

In an election 3 persons are to be elected from votes but not more than the candidates to be

(D) 210

elected. In how many ways can he cast his vote? Q.16

The number of committees formed by taking

(A) 41

5 men and 5 women from 6 women and 7 men are (A) 252 Q.17

(B) 125

(C) 126

(D) 64

Q.25

Q.18

Q.19

can cast his vote in 30 ways, then the number of

(D) 80

the candidates is-

The total number of ways to purchase one or

(A) 4

(B) 5

more books from 4 books by a student are-

(C) 6

(D) None of these

(A) 15

(B) 16

(C) 14

(D) None of these

Q.26

chemistry when each group contains at least one

In a hall there are 10 bulbs and their 10 buttons.

(A)

102

(B) 1023

(C) 210

(D) 10!

book of each subject(A) 3255 Q.27

(D) 1560

A father with 7 children takes 4 of them at a time to the zoo, as often as he can without taking the

can be put into two different boxes so that no box

same four children together more than once. How

remains empty, is-

often will he go? How often will each child go?

(A) n 2 – 1

(B) n2 – 2

(C) 2n – 1

(D) 2n – 2

A candidate is required to answer 6 questions out

(A) 30, 10

(B) 35, 15

(C) 30, 20

(D) 35, 20

Two groups of players consist of 6 and 8 players. In how many ways can a team of 11 players be

of 10 which are divided into two groups each

selected from these two groups if at least

consisting of 5 questions. In how many ways can

4 players are to be included from the first group?

he attempt 6 questions if he is not allowed to

(A) 334

answer more than 4 questions from each section?

Q.22

(B) 1670 (C) 4820

The number of ways in which n distinct things

Q.28 Q.21

The number of groups formed from 3 books of physics, 4 books of mathematics and 5 books of

In how many ways this hall can be enlightened?

Q.20

In an election the number of candidates is one

more than the candidates to be elected. If a voter

boys are to be in majority? (C) 100

(D) 6

A voter can cast any number of votes but not

formed. How many such groups are possible if (B) 90

(C) 15

more than the number of members to be elected.

Out of 6 boys and 4 girls a group of 7 is to be

(A) 120

(B) 20

(A) 100

(B) 200

(C) 300

(D) 400

If

n 2 n

C2 =

n 2 n

C10 , then n equals-

Q.29

(B) 344

(C) 120

(D) 168

A man has 7 relatives, of which 4 are ladies and 3 gents, his wife has also 7 relatives, of which 3 ladies and 4 gents. In how many ways can they invite 3 ladies and 3 gents when there are 3 relatives of man and 3 relatives of his wife-

(A) 12

(B) 4 only

(A) 324

(B) 485

(C) –3 only

(D) 4 or – 3

(C) 458

(D) None of these


2

Q.30

Q.31

Given five different green dyes four different blue

On the occasion of Dipawali festival each student

Q.38

dyes and three different red dyes how many

of a class sends greeting cards to the others. If

combinations of dyes can be chosen taking at

there are 20 students in the class, then the total

least one green and one blue dyes-

number of greeting cards exchanged by the

(A) 1345

(B) 3720

students is-

(C) 4675

(D) 7943

(A) 20C2

(B) 2. 20C2

(C) 2. 20P2

(D) None of these

If  = mC2, then C2 is equal to (A) m+1 C4

(B) m–1C4

(C) 3. m+2C4

(D) 3.m+1C4

A bag contains 6 different white and 5 different

Q.39

black balls. 4 balls are drawn at a time. The number of ways in which all the four will be of

Q.32

Q.33

A committee of 5 is to be formed out of 6 gents

the same colour is-

and 4 ladies. In how many ways this can be done

(A) 25

(B) 20

when atmost two ladies are included?

(C) 16

(D) none of these

(A) 186

(B) 286

(C) 386

(D) None of these

Question based on

Everybody in a room shakes hands with every body else. If total number of hand-shaken is

Q.40

(A) 11

(B) 12

(C) 13

In a plane there are 10 points out of which 4 are collinear, then the number of triangles that can be

66 then total number of persons in the room is-

Q.34

Geometrical Combination

formed by joining these points are-

(D) 14

In a class tournament every student has to play a

(A) 60

(B) 116

(C) 120

(D) None of these

game with one another. If 2 students fell ill after playing 3 games each, (they never play with each

Q.35

Q.41

The straight lines I 1, I2, I3 are parallel and lie in

other) and in total 84 games were played, then the

the same plane. A total number of m points are

number of students in the class is -

taken on I1; n points on I 2, k points on I 3. The

(A) 15

(B) 10

maximum number of triangles formed with

(C) 20

(D) None of these

vertices at these points are (A)

In a football championship, 153 matches were

m + n + k C 3

played. Every team played one match with each

(B) m + n + k C3 – mC3 – nC3 – k C3

other. The number of teams participating in the

(C) mC3 + nC3 + k C3

championship is-

(D) None of these

(A) 17

(B) 18

(C) 9

(D) None of these

Q.42

There are 12 points in a plane. The number of the straight lines joining any two of them when 3 of

Q.36

In how many ways can a game of tennis be

them are collinear is-

played from 3 men and 4 women when each team contains one man and one woman(A) 72

(B) 36

(C) 42

(D) 144

Q.43

(A) 60

(B) 62

(C) 64

(D) 66

There are 12 points in a plane, and 5 of them are in a straight line the number of triangle formed

Q.37

In how many ways can a mixed double tennis game be arranged from 7 married couples, if no husband and wife play in the same game? (A) 28

(B) 70

(C) 210

(D) 420

these points are: (A) 220

(B) 215

(C) 210

(D) 205


3

Q.44

m parallel lines in a plane are intersected by a family of n parallel lines. The total number of

Question based on

Combinations of identical objects

parallelograms so formed is(A) (C) Q.45

Q.46

( m  1) (n  1) 4

nm(m  1) (n  1) 2

(B) (D)

Q.51

mn

can be selected from 5 oranges, 4 apples, and

4

3 bananas is-

nm( m  1) (n  1) 4

The number of squares on a chess board is(A) 64

(B) 160

(C) 224

(D) 204

Q.52

(A) 1296

(B) 72

(C) 1200

(D) None of these

Q.53

(C) 60

(D) 120

The total number of donations which can be given

(A) 19

(B) 29

(C) 30

(D) 20

The number of ways in which at least one coin can be selected from 3 coins of one r upee, 4 coins

The three sides AB, BC and CA of any ABC are

of fifty paise, 5 coins of twenty paise and 6 coins of ten paise is -

Then the number of triangles formed by taking these points as the vertices are-

(A) 639

(B) 840

(A) 185

(B) 205

(C) 839

(D) None of these

(C) 230

(D) 215

Q.54

In a library there are p copies of each n different

If 4 parallel lines intersect another set of 3 parallel

books. The number of ways of selecting of one or

lines, then number of parallelograms formed with

more books from these i s-

these lines is-

(A) (p+1)n –1

(B) (p+1) n

(C) (p–1)n – 1

(D) (p–1)n

(A) 6

(B) 9

(C) 12

(D) 18 Q.55

Q.49

(B) 119

(at least one coin must be donated)-

The total number of rectangles on a chess board

having 3, 4, and 5 internal points respectively.

Q.48

(A) 59

from 5 one rupee coins and 4 fifty paise coins, is

is-

Q.47

The number of ways in which at least one fruit

The number of divisors of 9600 is-

The number of triangles whose vertices are at the

(A) 46

(B) 48

vertices of an octagon but none of whose sides

(C) 58

(D) 60

happen to come from the sides of the octagon is (A) 24 Q.50

(B) 52

(C) 48

(D) 16

Q.56

(A) 45

There are m points on one straight line AB and n points on another straight line AC, none of them

The number of different proper factors of 3780 is-

Q.57

The

(B) 46 total

number

(C) 47 of

being A. How many triangles can be formed with

(including 1 and 1998) is-

these points as vertices?

(A) 18

(A) (B) (C)

mn 2 mn 2 mn 2

(m + n)

Q.58

(B) 16

(C) 12

(D) 48 factors

of

1998

(D) 10

Number of divisors of the form 4n + 2 (n  0) of the integer 240 is -

(m + n –2) (m + n –1)

(D) None of these

(A) 4 Q.59

(B) 8

(C) 10

(D) 3

If a, b, c, d, e are prime integers, then the number of divisors of ab2c2de excluding 1 as a factor, is (A) 94

(B) 72

(C) 36

(D) 71


4

Q.60

A basket contains 4 oranges, 5 apples and 6 mangoes. In how many ways can a person make a selection of fruits, if atleast one fruit have to be

Question based on

Q.66

selected ?

Permutation ( P r)

If 15Pr = 2730, then the value of r is(A) 2

(A) 210

(B) 209

(C) 120

(D) None of these

Q.67

(B) 3

Q.61

Arrangement (n!)

Q.68

(D) 5

If nP3 = 120, then the value of n is(A) 6

Question based on

(C) 4

nP is n

(B) 8

(C) 10

(D) 12

equal to-

The number of ways in which 8 answer books be

(A) nP1

(B) nPn–1

arranged so that the best and the worst do not

(C) nP0

(D) None of these

occur together is-

Q.62

(A) 8!

(B) 7!

(C) 7(7) !

(D) 6 (7)!

There are 5 books on Mathematics, 4 on Physics. In how many ways these be placed on shelf if the books on the same subject are to be together?

Q.63

Q.69

Q.70

r ..n–1 Pr–1 equals-

(A) nPr

(B) n+1Pr+1

(C) n–1Pr+1

(D) None of these

nP is r

equal to -

(A) 4592

(B) 5760

(A) n.nPr–1

(B) n.

(C) 4800

(D) 2672

(C) (n-1) nP r–1

(D) None of these

There are 10 students in a class in which three A,

Q.71

them in a row when any two girls out of three never comes together(A) 7! × 8P3

(B) 7! × 3P3

(C) 10! × 3P3

(D) None of these

Q.72

n–1P r–1

If nPn= 720, then n equals(A) 2

B, C are girls. The number of ways to arrange

Q.64

n–1P + r

If

(m+n)

(B) 4 P2 = 56 and

(C) 6 m–nP 2

(D) 8 = 12 then (m, n)

equals(A) (5, 1) Q.73

A shelf contains 20 books, of which 4 are single

(B) (6, 2) (C) (7, 3)

(D) (9, 6)

For what value of r, nPr = 720 and nCr = 120 ? (A) 6

(B) 5

(C) 4

(D) 3

volume and the others are 8,5 and 3 volumes respectively. In how many ways can these books

Q.74

sides of a long table with 8 chairs on each side. 4

be arranged on the shelf so that the order of the

men wish to sit on one particular side and 2 on the

volumes of same work is maintained ?

Q.65

(A) 2.7!

(B) 7!

(C) 8. 7!

(D) None of these

other side. In how many ways can they be seated ?

There are 5 different books on mathematics, 2 different books on chemistry and 4 different

A tea party is arranged of 16 persons along two

Q.75

(A) 8P4 × 8P2

(B) 8P4 × 8P2 × 10!

(C) 8P4 × 10!

(D) None of these

Eight chairs are numbered from 1 to 8. Two

number of ways of

women and three men wish to occupy one chair

arranging these books on a shelf so that books of

each. First women choose the chairs from

the same subject are stacked together, is-

amongst the chairs marked 1 to 4; and then the

(A) 34560

(B) 11!

men select the chairs from the remaining. The

(C) 17285

(D) none of these

books on physics. The

number of possible arrangements is(A) 6C3 × 4C2

(B) 4P3 × 4P3

(C) 4C2 × 4P3

(D) None of these


5

Q.76

Eleven animals of a circus have to be placed in eleven cages, one in each cage. If four of the cages are too small for six of the animals, the

Question based on

Permutation

 n !   p ! q ! r !   

number of ways of caging the animals is(A) 7P6. 5!

(B) 6P4. 7!

(C) 11C4. 7!

(D) None of these

Q.83

The number of permutations of the letters x, x, y, y, y, y, z, z, z will be (A)

Q.77

The number of ways in which three persons can dress themselves when they have 4 shirts. 5 pants

(C)

and 6 hats between them, is(A) 4C3 × 5C3 ×6C3 (C)

Q.78

15! 4! 5! 6 !

(B) 4P3× 5 P3 × 6 P3 (D)

Q.84

15!

different flags when at a time 4 flags are used,

Q.79

(B) 10C4

(C) 4

(D) 40

9! 4 ! 3!

(B)

9! 2! 4! 3!

(D) 9!

In how many ways can 21 identical white and 19 black balls are together?

How many signals can be given by means of 10

(A) 10P4

2! 4!

identical black balls be put in a row so that no two

(3!) 3

one above the other?

9!

Q.85

(A) 1470

(B) 1540

(C) 735

(D) None of these

If out of 8 flags, 5 flags are white (identical) and 3 flags are of red colour (identical), then how many signals can be given by using all of them at a

A boat crew consist of 8 men, 3 of whom can

time?

only row on one particular side and 2 only on the

(A) 15

(B) 28

other. The number of ways in which the crew can

(C) 56

(D) 126

be arranged is-

Question based on

Q.80

(A) 1728

(B) 576

(C) 72

(D) None of these

Q.86

ways can we obtain 5 head and 5 ta ils-

Permutation [(n) ]

In how many ways can six different rings be wear

Q.87

in four fingers?

Q.81

(A) 25

(B) 252

(C) 52

(D) 22

Six identical coins are arranged in a row. The number of ways in which the number of tails is

(A) 6P4

64

(B)

(C) 46

(D) 6C4

equal to the number of heads is(A) 20

(B) 9

(C) 120

(D) 40

The number of ways in which n prizes can be distributed among n students when each student is eligible to get any number of prizes is-

Q.82

A coin is tossed 10 times. In how many different

(A) n n

(B) n!

(C) n n - n

(D) None of these

Question based on

Q.88

Application of permutation

How many words can be formed from the letters of the word ‘BHOPAL’ -

The number of ways of distributing n prizes

(A) 124

(B) 240

(C) 360

(D) 720

among n boys when any of the student does not Q.89

get all the prizes is(A) n n

(B) n!

(C) n n – n

(D) None of these

In how many ways can the letters of the word ‘SIMPLETON’ be rearranged? (A) 9!

(B) 9! – 1

(C) 9! – 2

(D) 8! – 1


6

Q.90

Q.91

The number of different words formed with all

the word “ALLAHABAD” in which vowels

(A) 2500

(B) 2460

occupy even positions, are-

(C) 2520

(D) None of these

(A) 1440

(B) 7560

(C) 240

(D) 60

The total number of words formed with the letters

Q.99

How many words can be formed by using the letters of the word ‘INSURANCE’, when vowels

(A) 720

(B) 180

(C) 360

(D) None of these

always remain together?

The number of permutations formed without changing the position of vowel and consonants of

Q.93

The number of different words from the letters of

the letters of the word ‘MISISSIPI’ is-

of the word “SERIES’ is-

Q.92

Q.98

Q.100

(A) 8640

(B) 17280

(C) 720

(D) None of these

The number of words which can be formed by

the letters of word ‘ALGEBRA’-

using the letters of the word ‘INDEPENDENCE’ so

(A)144

(B) 70

that both D occur together is-

(C) 360

(D) 72

(A)

The number of words which can be formed from (C)

the letters of the word ‘JODHPUR’ so that P,U,R

12! 4! 3! 2 !

(B)

11! 2!

11! 4! 3!

(D) None of these

4! 3!

always remain together isQ.101

(A) 4320

(B) 120

(C) 720

(D) None of these

How many different permutations can be formed from the letters of the word ‘MATHEMATICS” which starts from C-

Q.94

The number of words from the letters of

(A)

‘BHARAT’ is where B and H will never come together(A) 240 Q.95

Q.96

(B) 120

(C) 140

How many words can be formed from the letters

Q.102

2! 2! 2!

10! 2! 2! 2!

(B)

10! 2! 2 !

(D) None of these

The number of words from the letters of the word

of the word ‘GANESH PURI’ when P and I

‘INSTITUTION’ when first two letters are N-

occupy the first and last place respectively-

(A) 5040

(B) 32240

(A) 2! × 8!

(B) 8!

(C) 20160

(D) 10080

(C) 10!

(D) None of these

The number of words which can be formed from the letters of the word ‘SCHOLAR’ which begin with A and end with S is-

Q.97

(C)

(D) 40

11!

(A) 120

(B) 720

(C) 1440

(D) 5040

The number of words formed with the letters of the word ‘JODHPUR’ in which P, U, R never

Q.103

How many words can be formed from the letters of word ‘ASSASSINATION” when four S remains together(A)

(B)

(C)

come together, is(A) 720

(B) 5040

(C) 4320

(D) None of these

(D)

10! 3! 2 ! 10 ! 4! 13! 4! 3! 2! 13! 4! 4!


7

Q.104

How many different words can be formed from the letters of ‘CONSTANTINOPLE’ in which all

Q.112

3, 4, 5, 6, 7 which are greater than 1000

three N come together? (A)

(C)

Q.105

14! 2! 3! 2!

12! 2 ! 3! 2 !

(B)

(Repetition not allowed) ?

12! 2! 2!

Q.113

Q.114

(A) 8!

(B) 5040

(C) 1080

(D) None of these

allowed) (A) 240

(B) 150

(C) 720

(D) 360

Q.115

repetition of digits is not allowed (D) 65

Q.116

digit can be repeated any number of times? (C) 44

(D) 54

(C) 1024

(D) None of these

How many five digit even numbers can be formed

(A) 72

(B) 60

(C) 54

(D) 36

The number of 4-digits numbers formed with

(A) 42

(B) 36

(C) 48

(D) 24

The number of 4 digits odd numbers formed with (Repetition not allowed)

can be formed with the digits 2, 3, 5, 6, 9 if each (B) 55

(B) 625

the digits 0, 1, 2, 3, 4 and 5 is

How many numbers between 30000 and 40000

(A)45

(A) 125

(Repetition not allowed)

How many numbers lying between 100 and 1000

(C) 64

How many 4 digit numbers can be formed with

0, 2, 3, 4, 5, and divisible by 5 is

can be formed with the digits 1, 2, 3, 4, 5 if the

Q.108

(D) 5

(Repetition not allowed)?

from the digits 1, 2, 3, 4, 5, 6 (repetition is not

(B) 60

(C) 24

by using the digits 0, 2, 3, 4, 5

How many numbers of 4 digits can be formed

(A) 62

(B) 120

digits 1, 2, 3, 4, 5 when digits may be repeated ?

How many words with the letters of the word R-

Q.107

(A) 240

(D) None of these

‘CARAVELLE’ can be formed which starts with

Q.106

How many numbers can be formed with the digits

Q.117

(A) 54

(B) 144

(C) 180

(D) 360

How many numbers can be formed with the digits 2,3,5,7,0

Q.109

which

are

greater

How many numbers consisting of 5 digits can be

(Repetition not allowed) ?

formed in which the digits 3,4 and 7 are used only

(A) 24

(B) 12

once and the digit 5 is used twice-

(C) 120

(D) 6

(A) 30

(B) 60

(C) 45

(D) 90

Q.118

than

70,000

The number of numbers can be formed by taking any 2 digits from digits 6,7,8,9 and 3 digits from

Q.110

Q.111

The number of 5 digit even numbers formed with

1, 2, 3, 4, 5 is -

the digits 2, 3, 5, 7, 9 is (Rept. not allowed)

(A) 5C3 × 4C2 × 3! × 2!

(A) 12

(B) 24

(C) 120

(D) None of these

Using digits 3, 4, 5, 6, 7, 8 how many numbers between 3000 and 4000 can be formed which are divisible by 5 and same digit is not repeated ? (A) 60

(B) 12

(C) 120

(D) 24

(B) 5P3 × 4 P2 × 5! (C) 5C3 × 4C2 × 5! (D) 5C3 × 4C2 ×

5! 2!


8

Question based on

Q.119

Q.120

Question based on

Circular Permutations

The number of ways in which 20 persons can sit

Q.125

Division into groups

The number of ways to make 5 heaps of 3–3

on 8 chairs round a circular table is-

books each from 15 different books a-

(A) 20 P8

(B) 19P8

(A)

(C) 1/8 ( 20P8)

(D) None of these

(3!) 5

(D) 15P5

Q.126

150 students take admission. They are to be put in three sections A, B, C of equal size. The number

neighbours of the president?

of ways in which this can be done is-

(A) 8! × 3!

(B) 10!

(A)

(C) 8! × 2!

(D) 7! × 2!

In how many ways 7 different beads be strung together? (A) 240

Q.127

3!(50!) 150 ! (50!)

3

3

× 150!

(B)

150 ! (50!) 3

(D) None of these

divided equally among four players in order to(B) 720

(C) 120

(D) 360

The number of necklaces which can be formed b y selecting 4 beads out of 6 beads of different

(C)

coloured glasses and 4 beads out of 5 beads of different metal, is(A)

150 !

In how many ways can a pack of 52 cards be

(A)

6P × 5P × 4 4

(C) 6C4 × 5C4 ×

Q.124

15 !

that the secretary and joint secretary are always

into a ring so that two particular beads are always

Q.123

(B)

Eleven members of a committee sit round a

(C)

Q.122

5! (3! )

5

(C) 15C3

circular table. In how many ways can they sit so

Q.121

15 !

7! 2! 8! 2!

(B)

6C × 5C × 4 4

7!

Q.128

(13! )

(B)

4

52 ! (17 ! )

4

3!

52! (13!) 4 4!

(D) None of these

The number of ways in which six different prizes can be distributed among three children each

2!

receiving at least one prize is-

(D) 6C4 × 5C4 × 7!

If two specific beads are kept together, then in

52!

Q.129

(A) 270

(B) 540

(C) 1080

(D) 2160

The number of ways in which 20 volunteers can

how many ways can seven different beads be

be divided into groups of 4, 7 and 9 persons is-

strung in one garland-

(A) 16C7× 13C2

(B) 20C7 × 11C4

(C) 20C4 × 16C7

(D) 20C9 × 13C9

(A) 5!

(B) 7!

(C) 5! × 2!

(D)

7! 2!

There are 20 persons among whom two are

Q.130

The number of ways in which mn students can be distributed equally among n sections is(A) (mn)n

(B)

brothers. The number of ways in which we can arrange them round a circle so that there is exactly one person between the two brothers, is(A) 18!

(B) 2(18!)

(C) 2 (19!)

(D) None of these

(C)

(mn) ! m!

(D)

( mn) ! ( m !) n (mn ) ! m! n !


9

Q.131

In how many ways two garlands of 6 flowers each

Q.138

can be made from 12 different flowers(A)

(C)

Q.132

12! 2

( 6 !) . ( 2 ! ) 12! ( 6!) 2 . (2!)

2

3

.(5!) (B)

boxes of colours same as those of the balls. The

12!.(5!) 2

number of ways in which the balls, one in each

( 6 !) 2

box, could be placed such that a ball does not go to box of its own colour is-

12!

2

. (5!) (D)

There are four balls of different colours and four

(6!) 2 . ( 2!)

(A) 8

(B) 7

(C) 9

(D) None of these

In how many ways 8 different balls can be distributed among 3 children so that one child

Question based on

Exponent of prime number in (n!)

gets 4 balls and two children get 2 balls each?

Q.133

(A) 210

(B) 240

(C) 420

(D) 1260

In how many ways can 6 prizes be distributed equally among 3 persons? (A) 6C2 × 4C2 (C) 3

Question based on

Q.134

Q.139

(B) 6P2 × 4P2 (D)

Q.140

36

Sum of numbers

Q.141

The sum of numbers formed by the digits

The exponent of 7 in 100! is (A) 14

(B) 15

(C) 16

(D) none of these

The number of zeros at the end of 70 ! is (A) 16

(B) 5

(C) 7

(D) 70

The number 24 ! is divisible by (A) 624

(B) 24 6

(C) 1212

(D) 485

1, 3, 5, 7, 9 is-

Q.135

Q.136

(A) 666600

(B) 6666600

(C) 666660

(D) None of these

The sum of all numbers greater than 1000 formed

Question based on

Q.142

all possible orders and these words are written

any number is-

out as in a dictionary then the rank of the word

(A) 106656

(B) 101276

RANDOM is -

(C) 117312

(D) 811273

(A) 614

(B) 615

(C) 613

(D) 616

The sum of the digits in the unit place of all the Q.143

If the letters of the word MOTHER are written in all possible orders and these words ar e written out

all at a time is-

Q.137

The letters of the word RANDOM are written in

by using the digits 1,3,5,7 no digit is repeated in

numbers formed with the help of 3, 4, 5, 6 taken

Question based on

Rank of words in dictionary

(A) 18

(B) 108

as in dictionary, then the rank of the word

(C) 432

(D) None of these

MOTHER is -

Derangement Theorem

The number of ways to put five letters in five envelopes when any one letter is kept in right envelope and four letters in wrong envelopes are(A) 40

(B) 45

(C) 30

(D) 70

Q.144

(A) 240

(B) 261

(C) 308

(D) 309

All letters of the word 'AGAIN' are permuted in all possible ways and the words so formed (with or without meaning) are written as in dictionary, then the 50th word is (A) NAAGI

(B) NAAIG

(C) IAANG

(D) INAGA.


10

LEVEL- 2 Q.1

A box contains two different white balls, three

Q.8

The number of words formed from letters of the

different black balls and four different red balls.

word ‘EAMCET’ so that no two vowels come

In how many ways can three balls be drawn from

together, is-

the box if at least one black ball is to be included

(A) 360

(B) 144

(C) 72

(D) 54

in the draw? (A) 129

(B) 84

(C) 64

(D) None of these

Q.9

How many 5 digit odd numbers can be formed with the help of digits 0, 2, 3, 4 and 6 (Rept. not allowed) ?

Q.2

The number of word groups by taking at least 1 letters of each words ‘PATH’, ‘GROW’ and ‘SKIN’ are-

Q.3

(A) 153 –1

(B) 15 3

(C) 163 – 1

(D) 163

(A) 1296

(B) 625

(C) 671

(D) None of these

consonants cannot occur together, is(A) 4!

(B) 3! × 4!

(C) 7!

(D) None of these

(B) 24

(C) 96

(D) 120

How many four digit numbers from the digits 0, 1, 2, 3 will contain 3 at unit place (Repetition not allowed)

Q.11

(A) 6

(B) 18

(C) 4

(D) None of these

The number of numbers of 4 digits which are not divisible by 5 are (when repetition is allowed)-

The number of words which can be formed from the letters of the word MAXIMUM, if two

Q.5

Q.10

Four dice are rolled. The number of possible outcomes in which at least one die shows 2 is-

Q.4

(A) 18

Q.12

(A) 7200

(B) 3600

(C) 14400

(D) 1800

How many 5 digit numbers be formed by the digits 1, 2, 3, 4, 5 which are divisible by 4 (Repetition not allowed) ?

How many six letter words be made out of the

(A) 24

(B) 120

(C) 72

(D) None of these

letters of ‘ASSIST? In how many words the alphabet S alternates with other letters?

Q.6

The total number of 5- digit num bers formed with

(A) 120, 6

(B) 720, 12

the digits 0, 1, 2, 3, 4 and 5 which are divisible by

(C) 120, 12

(D) 720, 24

3, is (Repetition not allowed) -

In how many ways the alphabets of the word ‘MULTIPLE’ can be ordered other than itself, when the order of vowels is not changed?

Q.7

Q.13

(A) 3360

(B)3359

(C) 6720

(D) 20160

The total number of words which can be formed using letters of the word ‘FAILURE’ so that consonants always occupy odd places, is(A) 144

(B) 576

(C) 5040

(D) None of these

Q.14

(A) 216

(B) 240

(C) 600

(D) 3125

How many 6 digit different number can formed with help of the digits of numbers 121 and 202?

Q.15

(A) 25

(B) 50

(C) 100

(D) None of these

The number of 4 - digit numbers formed with the digits 1, 2, 3, 4, 5, 6, 7 which are divisible by 25 is (Repetition not allowed) (A) 20

(B) 30

(C) 40

(D) None of these


11

Q.16

Q.17

Q.18

The number of six digit numbers that can be

made

terminal digits are even is-

‘MATHEMATICS’?

(A) 72

(B) 720

(A) 136

(B) 330

(C) 144

(D) 288

(C) 70

(D) None of these

Using all digits 2, 3, 4, 5, 6, how many even

Q.21

Q.25

out

of

the

letters

of

the

word

The number of words which can be formed by

numbers can be formed?

taking two same and two different letters from the

(A) 24

(B) 48

letters of the word ‘COMBINATION’ is-

(C) 72

(D) 120

(A) 756

(B) 1512

(C) 252

(D) None of these

How many non zero numbers can be formed with Q.26

Taking three same and one different letters from

repeated in any number?

the letters of the word ‘PROPORTION’, the

(A) 260

(B) 336

number of words which can be formed is-

(C) 410

(D) None of these

(A) 18

(B) 360

(C) 20

(D) None of these

How many four digits numbers can be formed with digits 1, 2, 3, 4, 5, 6 when it includes 1 and 2

Q.20

In how many ways can a selection of 4 letters be

formed from the digits 1, 2, 3, 4, 5, 6 and 7 so that

the help of digits 0, 1, 2, 3, 4, when no digit is

Q.19

Q.24

Q.27

How many words can be formed by taking three

necessarily (Repetition not allowed) ?

letters from the letters of the word ‘SERIES’?

(A) 6

(B) 288

(A) 24

(B) 18

(C) 144

(D) 48

(C) 42

(D) None of these

The total number of seven digit numbers the sum

Q.28

The number of words which can be formed using

of whose digits is even is-

4 letters of the word ‘EXAMINATION’ is-

(A) 9000000

(B) 4500000

(A) 1896

(B) 2136

(C) 8100000

(D) None of these

(C) 2454

(D) None of these

The number of times the digit 3 will be written when listing 1 to 1000 is ……

Q.22

(A) 300

(B) 271

(C) 302

(D) 269

The number of ways in which any four letters can be selected from the word ‘CORGOO’ is-

Q.23

(A) 15

(B) 11

(C) 7

(D) None of these

The total number of ways of selecting five letters from the letters of the word ‘INDEPENDENT’ is(A) 72

(B) 3320

(C) 120

(D) None of these


12

LEVEL- 3 Q.1

The number of rectangles in the adjoining figure

Q.7

number of ways of selecting three of these objects

is -

so that no two of them are next to each other is-

(A) 5 × 5 (B) 5P2 × 5P2 (C) 5C2 × 5C2 (D) None of these Q.2

If n objects are arranged in a row, then the

Q.8

(A) n – 2 C3

(B) n – 3C2

(C) n – 3C3

(D) None of these

Between two junction stations A and B there are 12 intermediate stations. The number of ways in

In a plane there are 37 straight lines, of which 13

which a train can be made to stop at 4 of these

pass through the point A and 11 pass through the

stations so that no two of these halting stations are

point B. Besides, no three lines pass through one

consecutive is-

point, no line passes through both points A and B, and no two are parallel. Then the number of

(A) 8C4

(B) 9C4

intersection points the lines have is equal to-

(C) 12C4 – 4

(D) None of these

(A) 535

(B) 601

(C) 728

(D) None of these.

Q.9

The number of integral solutions of x + y + z = 0 with x  – 5, y  – 5, z  – 5 is-

Q.3

The number of numbers between 1 and 1010

(A) 135

(B) 136

which contain the digit 1 is-

(C) 455

(D) 105

(A) 1010 – 910 – 1

(B) 910

(C) 1010 – 810

(D) None of these.

Q.10

The number of non-negative integral solutions of x + y + z  n, where n  N is -

Q.4

A set contains (2n + 1) elements. If the number of

(A) n + 3C3

(B) n + 4C4

subsets of this set which contain at most

(C) n + 5C5

(D) None of these

n

elements is 4096, then th e value of n is(A) 6

(B) 15

(C) 21

(D) None of these.

Q.11

The number of ways is which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question is -

Q.5

How many different nine digit numbers can be formed

from

the

number

223355888

by

rearranging its digits so that the odd digits occupy even positions ?

Q.6

(A) 16

(B) 36

(C) 60

(D) 180

Q.12

of factors out of the total obtained which are multiples of 5 is(A) 5040

(B) 7180

(C) 8150

(D) None of these

(B) 21C8

(C) 21C9

(D) 21C10

Number of ways of placing 5 identical balls in 3 identical boxes (no box remains empty), is-

All possible two factors products are formed from numbers 1, 2, 3, 4, ........, 200. The number

(A) 21C7

Q.13

(A) 6

(B) 2

(C) 3

(D) None of these.

Number of ways of placing 5 identical balls in 3 different boxes (no box remain empty), is– (A) 6

(B) 12

(C) 150

(D) None of these.


13

Q.14

Number of ways of placing 5 different balls in 3 identical boxes (no box r emains empty), is-

Q.15

(A) 50

(B) 10

(C) 25

(D) none of these

Number of ways of placing 5 different balls in 3 different boxes (no box remains empty), is(A) 10

(B) 15

(C) 25

(D) 150

Questions based on statements (Q. 16 - 18) Each

of

the

questions

given

below

consists

of

Statement – I and Statement – II. Use the following Key to choose the appropriate answer. (A) If both Statement - I and Statement - II are true, and Statement - II is the correct explanation of Statement- I. (B) If both Statement-I and Statement - II are true but Statement - II is not the correct explanation of Statement-I. (C) If Statement-I is true but Statement - II is false. (D) If Statement-I is false but Statement - II is true. Q.16

Statement

I :

( r  2) ! ( r  1)!

is divisible by 6.

Statement II : Product of three consecutive

integers is divisible by 3!. Q.17

Statement I: The exponent of 7 in 100C50 is 4. Statement II : The number of ways in which

we can post 5 letters in 12 boxes is 12 5. Q.18

Statement I : The number of ways of dividing

n identical objects among r groups is

n+r–1C

n.

Statement II : The number of ways of dividing

n identical objects among r groups is equal to number of arranging n identical objects of one kind and (r – 1) identical objects of other kind in a row. Therefore, it is equal to ( n  r  1) ! n! (r  1)!

= n+r–1Cn.


14


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

B

C

D

C

D

B

C

D

B

D

A

C

C

A

C

C

C

A

B

D

Q.No. 21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

B

D

A

A

B

A

D

B

B

B

D

A

B

A

B

B

D

B

B

B

Q.No. 41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

B

C

C

D

D

A

B

D

D

B

B

B

C

A

B

B

B

A

D

B

Q.No. 61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

D

B

A

C

A

B

A

B

A

B

C

B

D

B

D

A

B

A

A

C

Q.No. 81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99 100

A

C

B

B

C

B

A

D

B

C

B

D

C

A

B

A

C

D

A

Ans. Ans. Ans. Ans.

B

Q.No. 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 Ans.

C

D

B

B

B

D

B

D

B

B

B

A

B

B

A

B

A

C

C

C

Q.No. 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 Ans.

C

B

A

B

A

B

A

B

C

B

A

D

A

B

A

B

B

C

C

A

Q.No. 141 142 143 144 Ans. B,D A

D

B

LEVEL- 2 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans. C

B

C

A

C

B

B

C

A

C

A

A

A

B

C

B

C

A

C

B

21

22

23

24

25

26

27

28

Ans. A

C

A

A

A

C

C

C

Q.No.

LEVEL- 3 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

Ans. C

A

A

A

C

B

A

B

B

A

A

B

A

C

D

A

D

A


15

IIT – ian’s

P A C E



LEVEL –1 Q.8

Binomial Theorem for positive integral Index

equal then (where r > 1, n > 2), positive

Q.2

 a   9 b  is 3 

integer)-

Fourth term in the expansion of  (A) 40 a 7 b3

(B) 40a 3 b7

(C) 1890 a 6 b4

(D) 1890a4 b6

(A) r = n/2 (C) r =

Second term in the expansion of (2x + 3y) 5 will

Q.9

be (A) 46 x2y3

(B) 30 x 3y2

(C) 240 x4 y

(D) 810 xy4

The coefficient of (3r) th term and coefficient of (r + 2)th term in the expansion of (1 + x) 2n are

10

Q.1

Binomial Theorem

Q.10

n 1 2

(B) r = n/3 n 1

(D) r =

2

The coefficient of a 2 b3 in (a + b) 5 is(A) 10

(B) 20

(C) 30

(D) 40

The coefficient of x7 and x8 in the expansion of n

Q.3

Q.4

 x   2   are equal, then n is equal to 3 

The 5 term of the expansion of (x – 2) is th

8

(A) 8C5x3( –2)5

(B) 8C5x3 25

(C) 8C4x4 (–2)4

(D) 8C6x2 (–2)6

The number of terms in expansion of (x – 3x 2 +

Q.11

3x3)20 is(A) 60 Q.5

(B) 61

(C) 40

(D) 41

(C) 55

(D) None of these

The coefficient of x5 in the expansion of (A) 12C525, 37

(B) 12C626.36

(C) 12C527.35

(D) None of these

The term with coefficient 6C2 in the expansion

n

Q.12

(A) T1 and T3

(B) T2 and T4

(C) T3 and T5

(D) None of these

1   If the expansion of  x 2   , the coefficient 4   of third term is 31, then the value of n i s-

If n is a positive integer, then r th term in the expansion of (1–x)n is-

Q.7

(B) 45

(2 + 3x)12 is-

of (1+ x)6 is-

Q.6

(A) 35

Q.13

(A) 30

(B) 31

(C) 29

(D) 32

If A and B are coefficients of x r and xn–r

(A) nCr (–x)r

(B) nCr xr

respectively in the expansion of (1+ x) n, then-

(C) nCr-1(–x) r–1

(D) nCr–1xr–1

(A) A = B (B) A  B

 

If the 4th term in the expansion of  ax  5 2

1 

(C) A =  , B for some 

n

 is x 

, then the values of a and n are-


If (1 + by)n = (1 + 8y +24y2 + …..) then the value of b and n are r espectively-

(A) 1/2, 6

(B) 1, 3

(A) 4, 2

(B) 2, –4

(C) 1/2, 3

(D) can not be found

(C) 2, 4

(D) –2, 4


1

Q.15

Q.23

The number of terms in the expansion of 9

10

 x 3    2  is  2 x 

9

(1 + 5 2 x) + (1 – 5 2 x) is-

Q.16

(A) 5

(B) 7

(C) 9

(D) 10

(A)

The number of non zero terms in the expansion

(C)

of [(1 + 3 2 x )9 – (1 – 3 2 x)9] is -

Q.17

(A) 9

(B) 10

(C) 5

(D) 15

Q.24

(B) 5

(C) 4

(D) 3

256

(B)

450

504 259

(D) None of these

263

The coefficient of x –26 in the expansion of 11

in the expansion of (x + 2 ) + (x – 2 ) is(A) 10

405

 2 2   x  4  is x  

After simplification, the total number of terms 4

The coefficient of x 4 in the expansion of

(A) 330 × 2 6

(B) – 330 × 26

(C) 330 × 2 7

(D) – 330 × 27

4

Q.25

The term independent of x in the expansion of 10

Q.18

 x 3   will be   3 2x 2   

The number of terms in the expansion of [(x – 3y)2 (x + 3y) 2]3 is-

Q.19

(A) 6

(B) 7

(A) 3/2

(B) 5/4

(C) 8

(D) None of these

(C) 5/2

(D) None of these

The number of terms in (x + a) 100 + (x – a) 100

Q.26

8

(A) 202

(B) 51

 1  expansion of  y1/ 3  y 1/ 5  is  2 

(C) 101

(D) None of these

(A) sixth

(B) seventh

(C) fifth

(D) None of these

after solving the expansion is -

Q.20

The term independent of y in the binomial

The coefficient of x 5 in the expansion of (1+ x2)5 (1+ x)4 is (A) 30

(B) 60

(C) 40

(D) None of these

Q.27

If x4 occurs in the r th term in the expansion of 15

 4 1   x  3  , then r equalsx   (A) 7

Q. 21

(B) 8

Q.28

(A) 6

(B) 22

(C) – 6

(D) 8

The term containing x in the expansion of 5

 2 1   x   is x   (A) 2nd

(B) 3 rd

(C) 4 th

(D) 5

th

The coefficient of x 4 in the expansion of 6

(1+ x + x 2 + x3)11 is-

Question based on

(D) 10

The coefficient of x in the expansion of (1 + x)3.(1 – x)6 is -

Q.22

(C) 9

5

(A) 990

(B) 495

(C) 330

(D) None of these

Q.29

1   The term independent of x in  2x   is 3x   (A) 160/9

(B) 80/9

(C) 160/27

(D) 80/3

Particular term in the expansion


2

Q.30

The term independent of x in the expansion of

8

Q.37

10

 x 3    is 2   3 2 x   (A) 10C1

(B) 5/12

(C) 1

(D) None of these

Q.38

 x a  The middle term of the expansion    is a x  (A) 56a2/x2

(B) – 56a 2/x2

(C) 70

(D) –70

The

middle

term

in

the

expansion

of

10

Q.31

 3 1   x  3  isx  

If 9 term in the expansion of (x + x ) does th

1/3

-1/3 n

not depend on x, then n is equal to(A) 10 Q.32

The

constant

 1  x   x  (A)

Q.33

(B) 13

2n ! n!

(C) 16 term

in

(D) 18

the

expansion

of

2n

Q.39

is-

(A) 252

(B) – 252

(C) 210

(D) – 210

If the middle term in the expansion of n

n!

(B)

2n !

(C)

2n ! 2 !n !

(D)

2n !

 2 1   x   is 924 x6, then n = x  

n !n !

(A) 10

(B) 12

(C) 14

(D) None of these

The coefficient of the term independent of y in 3n

Q.40

 1   is the expansion of  y  2    y 

The greatest coefficient in the expansion of (1+ x)10 is(A)

(A) 3nCn–1 (–1)n-1

(B) 3n Cn

(C) 3n Cn(–1)n

(D) None of these (C)

Q.34

The number of integral terms in the expansion of (51/2 + 7 1/6)642 is -

Question based on

Q.35

(A) 106

(B) 108

(C) 103

(D) 109

Q.41

The

10! 5! 6!

(B)

10!

10! (5! )

2

(D) None of these

5! 7! middle

term

in

the

expansion

of

(1 – 3x + 3x – x ) is 2

3 6

(A) 18C10x10 (B) 18C9(–x)9

Middle Term

(C) 18C9 x9 (D) – 18C10 x10

Middle term in the expansion of (x 2 – 2x)10 will be (A) 10C4x17 24

(B) – 10C5 25 x15

(C) – 10C4 24 x17

(D) 10C5 24 x15

Question based on

Q.42 Q.36

The

middle

term

in

the

expansion

9

 3 x 3    is  x2  6    (A)

189 8

x2,

21 16

x7

189 2 21 7 (C) – x , – x 8 16

(B)

189

21 x2, – x7 8 16

of

Term from end

The 5th term from the end in the expansion of 9

 x 3 2     is  2 x3    (A) 63x3 (C)

(D) None of these

672 x18

(B) –

252 x3

(D) None of these


3

n

Q.43

1   If in the expansion of  21 / 3  1/ 3  , the ratio 3  

Q.51

C0C1 + C 1C2 + C2C3 + ....+ Cn–1Cn is equal to-

of 6 th terms from beginning and from the end is

(A)

1/6, then the value of n is (A) 5

Q.44

(C)

(D) None of these

Binomial Coefficient

Q.52

If (1+ x) n = C 0 + C 1x + C 2x2 + C 3x3 + ....+ C nxn, (A) 2n+1

(B) 2n–1

(C) 2 + 1

(D) 2 – 1

n

If (1+ x) n = C 0 + C 1x + C 2x2 + ...+ C nxn, then the

2n !

(D)

(n  1)! n !

n 1 1 Cn C0 – nC1 + n C2 – ......+ (–1) n = 2 3 n 1

(B) 1/n 1

n 1

(D)

1 n 1

 1  In the expansion of (1 + x) 1   , the term  x  n

independent of x is-

(A) 2n (n + 1)

(B) 2 n-1 (n + 1)

(A) C 02 + 2 C12 + .....+ (n +1) C 2n

(C) 2n-1 (n + 2)

(D) 2 n (n + 2)

(B) (C0 + C1 + …..+ C n)2 (C) C 02 + C12 + .....+ C 2n

If C0, C 1, C 2, ....., C 15 are coefficients of different

(D) None of these

(B) 2 14

(C) 2 7

Q.54

(D) 28

If (1+ x)n = C0 + C1x + C2x2 +...+ Cnxn, then C0Cr + C1Cr +1 + C2Cr+2 + .....+ C n–r Cn is equal to-

If (1+ x)n = 1 + C 1x + C2x2 + ....+ C nxn, then

(A)

C1 + C3 + C5 + ..... is equal to(A) 2n

(B) 2 n – 1 (C) 2n + 1

 1

n! 

 n !



1 2 ! ( n  2) !



1 4!(n  4) !

(D) 2 n–1

 ...... 

(C)

(B) 2 n–1

1! (n  1) ! (A)

2

+

1 3!( n  3)!

(C) 2n+1

+

1 5!( n  5)!

n

n!

(C) 0

( n  r )!(n  r )! 2n! n !(n  r )!

(B)

(D)

2n! n !( n  r )! 2n! (n  1) !(n  1)!

 is

n ! 

Q.55

If (1+ x)n = C0 + C1x + C2x2 + ...Cn. xn then the value of C0 + 3C 1 + 5C2 + .....+ (2n + 1) C n is-

(A) 2n 1

2n!

1 

equal to -

Q.50

(n  1)! (n  1) !

n !( n  1)!

value of C0 + 2C 1 + 3C2 + ....+(n +1)C n is -

(A) 215

Q.49

2n !

2n !

n

C0+ C2+ C4 + ...+ C 14 is equal to-

Q.48

(B)

n

terms in the expansion of (1 + x) 15, then

Q.47

n

(C)

Q.53

Q.46

n! n!

(A) n

then the value of C 1+ C2+ C3 + ...+ Cn is-

Q.45

2n !

(B) 7

(C) 9 Question based on

If (1+ x)n = C0 + C1x + C2x2+ ...+ Cnxn, then

(B)

2

(D) 2 –n+1

+ .......=

n 1

n!

(D) None of these

The value of 8C0 + 8C2 + 8C4+ 8C6+ 8C8 is(A) 32

(B) 64

(C) 128

(D) 256

Q.56

(A) n.2n

(B) (n –1). 2 n

(C) (n + 2).2 n–1

(D) (n + 1).2 n

If C0, C1, C2.......Cn are binomial coefficients in the expansion of (1 + x)n, n  N, then C1 2 (A) (C)

+

C3 4

+

2 n 1  1 n 1 2n  1 n 1

C5 6

+…..+

Cn n 1

is equal to-

(B) (n + 1) . 2 n+1 (D) None of these


4

Q.57

If (1 + x) n = C0 + C1x + C2x2 + ....+ C nxn, then for n odd, C12 + C32 + C52 + .....+ C n2 is equal to (A) 22n-2

Q.58

(B) 2 n

(C)

( 2n)! 2( n!)

2

(D)

(2n )! ( n!)

2

If (1+ x + x 2)2n = a0+ a1x + a2x2 +....then the value of a0 – a 1 + a2 – a3 + .... is(A) 2n

Q.59

(B) 3n

(C) 1

(D) 0

The sum of the coefficients in the expansion of (a + 2b + c) 10 is (A) 410

Q.60

(B) 310

(C) 2 10

(D) 104

The sum of coefficients of even powers of x in the expansion of (1+ x + x 2 + x3)5 is (A) 512

(B) – 512

(C) 215

(D) None of these


5

LEVEL- 2 Q.1

If (1 + x – 2x 2)6 = 1 + C 1x + C2x2 + C3x3 + ....+ C12

x 12,

Q.8

then the value of C 2+ C 4+ C 6 + ...+ C 12

is -

Q.2

Q.3

(A)

(A) 30

(B) 32

(C) 31

(D) None of these

(C) Q.9

If C0, C1, C2, ....C n are binomial coefficients of

n 1

x

r

(B)

n  r  1 r

x

r n  r

(D)

r  1

x

x

If (1+ x) n = C0 + C1x + C2x2 + ...+ C nxn, then the (A) n(n + 1) 2 n–2

(B) n(n + 1) 2 n–1

equals-

(C) n(n + 1) 2 n

(D) None of these

(A) –n.2n–1

(B) 0

(C) 2n–1. (2 – n)

(D) None of these

If C r

Q.10

Q.11

a/2 n

The sum of coefficients of odd powers of x in the expansion of (1+ x) n is -

stands for nCr , then the sum of first (n+1)

(A) 2n + 1

(B) 2n – 1

(C) 2n

(D) 2n–1

The sum of C02 – C12 + C22 – ....+ (–1) n Cn2 where n is an even integer, is-

(B) na

(C) 0

(A) 2nCn

(D) None of these

(B) (–1)n

5

(A) 252

(B) 252

(C) 452

(D) 532

Q.12

(A) 0, 6

(B) 0, 2 6

(C) 1, 6

(D) 0

b  3

C2,.....Cn denote the binomial

 (r  1) C r is -

Q.13

(A) n 2n

(B) (n+1) 2 n–1

(C) (n +2) 2 n–1

(D) (n + 2) 2 n–2

If 1  r  n–1 then

n–1C

r

+

n–2C

r

+ .....+ r Cr

equalsterm in the expansion of contains

same

powers

of Q.14

(A) 9

(B) 10

(C) 8

(D) 6

(A) nCr

(B) nCr+1

(C) n+1Cr

(D) None of these

The coefficient of 1/x in the expansion of

 

(1+ x)n 1 

(A)

If n is odd, then C 02 – C12 + C 22 – C 32 + ....+ (–1)n C 2n = (B) 1

C1 ,

r  0

a and b, then the value of r is -

(A) 0

C0 ,

n

21

a 

If

value of

values of a0 and a6 are -

 a 3  b  

(D) (–1) . Cn/2

coefficients in the expansion of (1+x) n, then the

If (2x – 3x 2)6 = a 0 + a 1x + a 2x2 + ...+a 12x12, then

If the (r

n

5

The value of ( 5 + 1) –( 5 – 1) is -

+1) th

2nC

n/2 n

(C) 2nCn–1

Q.7

n  r  1

C0 – 2.C1+ 3.C2 – 4.C 3 + .....+ (–1) n .(n + 1)C n

(A)

Q.6

is equal to

value of 12 C1 + 22 C2 + 32 C3 + ...+ n 2 Cn is

aC0 – (a + d) C1+ (a + 2d)C2 – (a + 3d)C3 + ... is-

Q.5

Tr

different terms in the expansion of (1+ x) n then

terms of the series

Q.4

Tr 1

In the expansion of (1+ x) n,

(C) 

(D)

(C) n!

1 

n

 is -

x 

n! ( n  1)!( n  1)!

( 2n ) ! (2n  1)!(2n  1)!

(B)

(2n ) ! ( n  1)!( n  1)!

(D) None of these

2

( n / 2) !


6

Q.15

If 6th term in the expansion of

Q.21

8

consecutive terms in the expansion of (1+ x) n,

 1   8 / 3  x 2 log10 x  is 5600, then x i s equal to x 

Q.16

(A) 8

(B) 10

(C) 9

(D) None of these

(C)

Q.22

second term, then the value of n is (B) 9

(C) 10

(D) None of these

a2  a3

is equal to-

(B)

(D)

1

a2

2 a2  a3 2a 3 a2  a3

If the coefficients of four consecutive terms in the

a1 a1  a 2

,

a2 a2  a3

,

a3 a3  a 4

are in -

series

(A) A.P.

(B) G.P.

[3.nC0 – 8.nC1 + 13. nC2 – 18. nC3 + .....+ (n +1)]

(C) H.P.

(D) None of these

The

sum

of

the

terms

of

the

Q.23

(A) 3. 2 n –5n.2n-1 (C)

3.2 n +

5n.2n-1

(B) 0

If (1+ x) n = C0 + C1x + C2x2 + .....+ C nxn, then C0C2 + C1C3 + C2C4 + ......+ C n–2 Cn =

(D) None of these (A)

If sum of all the coefficients in the expansion of (x3/2 + x-1/3) n is 128, then the coefficient of x 5

(C)

is (A) 35

(B) 45

(C) 7

(D) None of these

Q.24

(A)

(C)

is -

 (2n )!    n! 

(B)

2n ! (n  2) ! (n  2)!

(B)

(D)

2n ! n !( n  2)! 2n ! (n  1) ! (n  2)!

2n ! ( n !) 2

(B)

(2n  2) ! n ! (n  1)!

(D)

( 2n  2)! [(n  1)!]2 (2n )! n !(n  1)!

( 2n )! ( n )! ( n )!

Q.25

2

(C) 

( n  1)!( n  2)!

(1+ x) 2n+2 is -

If (1 + x) n = C0 + C1x + C 2x2 + ....+ C n.xn then

(A) 1

2n !

The greatest coefficient in the expansion of

the value of C 0Cn + C 1Cn–1 + C2Cn–2 + ....+ C nC0

(D) (2n)2

1   The middle term in the expansion of  x    2 x  1.3.5.....(2n  3) n!

Find the value (183  73  3.18.7.25)

(C)

6

3  6.243.2 15.814 20.27.8 15.9.16 6.3.32 64 (A) 1

(B) 5

(C) 25

(D)100

2n

is(A)

Q.20

a2  a3 2a 2

a3  a4

expansion of (1+ x)n are a 1,a2,a3 and a4, then

(A) 8

is-

Q.19

a2

If the coefficient of third term in the expansion of

a3

+

a1  a 2

(A)

n

Q.18

a1

then

1   2 / 3  x  1/ 3  is 27 more than the coefficient of x  

Q.17

If a1, a2, a3, a4 are the coefficients of any four

Q.26

(B)

1.3.5.....(2n  1) n!

1.3.5.....(2n  1) n!

(D) None of these


3

 1   1   x    x   is x   x  (A) –3

(B) 0

(C) 1

(D) 3


7

Q.27

If the sum of the coefficients in the expansion of (2 x 2 – 2x +1)51 vanishes, then the value of 

n

Q.34

If (1 + x) =

Q.28

  

then 1 

(B) –1

(C) 1

(D) –2

If the sum of coefficients in the binomial

(A)

expansion of (x + y)n is 4096 then greatest coefficient in the expansion is (A) 922

(B) 942

(C) 787

(D) 924

(C)

Q.35 Q.29

The value of

(nC2 –2. nC3+3.nC4 –4.nC5 +

.....) is

equal to -

Q.30

(A) 1

(B) 0

(C) –1

(D) None of these

n

Cr .( x ) r

r  0

is (A) 2



n

n

n   C1   n C2   . 1   ..... 1  C n  = n  nC  C0   n C1  n – 1   n

n – 1

( n – 1) ! (n  1)

(B)

n

n!

(D)

( n  1)

n – 1

( n – 1) ! (n  1)

n 1

n!


x 1 x  1     2 / 3 1/ 3  is  x  x  1 x  x1/ 2  (A) T4 = 180

(B) T 5 = – 210

(C) T4 = – 180

(D) T5 = 210

The sum of 12 terms of the series 12C

1.

1 3

+ 12C2 .

1 9

12

 4  (A)   – 1  3 

+ 12C3 .

1 27

+ .... is -

12

 3  (B)   – 1  4 

12

 3  (C)   + 1  4  Q.31

(D) None of these

If n = 10 then ( C 02 – C12 + C 22 – C 32 + ..... + (–1)n ( C 2n )) equals-

Q.32

(A) (–1)5.10C5

(B) 0

(C) 10C5

(D) (–1)6 10C6

If n = 11 then ( C 02 – C12 + C 22 – C 32 + ..... + (–1)n ( C 2n ) equals-

Q.33

(A) (–1)5. 10C5

(B) 0

(C) 10C6

(D) None of these

The sum of (n +1) terms of the series C02+ 3C12 + 5C22 + ..... is (A) 2n–1Cn-1

(B) 2n–1 Cn

(C) 2(n +1) 2n–1Cn

(D) None of these


8

LEVEL- 3 Q.1

If the coefficients of T r , Tr+1, T r+2 terms of

Q.9

integral terms is -

(1+ x)14 are in A.P., then r (A) 6 Q.2

(B) 7

(C) 8

The value of x , for which the



log expansion of  2 2

( 9 x 1  7 )

(D) 9 6 th term

2

Q.10

in the

1





(1 / 5) log 2 ( 3

x 1

  1)  

(A) 4, 3

(B) 0, 3

(C) 0, 2

(D) 1, 2

Q.11

Q.4

(C) 31C6 – 21C6

(D) 30C5 + 20C5

Q.12

(x2 – x – 2) 5 is-

Q.5

(B) – 82

(C) – 81

(D) 0

Q.13

In the expansion of (1+3x+2x 2)6 the coefficient of x11 is (A) 144

(B) 288

(C) 216

(D) 576

Let R = (5

5 + 11)2n+1 and ƒ = R – [R], where [.]

(A) 42n+1

(B) 42n

(C) 42n–1

(D) 4 –2n

The greatest integer less than or equal to

(A) 196

(B) 197

(C) 198

(D) 199

The number of integral terms in the expansion of

Q.14

(A) 106

(B) 108

(C) 103

(D) 109

The remainder when 599 is divides by 13 is (A) 6

(B) 8

(C) 9

(D) 10

When 2301 is divided by 5, the least positive remainder is (A) 4

Q.6

Coefficients of of (x +

x r [0

3) n–1

(B) 8

(C) 2

(D) 6

 r  (n–1)] in the expansion

+ (x + 3)n–2 (x + 2) +

Q.15

If the sum of the coefficients in the expansion of

(x + 3)n–3 (x + 2)2 + ........ + (x + 2) n–1 -

(1 – 3x + 10x2)n is a and if the sum of the

(A) nCr (3r – 2n)

(B) nCr (3n–r –2n–r )

coefficients in the expansion of (1 + x 2)n is b,

(C) nCr (3r + 2n–r )

(D) None of these

then -

n

Q.7

(D) 131

(51/2 + 71/6)642 is -

The coefficient of x5 in the expansion of (A) – 83

(C) 130

( 2 + 1)6 is -

(1+ x)21 + (1+ x)22 + ...........+ (1+ x) 30 is (B) 9C5

(B) 129

R.ƒ is -

The coefficient of x5 in the expansion of (A) 51C5

(A) 128

denotes the greatest integer function. The value of

7

is 84 is equal to -

Q.3

In the expansion of (51/2 + 71/8)1024, the number of

If x + y = 1, then

 r 2 n Cr x r y n r equals -

(A) a = 3b

(B) a = b3

(C) b = a3

(D) none of these

r  0

Q.8

(A) nxy

(B) nx (x+ yn)

(C) nx (nx+y)

(D) None of these.

10

Q.16

 20C k =

k  0

(1 + x)n –nx –1 is divisible by (where n  N)–

(A) 219 +

(A) 2x3

(C) 20C10

2

(C) x

(B) 2x

1 2

20

C10

(B) 219 (D) none of these

(D) All of these


9

Q.17

If n  N such that (7 + 4 3 )n = I + F,

Q.22

where I  N and 0 < F < I, Then the value of

expansion of (3 + 5x) 11 when x =

(I + F) (1 – F) is (A) 0 Q.18

(C) 7 2n

(B) 1

The value of

95

C4 +

The value of numerically greatest term in the

(D) 2 2n

(A) 55  310

(B) 110  39

(C) 55  38

(D) 55  39

 100 j C 3 is -

j 1

Q.23 100

(A) C5

(B)

(C) 99C4

(D) 100C5

C4

The value of numerically greatest term in the expansion of (3x + 2) 9 when x = 3/2 (A)

(C) 7 

0  ƒ  1, then I equals (A) (B) (C) (D) Q.20

7  313

If (5 + 2 6 ) n = I + ƒ, where I  N, n  N and

1

5

5

95

Q.19

1

(B) 7 313

2 314

(D)

7  314 2

–f

 f

Question based on Statements (Q. 24 - 26)

1

Each of the questions given below consist of Statement – I

–f 1  f

and Statement – II. Use the following Key to choose the

1

appropriate answer.

–f 1  f 1 1  f

If (15Cr +

(A) If both Statement- I and Statement- II are true, and Statement - II is the correct explanation of

+f

Statement– I.

15

Cr–1) (15C15–r +

15

C16 –r ) = ( 16C13)2, then

the value of r is (A) r = 3

(B) r = 2

(C) r = 4

(D) none of these

(B) If both Statement - I and Statement - II are true but Statement - II is not the correct explanation of Statement – I. (C) If Statement - I is true but Statement - II is false (D) If Statement - I is false but Statement- II is true.

Passage Based Questions (Q. 21 - 23)

The numerically greatest term in the expansion of (x + a)n is given by

n 1 x 1 a

Q.24

(1 + x + x 2 + ......+ x 10 )5 is 51.

= k (say)

Statement II

to 22n–1 –

numerically greatest term (b) If k is not an integer. Let m is its integral part

Q.21

Q.25

1 5

two together) is equal

( 2n )! 2.(n !) 2

Statement

I

: The coefficient of x 5 in the

expansion of (1 + x 2)5 (1 + x)4 is 120.

The numerically greatest term in the expansion of (3 – 5x)15 when x =

: The sum of the products of

nC , nC , nC ..... nC (taken 1 2 n 0

(a) If k is an integer then Tk and Tk+1 are the

then Tm+1 is the numerically greatest term.

Statement I : The number of terms in

Statement II : The sum of the coefficients in the

expansion of (1 + 2x – 3y + 5z) 3 is 125.

is –

n

(A) T4

(B) T 5 & T6

(C) T4 & T5

(D) T6

Q.26

Statement I :

 K . (nCK )2  n . 2n–1Cn–1

K 1

Statement II : If 22003 is divided by 15 the

remainder is 1.


10

ANSWER KEY LEVEL- 1 Ques.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans. A

C

C

D

C

C

A

A

A

C

C

D

A

C

A

C

D

B

B

B

Ques. 21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

C

A

A

C

B

A

C

C

C

D

C

D

C

B

B

B

C

B

B

B

Ques. 41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

B

B

B

D

C

B

D

B

B

C

C

C

C

A

D

C

C

C

A

A

Ans.

Ans.

LEVEL- 2 Ques.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

C

B

C

B B

A

A

C

A

D

D

C

B

B

B

B

B

A

B

A

Ques.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Ans.

C

A

C

B

B

B

C

D

A

A

A

B

C

C

D

LEVEL- 3 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

D

D

C

C

D

B

C

C

B

A

B

B

B

C

B

A

B

B

C

A

21

22

23

24

25

26

C

D

A

D

D

C


11

IIT – ian’s

P A C E

216 - 217, 2nd floor, Shopper’s point, S. V. Road. Andheri (West) Mumbai – 400 058 . Tel: 26245223 / 09


Q.1

LEVEL –1

Mathematical definition of Probability

The probability P(A) of an event is a(A) real number (B) positive real number (C) non- negative real number (D) non- negative real number  1

Q.2

Q.8

Q.9

Q.5

Q.6

Q.7

(B) 1/18

(C) 1/12

(D) 5/36

The probability of drawing a black king from

Q.10

(D) 4/13

Three cards are drawn from a pack of 52 cards. The probability that they are of the same colour is-

In tossing a coin getting a head or tail is-

(A) 4/17

(B) 22/225

(A) experiment (B) exclusive event

(C) 3/17

(D) 2/17

Q.11

(C) joint event (D) None of these Q.4

together is(A) 1/9

(C) 2/13


The probability of getting difference of number as 5, when two dice are tossed

a pack of 52 cards is(A) 1/13 (B) 1/26

Winning a game by a player is(A) an experiment (B) an event (C) experiment and event both

Probability

A bag contains 6 blue, 4 white and 6 red balls. Two balls are drawn at random. The probability that both the balls are red is-

The correct statement for any event A is-

(A) 1/3

(B) 1/6

(A) 0  P(A)  1

(B) 0  P(A)  1

(C) 1/8

(D) 2/9

(C) P(A) = 1

(D) P(A) < 0

Q.12

A bag contains 20 tickets numbered with 1 to

A coin is tossed three times. The probability that in the second toss head does not occur,

20. Three tickets are drawn. The probability

is(A) 1 (C) 1/3

ticket number 18 is not included is-

that ticket number 7 is definitely included and

(B) 1/2 (D) 1/4

The probability of coming up an even (odd)

Q.13

(A) 51/380

(B) 1/20

(C) 3/20

(D) None of these

From a lottery of 30 tickets, marked 1, 2,

number in the throw of a die is-

3,...., 30, four tickets are drawn. The chance

(A) 1/6 (C) 1/3

that those marked 1 and 2 are among them is-

(B) 1/2 (D) 2/3

From a pack of playing cards three cards are drawn simultaneously. The probability that these are one king, one queen and one jack is(A) 64/5525 (B) 16/5525 (C) 128/5525 (D) 64/525

Q.14

(A) 413/145

(B) 2/145

(C) 1/145

(D) 4/145

The probability that a non leap year will have 53 Saturdays is(A) 1/7

(B) 2/7

(C) 6/7

(D) 5/7


1

Q.15

The probability that a non leap year will have

Q.22

52 Fridays is-

Q.16

(A) 1/7

(B) 2/7

(C) 5/7

(D) 6/7

The probability that a leap year will have 52

(A) 0 (C) 1/365 Q.23

Sundays is-

Q.17

(A) 1

(B) 5/7

(C) 2/7

(D) None of these

There are 13 men and 2 women in a party.

Q.24

(A) 2/105

(B) 1/105

(C) 1/14

(D) 1/7

tossed

together.

The

Two

(B) 1/4 dice

are

(C) 1/8 thrown

(D) 1/3 together.

The

be 10 is-

Q.25

(A) 1/6

(B) 1/12

(C) 2/3

(D) 1/4

An urn contains 5 white and 3 black balls and

A bag contains two pairs of shoes. Two shoes

4 balls are drawn at random. The probability

are drawn from it. The probability that it is a pair is(A) 1/3 (B) 1/2

of getting white and black balls equal in (A) 1/7

(B) 2/7

(C) 1/4

(C) 3/7

(D) None of these

number is-

(D) 2/3

If out of 20 consecutive whole numbers two

Q.26

From a book containing 100 page one page is

are chosen at random, then the probability

selected randomly. The probability that the

that their sum is odd, is-

sum of the digits of the page number of the

(A) 5/19

(B) 10/19

selected page is 11, is-

(C) 9/19

(D) None of these

(A) 2/25

(B) 9/100

(C) 11/100

(D) None of these

If the probabilities of boy and girl to be born are same, then in a 4 children family the probability of being at least one girl, is(A) 14/16 (C) 1/8

Q.21

are

probability that the sum of their two numbers

together is-

Q.20

coins

(A) 1/2

probability that the two women will sit

Q.19

Two

(B) 1 (D) 364/365

probability of getting two heads is-

They are seated round a circular table. The

Q.18

The probability that two persons have same date of birth is (in non-leap year)

(A) (B) (C) (D)

52 13 52 13 52 13 52

× × × ×

13 39 13 51 13 39 13 51

× × × ×

13 26 13 50 13 26 13 50

× × × ×

probability that both numbers are prime is-

Q.28

13 49 13 13 13 49

(A) 4/95

(B) 7/95

(C) 14/95

(D) 1/10

Two dice are thrown, the probability that the total score is a prime number is-

13 13

A bag contains 20 tickets marked with numbers 1 to 20. Two tickets are drawn, the

(B) 15/16 (D) 3/8

If 4 cards are drawn one by one from a pack of 52 cards, the probability that one will be from each suit, is13

Q.27

(A) 1/6

(B) 5/12

(C) 1/2

(D) None of these

×24 Q.29

×24

A box contains 25 tickets numbered 1, 2,....25. If two tickets are drawn at random then the probability that the product of their numbers is even, is (A) 11/50

(B) 13/50

(C) 37/50

(D) None of these


2

Q.30

A bag contains 8 white and 6 red balls. 5 balls

Q.37

A dice is thrown then the odds against of

are drawn from the bag at random. The

getting the digit 6 is-

probability that 3 or more balls are white will

(A) 5 : 1

(B) 4 : 1

be-

(C) 6 : 1

(D) None of these

(A) 658/1001

(B) 317/1001

(C) 205/1001

(D) 210/1001

Q.38

If one card is drawn from a pack of card then the odds in favour of getting Ace is-

Q.31

Out of 13 applicants for a job, there are 5

(A) 1 : 3

(B) 3 : 1

women and 8 men. It is desired to select 2

(C) 1 : 2

(D) 1 : 12

persons for the job, the probability that at least one of the selected persons will be a

Q.39

What are the odds in favour of drawing a

women is-

Question based on

(A) 25/39

(B) 14/39

(C) 5/13

(D) 10/13

white ball?

Q.40

(C) 2 : 3

(D) 3 : 5

Addition Theorem of probability

If A and B be any two events, then

(A) 9 : 1

(B) 10 : 1

P (A   equals-

(C) 11 : 1

(D) 12 : 1

(A) P(A) – P (B) (B) P (A) + P(B) + P (A  B)

From a pack of well shuffled cards, one card

(C) P (A) + P(B) – P (A  B)

is drawn randomly. A gambler bets that it is

(D) P (A) P (B)

either a diamond or a king. The odds in favour of his winning the bet is-

Q.34

(B) 2 : 5

The odd against throwing 10 with two dice in a throw are-

Q.33

(A) 3 : 2

Odds of an event Question based on

Q.32

A bag contains 3 black and 2 white balls.

(A) 4 : 7

(B) 4 : 9

(C) 9 : 4

(D) None of these

Q.41

P (A + B) means(A) probability of event A and B (B) probability of event A or B (C) probability of event B after happening of

A bag contains 4 red and 4 white balls. Three

event A (D) None of these

balls are drawn at random. The odd against these balls being all white are-

Q.35

Q.36

(A) 1 :13

(B) 13 : 1

(C) 12 : 1

(D) 14 : 1

Q.42

One of the two events must occur. If the

If P(A) + P(B) = P(A + B) then events A & B are(A) independent

chance of one is 2/3 of the other, then odds in

(B) mutually exclusive (C) dependent

favour of the other are-

(D) None of these

(A) 2 : 3

(B) 1 : 3

(C) 3 : 1

(D) 3 : 2

Q.43

If A and B are two events, then P (neither A nor B) equals-

If odds in favour of A is m : n then P(A)

(A) 1 – P (A  B)

equals-

(B) P ( A ) + P ( B )

(A) m/n

(B) m – n

(C) m / (m + n)

(D) m / (m – n)

(C) 1 – P (A) – P (B) (D) None of these


3

Q.44

For any two events A and B, P (A+B)(A) is always equal to P (A) + P (B)

Q.52

exhaustive

(B) never equals to P (A) + P (B) (C) equals P(A) + P(B) If A and B are

P(C) =

independent

(A) 1/3 (C) 1/6

(D) equals P (A) + P (B) If A and B are disjoint Q.45

If two dice are thrown, then the probability of getting the sum of digits even or less than 5 is(A) 1/2

Q.46

(B) 1/6

(C) 2/3

Q.53

(D) 5/9

Two dice are thrown together. The probability that the sum of their numbers be at least 5 is(A) 1/6

(B) 5/6

(C) 4/6

(D) None of these

Question based on

Q.54

Two dice are thrown then the probability of coming an odd number on one dice and an even number on other dice is(A) 1/3 (C) 1/2

Q.48

Q.50

A bag is containing 20 balls, which are arranged in order of their numbers. If one ball is drawn at random, then the probability that its number is multiple of 3 or 5 is(A) 9/20 (B) 1/20 (C) 3/50 (D) None of these

 B)

If P(A) = 0.25, P(B) = 0.50 & P (A  B) = 0.14, then the probability that neither A nor B occurs is(A) 0.39 (B) 0.25 (C) 0.11 (D) None of these

=

3 2

P(A),

P(B) then the value of P (A) is(B) 1/2 (D) None of these

A card is drawn from a well shuffled pack of 52 cards. Its probability of being an ace or a king or a queen or a jack is (A) 1/13 (B) 2/13 (C) 3/13 (D) 4/13 Conditional Probability

Let P(A) = 0.4 & P(B/A) = 0.5. The

Q.56

A pair of dice is thrown. If the two numbers appearing on them are different, the probability that the sum is 6, is(A) 2/15 (B) 1/9 (C) 5/36 (D) 1/12

Q.57

Two dice are thrown together. If 3 appears on at least one of the dice, then what is the probability that the sum is greater than 9 (A) 1/4 (B) 3/11 (C) 5/11 (D) zero

= 7/8, P(A) = 2P

(B) 5/12 (D) 7/12

P(B)

A pair of dice is thrown. If 5 appears on at least one of the dice, then the probability that the sum is 10 or greater, is(A) 11/36 (B) 2/9 (C) 3/11 (D) 1/12

If A & B are two events such that P(A  B) + P(A (B),then P(A)(A) 7/24 (C) 5/24

Q.51

(B) 9/13 (D) 4/13

3

and

Q.55

A card is drawn from a pack of 52 cards. The probability that the card drawn is neither a heart nor a king is(A) 35/52 (C) 17/52

Q.49


1

events

probability P( A  B ) is equal to(A) 0.8 (B) 0.7 (C) 0.6 (D) None of these

.

Q.47

A, B and C are three mutually exclusive and

Q.58

In a certain town, 40% of the people have brown hair, 25% have brown eyes and 15% have both brown hair and brown eyes. If a person selected at random has brown hair, the probability that he also has brown eyes is(A) 2/5 (B) 1/4 (C) 1/2 (D) 3/8


4

Q.59

A bag contains 7 red and 3 black balls. Three balls are drawn at random from the bag one

Q.66

The probability of getting head and tail alternatively in three throws of a coin (or in a

after the other. The probability that the first two are red and the third is black is-

throw of three coins) is(A) 1/3 (B) 1/4

(A) 21/40

(B) 1/5

(C) 1/5

(C) 7/50

(D) 7/40

Q.67

(D) 3/5

The probability of not getting tail in the first two times and getting a tail in the third time

Question based on

Q.60

Multiplication Theorem of Probability

by tossing a coin continuously is(A) 1/4 (B) 1/8

If A and B are two independent events then

(C) 3/8

(D) 7/8

P (A  B) equalsQ.68

(A) P(A) + P(B) (B) P(A) . P(B) (C) P(A/B) (D) None of these Q.61

that the first is a diamond and the second is a king, is-

If A and B are two independent events, then the probability that only one of A and B occur is(A) P(A) + P(B) – 2P(A  B)

(A) 1/52 (C) 1/4 Q.69

(B) P(A) + P(B) – P(A  B) (D) None of these For two given events A and B, the relation P (AB) = P(A) P(B) implies that A and B are(A) independent (B) mutually exclusive (C) dependent (D) None of these Q.63

Q.70

the probability of Q.71

(D) occurrence of only one

Q.65

(D) 1

A coin is tossed four times then the probability of obtaining at least one tail is(A) 1/16 (B) 14/16 (C) 15/16

(D) 1/4

(B) 11/13 (D) 42/121

The probability that A will pass in a examination is 2/5 and the probability that B will fail in the same examination is 3/4. The probability that only one of them will pass in the examination is-

A coin is tossed three times. The probability of getting all heads or tails only is(A) 0 (B) 1/2 (C) 1/4

A bag contains 6 black and 5 white balls, while the second bag contains 7 black and 4

(A) 2/11 (C) 20/121

independent events then (1 – p 1 – p2 + p1 p2) is

Q.64

If two cards are drawn from a pack of card one by one. If first drawn card is replaced

white balls. Two balls are drawn one from each bag, the probability of both being black is-

If p1 and p2 are the probabilities of two

(A) their joint occurrence (B) occurrence of at least one (C) occurrence of None of these

(B) 1/13 (D) 4/13

then the probability of getting two jacks is(A) 1/221 (B) 1/169 (C) 12/221 (D) 4/663

(C) P(A) + P(B)

Q.62

From a pack of 52 cards two cards are drawn in succession the first having been replaced before the second is drawn. The probability

(A) 3/20 (C) 9/20 Q.72


For solving a problem, odds against to A are 4 : 3 and odds in favour to B are 7: 5. The probability that the problem will not be solved is(A) 16/21 (B) 5/21 (C) 43/84 (D) 45/84


5

The probability of solving a problem by A and B are 1/4 and 2/3 respectively. If A and B work independently, then the probability that the problem will be solved by both of them is(A) 1/6 (B) 3/4 (C) 1/3 (D) 11/12

Q.79

Q.74

The probabilities that three boys will pass an examination are 1/6, 1/4 and 1/3 respectively. The probability that exactly one boy will pass the examination is (A) 31/72 (B) 7/12 (C) 41/72 (D) 11/12

Q.80

The probability that Krishna will be alive 10 years hence is 7/15 and Hari will be alive is 7/10. The probability that both Krishna and Hari will be dead 10 years hence is(A) 21/150 (B) 24/150 (C) 49/150 (D) 56/150

Q.75

If A and B are any two events such that

Q.81

From the records of a hospital, it is found that 20% patients died with the disease cancer. If two patients with cancer are admitted to hospital; then probability that at least one patient will be cured, is(A) 16/25 (B) 24/25 (C) 9/25 (D) None of these

Q.82

A draws two cards one by one (replacing previous one) from a pack of cards and B throws two dice together. The probability that both cards of A are of the same suit and the sum of digits of B is 6, will be(A) 1/4 (B) 1/44 (C) 5/144 (D)7/144

Q.83

India plays two matches each with West Indies and Australia. In any match the probability to get 0,1 and 2 point by India are 0.45, 0.05 and 0.50 respectively. If the results are independent, then the probability that India gets at least 7 points is(A) 0.8750 (B) 0.0875 (C) 0.6250 (D) 0.0250

Q.84

The probability that a man will remain alive for the next 25 years is 4/5 and the probability that his wife will remain alive for the sa me 25 years is 3/4. The probability that at least one of them will be alive 25 years hence, is(A) 19/20 (B) 3/5 (C) 3/20 (D) None of these

Q.73

Two- two balls are drawn two times. If balls are not replace once it is drawn then the probability that first two balls are black and second two balls are white is-

P (A + B) = 5/6, P (AB) = 1/3, P ( B ) =1/2, then the events A and B are(A) independent (B) dependent (C) mutually exclusive (D) exhaustive Q.76

Q.77

Q.78

A card is drawn from a pack of playing cards. It is replaced in the pack and the pack is shuffled, and again a card is drawn. This process as repeated six times, then probability of getting in sequence 2 heart, 2 diamond and 2 black cards is (A) (1/4)4

(B) (1/4)5

(C) (1/4)6

(D) None of these

A man and a woman appear in an interview for two vacancies in the same post. The probability of man’s selection is 1/4 and that of the woman’s selection is 1/3. What is the probability that none of them will be selected(A) 1/2 (B) 1/12 (C) 1/4 (D) None of these If the probabilities of three persons A, B & C hitting a target are 3/5, 2/5 and 3/4 respectively. If they hit at a time then the probability that two persons hit the target is(A) 9/50 (B) 9/20 (C) 11/20 (D) 41/50

A bag contains 4 black and 3 white balls.

(A) 4/49

(B) 2/35

(C) 1/35

(D) 3/35


6

Q.85

Q.86

Question based on

Q.87

Q.88

Q.89

Q.90

A piece of equipment will function only when all the three components A, B and C are working. The probability of A failing during one year is 0.15, that of B is 0.05 and that of C is 0.10. The probability that the equipment will fail before the end of the year is(A) 0.72675 (B) 0.27325 (C) 1 (D) 0.95 A & B are two horses. The probability of A winning a race is 1/3 and that of horse B winning the same race is 1/5. The probability that none of them will win is(A) 9/15 (B) 8/15 (C) 7/15 (D) 4/15 Binomial Probability distribution

The probability that an event A happens in one trial of an experiment is 0.4. Three independent trails of the experiment are performed. The probability that the event A happens at least once is(A) 0.936 (B) 0.784 (C) 0.904 (D) None of these A pair of dice is thrown four times. If getting the same number on both dice is considered as a success, the probability of getting two success is(A) 20/216 (B) 25/216 (C) 19/216 (D) None of these A cube is thrown 6 times, then probability of getting the digits 2 and 4 exactly three times each is(A) 1/5184 (B) 5/11664 (C) 1/46656 (D) 3/11664 A box of 100 bulbs has 90 bulbs right then in a sample of 8 bulbs, the probability that at least one bulb is defective is8

 9    10 

(A) 1 – 

 1  (C)    10 

8

 9    10 

8

The odds in favour of escape of an enemy ship are 4 : 1 . The probability that at least one ship out of three ships gets destroyed is(A) 1/125 (B) 16/125 (C) 61/125 (D) 64/125

Q.92

If X is binomial variate with parameters n and P (X  r )

p, where 0 < p <1 such that

P (X  n  r )

is

independent of n and r, then p equals(A) 1/2 (B) 1/3 (C) 1/4 (D) none of these Q.93

Let X denote the number of times heads occur in n tosses of a fair coin. If P (X = 4), P (X = 5) and P (X = 6) are in AP; the value of n is(A) 7 (B) 10 (C) 12 (D) 8

Q.94

If X follows a binomial distribution with parameters n = 8 and p = 1/2, then P (| X – 4 |  2) equals(A)

Q.95

A

118

128

119

(B)

random

128

(C)

variable

117

(D) none

128

has

the

following

probability distributionX :

0

P(X): 0

1

2

2p

2p

3

4

5

6

7

3p

p2

2p2

7p2

2p

The value of p is (A) 1/10 (C) –1/10 Q.96

(B) –1 (D) none of these

A random variable X has the distributionX 2 3 4 P(X = x)

0.3

0.4

0.3

Then, variance of the distribution is (A) 0.6 (C) 0.77 Q.97

(B) 0.7 (D) 1.55

A fair die is thrown twenty times. The probability that on the tenth throw the fourth six appears is-

(B) 

 1  (D) 1–    10 

Q.91

20

8

(A) (C)

C10  56 6

20

84  56 610

(B)

120  57 10

6

(D) none of these


7

Question based on

defective screws in three Q.103 The probability of defective

Compound Events

boxes A,B,C are Q.98

A bag contains 4 tickets numbered 1, 2, 3, 4

5 6 7

2, 4, 6, 7, 8, 9. One bag is chosen and a ticket

probability that that it came from box A is(A) 16/29 (B) 1/15

is drawn. The probability that the ticket bears the number 4 is-

(C) 27/59

(A) 1/48

(B) 1/8

(C) 5/24

(D) None of these

The chance of India winning toss is 3/4. If it wins the toss, then its chance of victory is 4/5 otherwise it is only 1/2. Then chance of India’s victory is(A) 1/5

(B) 3/5

(C) 3/40

(D) 29/40

Q.100 Three groups A, B, C are competing for

positions of d the Board of Directors of a

respectively. A box

is selected at random and a screw drawn from it at random is found to be defective. Then the

and another bag contains 6 tickets numbered

Q.99

1,1,1

Question based on

(D) 42/107

Dearrangement

Q.104 Three letters are written to three different

persons and their addresses are written of three envelopes. The probability that letters are placed in right envelopes without seeing the addresses is (A) 1/27

(B) 1/6

(C) 1/9

(D) None of these

company. The probabilities of their winning

Q.105 3 letters are placed in 3 envelopes randomly.

are 0.5, 0.3, 0.2 respectively. If the group A

The probability that all letters are not in right envelopes is(A) 1/6 (B) 1/2

wins, the probability of introducing a new product

is

0.7

and

the

corresponding

probabilities probabilities for group B and C are 0.6 and

(C) 1/3

(D) 5/6

0.5 respectively. The probability that the new product will be introduced, introduced, is(A) 0.18

(B) 0.35

(C) 0.10

(D) 0.63

Q.106 There are n letters and n addressed envelopes.

The probability that all the letters are not kept in the right envelope, is(A)

Question based on

1

(B) 1 –

n!

Baye's Theorem

(C) 1 –

1 n

1 n!

(D) None of these

Q.101 A bag A contains 2 white and 3 red balls and

bag B contains 4 white and 5 red balls. One ball is drawn at random from a randomly chosen bag and is found to be red. The probability that that it was drawn from bag B was(A) 5/14

(B) 5/16

(C) 5/18

(D) 25/52

Q.102 A man is known to speak the truth 3 out of 4

times. He throws a die and reports that it is a six. The probability that it is actually a six, is(A) 3/8

(B) 1/5

(C) 3/4

(D) None of these


8

LEVEL- 2 Q.1

The probability of getting a number chosen from 1, 2, ..... 100 as cube is(A) 1/25 (B)2/25 (C) 3/25

Q.2

(D) 4/25

Q.3

Q.4

A committee consists of 9 experts taken from three colleges, A, B and C; of which 2 are from A,3 from B and 4 from C. If three experts resign, then the probability that they belong to different different institutions is(A) 1/729 (B) 2/7 (C) 1/21 (D) 1/24

Q.11

A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to that of getting 9 heads, then the probability of getting 3 heads is -

(B) 3/8 (D) 5/8

A fair coin is tossed n times. If the probability that head occurs 6 times is equal to the probability that head occurs 8 times, then n is equal to (A) 15 (C) 12

(D) 2/3

Q.10

From a group of 5 boys and 3 girls, three persons are chosen at random. The probability that there are more girls than boys is(A) 4/7 (C) 2/7

(C) 1/4

A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Two cards are drawn at random. The probability probability that at least one of them is an ace is(A) 1/5 (B) 3/16 (C) 9/20 (D) 1/9

letters i,i,i,t,t,n,n,o,a,v i,i,i,t,t,n,n,o,a,v in row result in a word “Invitation” “Invitation” is(B) 1/1128800 (D) None of these

(B) 11/36

Q.9

The probability that a random arrangement of

(A) 1/151200 (C) 1/24

(A) 13/36

(A)

(B) 14 (D) 7

(C) Q.5

5 persons A, B, C, D and E are in queue of a shop. The probability that A and E always together, is(A) 1/4 (C) 2/5

Q.6

Q.7

(B) 1/72

(C) 1/54

(D) None of these

If an integer is chosen at random from first 100 positive integers, then the probability that the chosen number is a multiple of 4 or 6, is(A) 41/100 (C) 1/10

Q.8


Two dice are thrown together. The probability of showing odd number on any

7 212

(B)

35 214

(D) None of these

There are 6 positive and 8 negative numbers. Four numbers are chosen chosen at random random and multiplied. The probability that a product is a positive number is(A) 505/1001 (B) 420/1001 (C) 15/1001 (D) 70/1001

Q.13

If all letters of the word ‘MISSISSIPPI’ are arranged then the probability that all S come together will be(A) 1/165 (B) 4/165 (C) 8/165 (D) None of these

Q.14

From the word ‘POSSESSIVE’, a letter is chosen at random. The probability of it to be S is(A) 3/10 (B) 4/10 (C) 3/6 (D) 4/6

Three dice are thrown simultaneously. What is the probability of obtaining a total of 17 or (A) 1/9

212

Q.12

(B) 2/3 (D) 3/5

18-

35

one and multiple of 3 on the other is2624522 3 / 09 ; .www.iitianspace.com .www.iitians pace.com IIT - ian’s PACE ; ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223

9

Q.15

Q.16

Q.17

Three letters are selected at random from the word ‘SLEEPER’. The probability that at least two E’s occur is(A) 12/35 (B) 13/35 (C) 14/35 (D) 15/35 The probability of occurring of two events A and B are 0.21 and 0.49 respectively and of occurring both simultaneously is 0.16, then the probability that none of the two occur is(A) 0.30 (B) 0.46 (C) 0.14 (D) None of these

Q.23

wins. If A starts, what is his chance of winning ?

Q.24

 A2 ....  An)

= P(A1) + P(A2)

+...+ P(An) (B) P(A1



A2

.... 

(C) 1/4

(D) 3/4

A man alternately tosses a coin and throws a

(C) 1/3 Q.25

(D) None of these

Two dice are thrown thrice. The probability that first throw shows 10, second 11 and third 12 will be-

An) > P(A1) + P(A2)

 A2 ....  An) 

(B) 1/3

5 or 6 in the dice is (A) 3/4 (B) 1/2

+...+ P (An) (C) P (A1

(A) 2/3

dice beginning with the coin. The probability that he gets a head in the coin before he gets a

If A1,A2,......,An are any n events, then(A) P(A1

A and B toss a coin alternatively on the understanding that the first who obtains tail

P(A1) + P(A2)

(A) 1/216

(B) 1/432

(C) 1/7776

(D) 1/648

+...+ P (An) (D) None of these Q.18

Q.19

Q.20

Q.21

Q.22

Q.26

well shuffled pack of cards and at the same time B throws a pair of dice. The probability that A gets two kings and B gets a doublet is-

Two dice are thrown, the probability of getting sum as neither 7 nor 11, is(A) 8/9 (B) 2/9 (C) 7/9 (D) 1/18 Three athlete A, B and C participate in a race competition. The probability of winning A and winning of B is twice of winning C. Then the probability that the race win by A or B, is(A) 2/3 (B) 1/2 (C) 4/5 (D) 1/3 Two dice are thrown. What is the probability that the sum of the numbers appearing on the two dice is 11, if 5 appears on the first(A) 1/36 (B) 1/6 (C) 5/6 (D) None of these A and B throw two dice, if A throws 8, the probability that B will throw a higher number is(A) 5/18 (B) 3/18 (C) 7/18 (D) 1/18 Two integers are selected selected from integers 1 to 11. If their sum is even then the probability that both are odd will be(A) 2/5 (B) 3/5 (C) 4/5 (D) None of these

A draws two cards with replacement from a

(A) 1/69 (C) 1/1014 Q.27

Two coins and a die are tossed. The probability that both coins fall heads and the die shows a 3 or 6 is(A) 1/8 (C) 1/16

Q.28



A bag contains 3 white and 3 black balls. Balls are drawn one by one with out replacing them in the bag. The probability that drawing ball will be in alternate alternate colours colours is(A) 1/10 (B) 5/21 (C) 1/2

Q.29

(D) None of these

A bag contains 5 black and 3 blue balls. Balls are drawn (without replacement) one by one. The probability of getting blue ball first time in fifth draw is(A) 1/56 (C) 4/56

(B) 3/56 (D) 5/56


10

Q.30

Q.31

Q.32

Q.33

Q.34

Q.35

Q.36

A box contains 2 black, 4 white and 3 red balls. One ball is drawn at random from the box and kept aside. From the remaining balls in the box, another ball is drawn at random and kept aside the first. This process is repeated till all the balls are drawn from the box. The probability that the balls drawn are in the sequence of 2 black, 4 white and 3 red is(A) 1/1260 (B) 1/7560 (C) 1/126 (D) None of these Cards are drawn one after another till an ace is obtained. The probability of not getting ace in first 26 cards is(A) 46/153 (B) 109/53 (C) 23/27 (D) None of these In order to get at least once a head with probability  0.9, the minimum number of times a coin need to be tossed is(A) 3 (B) 5 (C) 4 (D) None of these The chance of winning a test match by India against West Indies is 1/2. A series of 5 test matches is played and the results of all test matches are independent, then the probability of second win by India is third test match is (No match ends in draw) (A) 2/3 (B) 1/2 (C) 1/4 (D) 1/8 Odds 8 to 5 against a person who is 40 years old living till he is 70 and 4 to 3 against another person now 50 till he will be living 80. Probability that one of them will be alive next 30 years(A) 59/91 (B) 44/91 (C) 51/91 (D) 32/91 Let E and F be two independent events. The probability that both E and F happens, is 1/12 and the probability that neither E nor F happens is 1/2, then(A) P(E) = 1/3, P(F) = 1/4 (B) P(E) = 1/3, P(F) = 1/6 (C) P(E) = 1/6, P(F) = 1/2 (D) P(E) = 1/4, P(F) = 1/5 A digit is chosen out of digits 1, 2, 3, 4 and 5. A second digit is chosen from the remaining four digits, then the probability that an odd digit will be selected both the times is-

(A) 1/10 (C) 3/200 Q.37


The items produced by a firm are supposed to contain 5% defective items. The probability that a sample of 8 items will contain less than 2 defective items, is(A)

(C)

27 20

7

 19     20 

153 20

(B)

7

 1     20 

(D)

533 400 35 16

 19     20 

 1     20 

6

6

Q.38

Cards are drawn one-by-one at random from a well shuffled pack of 52 playing cards until 2 aces are obtained for the first time. The probability that 18 draws are required for this, is(A) 3/34 (B) 17/455 (C) 561/15925 (D) None of these

Q.39

A letter is taken out at random from ‘ASSISTANT’ and another is taken out from ‘STATISTICS’. The probability that they are the same letters is(A) 1/45 (B) 13/90 (C) 19/90 (D) None of these

Q.40

A six faced die is a biased one. It is thrice more likely to show an odd number than show an even number. It is thrown twice. The probability that the sum of the numbers in the two throws is even, is(A) 5/9 (B) 5/8 (C) 1/2 (D) None of these

Q.41

Two dice are tossed 6 times. Then the probability that sum 7 will show an exactly four of the tosses is(A) 225/18442 (B) 116/20003 (C) 125/15552 (D) None of these

Q.42

The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. What is the probability that out of 5 such bulbs none will fuse after 150 days of use5

 19  (A) 1–    20   3  (C)    4 

5

 19  (B)    20 

5

 1  (D) 90    4 

5


11

LEVEL- 3 Q.1

The probability that the birth days of six

If an even face has turned up, then the

persons will fall in exactly two calendar months

probability that it is face 2 or face 4, is

is-

(A) 0.25

(B) 0.42

(C) 0.75

(D) 0.9

12

(A) 1/6

(B) C2× 2

12

(C) C2 ×

6

1

12 6

(D)

26 12 6

Q.6

341

For three events A, B and C, P (exactly one of the events A or B occurs) = P (exactly one of

125

the events B or C occurs) = P (exactly one of Q.2

1  4 p 1  p 1  2 p , , are probabilities of 4 4 2

If

the events C or A occurs)= p and P (all the three events occur simultaneously) = p 2 ,where 0

three mutually exclusive events, then(A) (C)

1 3

1

 p 

1

2

 p 

6

(B)

1 2

 p 

< p < 1/2. Then the probability of atleast one of

2

the three events A, B and C occurring is-

3

(A)

1

(D) none of these

2

(C) Q.3

A

bag

contains

50

tickets

numbered

1, 2, 3, ..., 50 of which five are drawn at

Q.7

3 p  2 p 2 2 3 p  p 2 2

(B) (D)

p  3 p 2 2 3 p  2 p 2 4

If A and B are two events such that P(A) = 1/3,

random and arranged in ascending order of

 B   = A  

P(B) = 1/4 and P(A  B) = 1/5, then P 

magnitude (x1 < x2 < x3 < x4 < x5). The probability

that

x3 = 30 is20

(A)

C2

C5

C2 

20

Q.4

Q.8

50

C5

C2

(D) none of these

C5

There are 9999 tickets bearing numbers 0001, these tickets at random, the probability that the number on the ticket will consist of all different digits, is-

A man takes a step forward with probability probability that at the end of eleven steps he is one step away from the starting point is11

5

11

(A) C6 (0.24) 6

(C) C6 (0.6) (0.4)

6

(B) C6 (0.4) (0.6) 5

Q.9

5

2

3

4

5

(B) 5000 / 9999

(C) 5030 / 9999

(D) none of these

All possible 6 letter words each containing all and are placed in a dictionary order.

probabilities for various faces to turn up are: 1

(A) 5040 / 9999

the letters of the word EMHORT are formed

(D) none of these.

A biased die is tossed and the respective

Face

(D) None

0002,...., 9999. If one ticket is selected from

29

0.4 and backward with probability 0.6.The

Q.5

(C) 23/40 C2

(B)

50

11

(B) 37/45

29

50

(C)

(A) 37/40

6

Probability : 0.1 0.24 0.19 0.18 0.15 0.14

10

consecutive words are now drawn at random. The probability that the word 'MOTHER' will be among the drawn words, is (A) 1/72

(B) 10/711

(C) 10/719

(D) none


12

Q.10

A number is chosen at random from the numbers 10 to 99. By seeing the number a

Q. 12

Statement-1 : The probability of being at

least one white ball selected from two balls

man will laugh if product of digits is 12. If he chooses three numbers with replacement then

drawn

the probability that he will laugh atleast once

containing 7 black and 4 white balls is

is3

 3  (A) 1     5 

 43    45 

 4    25 

(C) 1  

 43    45 

from

the

bag 34 55

.

Statement-2 : Sample space = 11C2 = 55 No.

3

(B) 

3

simultaneously

of favourable cases = 4C1 × 7C1 + 4C2 × 7C0 = 34

3

(D) 1  

34

required probability =

Statement type Questions

Q. 13

Each of the questions given below consists of Statement -1 and Statement-2. Use the following

If P(A) =

3 5

, P(B) =

Statement-1 :

key to choose the appropriate answer. (A) If both Statement- 1 Statement- 2 are true, and

Statement-2 :

55

2 3

then

A   P    5  B  2

4 15

9 10

 P(A  B) 

. 3 5

.

Statement-2 is the correct explanation of Statement- 1. (B) If Statement-1 and Statement-2 are true but Statement-2 is not the correct explanation of Statement- 1 (C) If Statement-1 is true but Statement- 2 is false (D) If Statement-1 is false but Statement- 2 is true. Q. 11

Statement-1 : Given Ei, i = 1,2,.........n are n

independent events, such that P( E i ) = 1



i

 n,

i 1 i

,

then the probability that none of

the n events occur is

n n 1

.

Statement-2 : Probability of occurrence of all

independent events together is equal to the product of the probabilities of these events.


13


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

D

B

B

B

B

B

B

B

B

A

C

A

B

A

D

B

D

A

B

B

Q.No.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Ans.

B

C

B

B

C

A

C

B

C

A

A

C

B

B

D

C

A

D

C

C

Q.No.

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

Ans.

B

B

A

D

D

B

C

B

A

D

A

A

D

A

C

A

D

D

D

B

Q.No.

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

Ans.

A

A

C

C

C

B

B

A

B

D

C

B

A

A

A

B

A

B

D

B

Q.No.

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

Ans.

B

C

B

A

B

C

B

B

B

A

C

A

A

B

A

A

C

C

D

D

Q.No. Ans.

101 D

102 A

103 D

104 B

105 D

106 B

LEVEL- 2 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

A

A

C

B

C

C

B

B

C

B

A

A

B

B

B

B

C

C

C

B

Q.No.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Ans.

A

B

A

A

C

C

B

A

B

A

D

C

C

B

A

B

A

C

C

B

Q.No.

41

42

Ans.

C

B

LEVEL- 3 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

Ans.

D

D

C

A

C

A

A

A

B

D

D

A

A


14

IIT – ian’s

P A C E


LEVEL- 1 Question based on

Q.1

Q.6

Kinds of vectors 

If a is a constant vector then -

(C) direction towards the origin (D) indeterminate direction



(A) the direction of a is constant 

(B) the magnitude of a is constant

The zero vector has(A) no direction (B) direction towards a particular point



(C) both direction and magnitude of a is constant (D) None of these Q.2



Question based on



If a = b , then

Q.7

(A) both have equal magnitude and collinear

(D) they have unequal magnitude but like vectors

Q.4

Q.5

(B)

ˆ (C) î  jˆ  k

(D)

is

a

pentagon,

then

(A) 3 AD

(B) 3 AC

(C) 3 BE

(D) 3 CE

 If a  2î  5 jˆ and b  2î  jˆ , then unit vector





in the direction of a  b is-

Q.9

(A) î  jˆ

(B)

2 ( î  jˆ )

(C) ( î  jˆ )/ 2

(D) ( î  jˆ )/ 2

If a and b are two unit vectors then vector ( a + b )

Which of the following is unit vectors(A) î  jˆ

ABCDE



Q.8

Two vectors will be equal when(A) they have same magnitude (B) they have same direction (C) they meet at a point (D) their magnitude and direction is same

If

AB + AE + BC + DC + ED + AC equals-

(B) both have equal magnitude and like vectors (C) both have equal magnitude

Q.3

Addition & subtraction of vectors

(A) is a unit vector (B) is not a unit vector (C) can be a unit vector or not

ˆ) (î  jˆ  k 2 ˆ) (î  jˆ  k

(D) is a unit vector when both a and b are

3

parallel



Unit vector in the direction of a is represented Q.10

by 

(A) 1. a

(B)

a |a |

 

(C) a | a |

(D)

a î

If a and b represent vectors of two adjacent sides AB and BC of a regular hexagon ABCDEF, then AE equals(A) a + b

(B) a – b

(C) 2 b

(D) 2 b – a


1

Q.11

If in a parallelogram PQRS, sides PQ and QR are

represented

by

vector

a and

Q.16

b

in order, then-

respectively then the side represented by a + b















(C) P  Q  R  0

(B) RS

(C) QS



(A) P  Q  R  0

is -

(A) PR

If three forces P, Q, R acting on a particle are represented by three sides of a triangle taken

(D) PQ

DA is

Q.13

(C)

(C) 2 AD

(D) 2 AB

Q.18

the origin O, then OA  OB  OC  OD =

Q.14





1 3

ˆ) ( 2î  jˆ  2k

1 3

(B)

1 5

ˆ) ( 2î  jˆ  2k

ˆ ) (D) None of these ( 2î  jˆ  2k



ˆ) (3î  6 jˆ  2k

(D) 0

7

ˆ (C)  î  2 jˆ  8k

ˆ (B) î  2 jˆ  8k

(D)

ˆ) ( î  2 jˆ  8k

If vector a , b represent two consecutive sides of

69

regular hexagon then the vectors

representing sequence are-

remaining four 

sides

in

Question based on

Vectors in terms of position vectors of end points



















 













to B is î  jˆ and that of B with respect to A













is î  jˆ . The position vector of C with respect





(A) a  b , a  b , a  b , a  b (B) a  b , a , b  a , b

Q.19

(C) a  b , – a , – b , a  b

(D) b  a , – a , – b , a  b Q.15



ˆ and î  2 jˆ  3k ˆ are two adjacent If 2î  4 jˆ  5k

(A)







is-

(B) AB  BC

(C) 2( AC  BD )



(D) P  Q  R  0

sides of a parallelogram, then the unit vector along the diagonal determined by these sides

If ABCD is a rhombus whose diagonals cut at

(A) AB  AC



 ˆ and b   î  jˆ  k ˆ , then unit If a  3î  2 jˆ  k

(A)

of the forces represented by BA , BC , CD and

(B) 2 AC



vector parallel to a  b is-

If ABCD is a quadrilateral, then the resultant

(A) 2 BA





Q.17



Q.12



(B) P  Q  R  0

to A is-

In the adjoining diagram vector

(A) 2 î

u – v is

represented by the directed line segmentD

Q.20

C





u

B

(A) BD

(B) AC

(C) DB

(D) CA

(B) – 2 î (C) 2 jˆ

(D) – 2 jˆ

If A, B, C are three points such that 2 AC =3 CB , then 2 OA + 3 OB equals-

v

A

The position vector of a point C with respect

Q.21

(A) 5 OC

(B) OC

(C) – OC

(D) None of these

If the position vector of the point A and B with respect to point O are respectively î  2 jˆ  3k ˆ and

ˆ then  2î  3 jˆ  4k

BA equals-

ˆ (A) 3î  jˆ  k

ˆ (B) 3î  jˆ  k

ˆ (C)  3î  jˆ  k

(D) None of these


2

Question based on

Q.29

Distance between two points

If 1 and 2 are lengths of the vectors î  2 jˆ  k ˆ and î  5 jˆ respectively, then-

Q.22

If the end points of AB are (3, –7) and

(A) 1= 2

(B) 1= – 2

(– 1, – 4), then magnitude of AB is(A) 2 (B) 3 (C) 4 (D) 5

(C) 1< 2

(D) 1> 2 

Q.23

If



ˆ and  î  2 jˆ  3k

a





ˆ then the b  2î  jˆ  3k

Q.30

equal magnitudes, then value of  is(A) 1 (B) 0 (C) 2 (D) 0 or 1



value of | a  b | is (A)

(B) 2 6

6

(C) 3 6 Q.24

(D) 4 6

Question based on

ˆ , î  3 jˆ  5k ˆ & 2î  jˆ  4k ˆ The vectors 3î  2 jˆ  k

form-

Q.31



(A)

ˆ and î  2 jˆ  k ˆ represents If vectors 2î  3 jˆ  2k

(A)

35 , 35

(B)

(C)

25 ,

(D) None of these

11

35 ,

11

(C) Q.32

(A)

1 6

(C) 1

6

ˆ ) is ( î  jˆ  2k

(B)

1 6

(B)

2



a  b

(D) None of these

2 



of the point dividing PQ in 2 : 5 is 

(A)



p  q 25





(C) Q.33

(B)



2 p  5q 25

25 

(D)



5 p  2q 

p  q 25

The position vector of the vertices of triangle ˆ then the position vector ABC are î , jˆ and k

of its orthocentre isˆ (A) î  jˆ  k

(D) None of these

If A = (1, 0, 3), B = (3, 1, 5), then 3 kg. wt.

The position vector of two points P and Q are

(C) Q.28

2





a  b

respectively p and q then the position vector

then this triangle is(A) right angled (B) equilateral (C) isosceles (D) None of these 1



b  a 

If position vectors of the vertices of a triangle ˆ , 5î  6 jˆ  4k ˆ and 6î  4 jˆ  5k ˆ are 4î  5 jˆ  6k

The length of vector



then the position vector of middle point of AB is -

the adjacent sides of any parallelogram then the length of diagonals of parallelogram are-

Q.27

If the position vector of points A and B with 

(D) None of these

Q.26

Position vector of dividing point

respect to point P are respectively a and b

(A) an equilateral triangle (B) an isosceles triangle (C) a right angle triangle Q.25

 ˆ and b  2î  jˆ   k ˆ are of If a  î   jˆ  2k

1 ˆ ˆ ˆ ( i  j  k ) 3

ˆ) (B) 2 ( î  jˆ  k

(D)

1 3

ˆ) ( î  jˆ  k

along AB is represented by the vectorˆ ˆ (A) 2î  2 jˆ  k (B) 2î  jˆ  2k ˆ (C) î  2 jˆ  2k

ˆ (D) î  jˆ  k


3

Q.34

If D, E, F are mid points of sides BC, CA and

Q.40

AB respectively of a triangle ABC, and î  jˆ , ˆ  î are p. v. of points A, B and C ˆ , k jˆ  k

respectively, then p. v. of centroid of

position vector of its fourth vertex isˆ) ˆ) (A) 6 ( î  jˆ  k (B) 7 ( î  jˆ  k

DEF

isî  jˆ  k ˆ

(A)

3

ˆ) (C) 2 ( î  jˆ  k

Q.35

ˆ (C) 2 jˆ  4 k

ˆ (B) î  jˆ  k

(D)

Q.41

ˆ) 2(î  jˆ  k

Q.36

If A, B, C, D be any four points and E and F be the middle points of AC and BD respectively,

then

AB  CB  CD  AD

is

equal to-

Q.37

(A) 3 EF

(B) 3 FE

(C) 4 EF

(D) 4 FE

If G is centroid of

ABC









(A) 1/2 ( a  b ) (C) 2/3 ( a  b )









Q.44

(C) 3î  2 jˆ

(D) None of these

The centroid of the triangle whose vertices ˆ isare î  2 jˆ , 2î  jˆ , î  jˆ  k



OE then x is equal to

(C)







(C) ( b  c  2a ) /2

3

(D) None of these

Q.45

4

(D) None of these

The position vector of the points A and B are 

a and b respectively. If P divides AB is the

ratio 3 : 1 and Q is the mid point of AP, then the position vector of Q is-

then AD equals(A) ( b  c  a ) /2

ˆ 4î  4 jˆ  k

ˆ (B) î  jˆ  k

î  jˆ  k ˆ







(B)

If p. v. of vertices of a tetrahedron are î  jˆ  k ˆ ,  î  jˆ  k ˆ ,  î  jˆ  k ˆ and î  jˆ  k ˆ,





2



If a , b , c be position vectors of A,B,C respectively and D is the middle point of BC, 

ˆ 4î  4 jˆ  k

(A) origin

If E is the intersection point of diagonals of parallelogram ABCD and



(B)  2î  3 jˆ

then its centre is-

(D) 1/6 ( a  b )

(where O represents origin)(A) 3 (B) 4 (C) 5 (D) 6 Q.39

(A) î  5 jˆ



and AB = a ,

(B) 1/3 ( a  b )

 OC OD= x



The orthocentre of the triangle whose vertices are 3î  2 jˆ , – 2 î + 3 jˆ and î + 5 jˆ is-

(C)

AC = b then AG equals-

OA OB



(D) b – 2 a

ˆ (A) 4î  4 jˆ  k



Q.38





Q.43





(B) a – 3 b

(C) 2 a – b Q.42

AD + BE + CF is equal to-

(D) 2 CA







BC, CA and AB of a triangle ABC, then

(C) 2 AB



Two points A and P are respectively a  2 b

(A) b

If D, E and F are midpoints of sides

(B) 2 BC

ˆ (D) 6î  8 jˆ  10 k

and a and P divides AB in the ratio 2: 3 then p.v. of B is-

3

(A) 0

If the position vectors of three consecutive vertices of any parallelogram are respectively î  jˆ  k ˆ , î  3 jˆ  5k ˆ , 7î  9 jˆ  11k ˆ then the













(B) ( a  c  2a ) /2

(A)

a  b 2 

(C)

(B)



5a  3 b 8





a  b 2





(D)

5a  3 b 8

(D) ( a  b  2c ) /2


4

Question based on

Q.52

Collinearity of three points

If

ˆ ), A(  î  3 jˆ  2k

ˆ) B(  4î  2 jˆ  2k

and

ˆ ) are collinear then the value of C ( 5î  p jˆ  qk

Q.46

If vectors (x – 2) î  jˆ and (x + 1) î + 2 jˆ are

p and q respectively(A) 5, 10 (B) 10, 5

collinear, then the value of x is(A) 3 Q.47

(B) 4

ˆ, points î  2k

If

(C) 5

(D) 0

ˆ and jˆ  k

(C) – 5, 10

î  µ jˆ are

Q.53

ˆ , î  jˆ  k ˆ & î  4 jˆ  2k ˆ , then A,B,C are 3î  2 jˆ  4k

= 2,  = 1 (B)  = 2,  = –1 (C)  = –1,  = 2 (D)  = –1,  = –2

are(A) vertices of a right angled triangle (B) vertices of an isosceles triangle (C) vertices of an equilateral triangle (D) collinear

If three collinear points A,B,C are such that AB = BC and the position vector of points A and B with respect to origin O are respectively a and 



(A)

2



(B)



(C) 2 b – a 



(A) 1 : 2 (C) 2 : 3

a  b 2

(B) 2 : 1 (D) 3 : 2

(D) None of these 

Question based on



Relation between two parallel vectors

If a , b and (3a  2 b) are position vectors of three points, then points are(A) collinear

Q.55

(A) 1









(B) –1

(C) 2

(D) –2





(A) like parallel (C) non-collinear

when(B) x + y + z  0 (C) x + y + z may or may not be zero (D) None of these



two vectors then a , b are-

xa  y b  zc  0

(A) x + y + z = 0

 ˆ and b  – 8 î  4 jˆ  6k ˆ are If a  4 î  2 jˆ  3k 

Three points A, B, C with position vectors 

ˆ , then  equals 2 î  3 jˆ  k



Q.56

a , b , c are collinear if

ˆ is parallel to sum of the vectors If î  2 jˆ  3k ˆ and 3 î   jˆ  2k

(B) vertices of a right angled triangle (C) vertices of an equilateral triangle (D) None of these Q.50

ˆ , 5î  2k ˆ& respectively î  2 jˆ  8k

are

ˆ then B, divides AC in the ratio11î  3 jˆ  7k





If A, B, C are collinear and their position vector

b then the position

vector of point C isa  b

Q.54





Q.49

If the position vectors of the points A, B, C

collinear, then(A)

Q.48

(D) 5, –10

Q.57

(B) unlike parallel (D) perpendicular

If position vectors of A, B, C, D are respectively î  2 jˆ  3k ˆ ,  5î  4 jˆ  2k ˆ and

ˆ, 2 î  3 jˆ  5k î  10 jˆ  10k ˆ,

thenQ.51

ˆ and  2î  p jˆ  qk ˆ are If the vectors 3î  2 jˆ  5k

collinear, then (p, q) is(A) (4/3, – 10/3) (B) (10, 4/3) (C) (–4/3, 10/3) (D) (4/3, 10/3)

(A) AB || CD (B) DC || AD (C) A, B, C are collinear (D) B, C, D are collinear


5

Q.58





ˆ and b   î  jˆ  k ˆ then the If a  3î  2 jˆ  k 

Q.64



three

unit vector parallel to a  b , is(A)

1 3

1

(C) Q.59

ˆ) ( 2î  jˆ  2k

(B)

1

points

whose







position



If A = (x + 1) a + (2y – 3) b and B = 5 a –

Question based on

(B) 3



Q.65

a , b are non zero non-collinear vectors then-

respectively 2î  jˆ , î  3 jˆ , 3î  2 jˆ and î   jˆ . If

Scalar or Dot product of two vectors

Q.61

Q.66

(B) 0 <  < /2 (D) 0    /2

(B) – 6





If the moduli of vectors a and b are 1 and 2  

respectively and a . b = 1, then the angle

(C) 8



between them is-

(D) – 8

Coplanar and non-coplanar vectors

(A)  = /6

(B)  = /3

(C)  = /2

(D)  = 2/3 



If



















Q.67

p =2 a –3 b , q = a –2 b + c , r =–3 a + b + 





(A) 17/3 (C) 4/3



vectors then the vectors –2 a +3 b – c is equal to (A)





7q  r 5





Q.68



(B) p – 4 q 



(C) 2 p – 3 q + r













and 2a  c then(A) PQ || RS (C) PQ 



 

Q.69

(D) None of these

Q.70

 





(A) 5

(B) 0

(C) 11

(D) None of these

.

ˆ and 2 î + x jˆ + k ˆ are If vectors 3 î + 2 jˆ + 8 k







If vector a  b is perpendicular to b and 







(A) | a | = 

2 | b |

Q.71















(B) | a |= 2| b |

(C) | b |= 2 | a |

(A) x + y + z + u = 0

(D) All correct





(2a  3 b)

then







vectors and x a + y b + z c + u d = 0 , then(B) x + y = z + u (C) x + z = y + u

them,

2 b + a is perpendicular to a , then-

If a , b , c , d are four linearly independent 

between

perpendicular then x is equal to(A) 7 (B) –7 (C) 5 (D) –4

(B) PQ = RS

RS





respectively 2a  4 c , 5a  3 3 b  4c ,  2 3 b  c 






If the position vectors of four points P, Q, R, 



(4a  b ) equals-

(D) 4 p – 2 r

S



If a and b are unit vectors and 60º is the angle





 ˆ , b  3î  4 jˆ  2k ˆ & c  î  2 jˆ  2k ˆ If a  2î  jˆ  k

then the projection of a  b on c is-

 





Q.63



 0 (A) 0     (C)  /2    

2 c , a , b , c being non zero, non coplanar

Q.62





If the angle between a and b is  then for 

AB || CD , then value of  is-

Question based on

(D) 1

The p. v. of four points A, B, C, D are

(A) 6



a . b

(A) x = 13/2, y = 0 (B) x = 0, y = 3 (C) x = –13/2, y = 0 (D) None of these Q.60

are



(C) 0



2 b are two vectors such that 2 A = 3 B & 

vector



are collinear, if m equals(A) 2







a – 2 b + 3 c , 2 a + m b – 4 c and –7 b + 10 c

ˆ ) (D) None of these ( 2î  jˆ  2k

3





ˆ) ( 2î  jˆ  2k

5





If a , b , c are non coplanar vectors then the

(D) | a | = | b | 







If | a | = | b |, then ( a + b ). ( a – b ) is(A) positive

(B) negative

(C) zero

(D) None of these


6

Q.72





If a and b are vectors of equal magnitude 2 and

 be

Q.78

the angle between them, then 

is -



magnitude of ( a + b ) will be 2 if -

(A) 2

(A)  = /3

(B)  = /4

(C)  = /2

(D)  = 2/3

Q.79



Q.73

 ˆ, ˆ and b  4î  2 jˆ  4k If a  î  3 jˆ  2k







Q.74

 9    91 

Q.75

Q.80

(C) 5

(D) –5

ˆ, If the points P, Q, R, S are respectively î  k

(A) 4/3

 9    11 

(B) cos –1 

 1  (D) cos –1    9 

Q.81

ˆ, 3î  2 jˆ  k

and

then

of four points A,B,C and D

(B) – 4/3 (C) 3/4

(D) –3/4 



If angle between vectors a and b is 120º and 







| a | = 3, | b | = 4, then length of 4 a – 3 b is-

Q.82

(A) 12 3

(B) 2 3

(C) 432




Vectors

a + b and





a – b are perpendicular,

respectively, then the angle between AB and

when-

CD is-

(A) a = 0

(B) a + b = 0 or a – b = 0

(C) b = 0

(D) None of these







(A) /4

(B) /2

(C) 

(D) None of these























ˆ moves a particle If the force F  î  2 jˆ  3k

(A) a and b are perpendicular

ˆ to 2î  jˆ  k ˆ , then the work done from î  jˆ  k

(B) a , b are parallel to each other

is(A) 6



If | a + b | = | a – b |, then 



Q.77

(B) –4

projection of PQ on RS is-

Q.83 Q.76

 

ˆ  î  2 jˆ , 2î  3k

ˆ , 2î  5 jˆ , 3î  2 jˆ  3k ˆ and î  6 jˆ  k ˆ If î  jˆ  k

be p.v.





a and b are vectors of equal magnitude and

(A) 4

ˆ is 12î  4 jˆ  3k

(C) cos –1 

(D) 3

5

then | a | equals-

ˆ and Angle between the vectors 2î  6 jˆ  3k

 1    10 

(C)





(B) –14 (D) None of these

(A) cos –1 

(B) 1

angle between them is 120º. If a . b = –8,

then (2a  b).(a  2 b) equals(A) 14 (C) 0

ˆ on x- axis The projection of vector î  2 jˆ  2k









0   b  0

(C) a (B) 5

(C) 4

ˆ Two forces P  2î  5 jˆ  6k

(D)

(D) 3 and

Q.84









|2 a + b | is-

forces displace the particle from point

(A)

ˆ ) to point B ( 6î  jˆ  3k ˆ ) . The A( 4î  3 jˆ  2k

21

(C) 21

work done by these forces is-

(C) 10 units (D) –10 units



120º. If | a | = 2, | b | = 1 then the value of

ˆ are acting on a particle. These Q  î  2 jˆ  k

(A) 15 units (B) –15 units



If the angle between two vectors a and b is

Q.85

(B) 13 (D) 13

For any vector 

ˆ , ( r .î ) î  ( r . jˆ) jˆ  ( r .k ˆ ) k ˆ equalsr  xî  y jˆ  zk 



(A) 0

(B) 2 r

(C) r

(D) 3 r






7

Q.86





If a and b be two non- zero vectors, then 







( a + b ) . ( a – b ) equals







(B) | a – b | 2



 2

(C) | a + b |2

(A) –15 unit (C) 0

2

(D) | a | – | b |

(C) 2

(D)

(B) /4

(C) /2

(A) sin –1

2

(C) cos –1

5

2

(A) 7

7





Q.97

Q.91

(B) 30º

(C) 60º

(C) 9

(D) 10 



If angle between two unit vectors a and b is 



(A) 2 | a – b |

(D) 90º (C)





ˆ on a particle it   jˆ  k

 then sin (/2) is equal to-

If angle between vectors a and b is 30º, then 



(B) 8



angle between vectors a  b and a  b is(A) 0º

ˆ,  iˆ  jˆ  k

The work done in moving an object along the

ˆ isF = 2î  jˆ  k

 ˆ and b  3î  jˆ  2k ˆ , then the If a  î  2 jˆ  3k



F3

F1

(B) 2 unit (D) None of these

7





ˆ,  î  2 jˆ  k

forces

ˆ , if the applied force is vector 3î  2 jˆ  5k

2

(D) cos –1

three

is(A) 1 unit (C) 0 unit Q.96

5

acting

displaces it from point A(4, –3, –2) to point B (6, 1, – 3) then the work done by the force

(D) /3

(B) sin –1

by



ˆ and The angle between the vectors 3î  jˆ  2k

2

If F2

3

ˆ is2î  2 jˆ  4k

Q.90



Q.95

ˆ ) isThe angle between ( î  jˆ ) and ( î  k

(A) 0 Q.89

(B) 16 unit (D) None of these

If sum of two unit vectors is again a unit vector, then modulus of their difference is(A) 1 (B) 2

Q.88

ˆ acting on a particle A force F  î  3 jˆ  5k

displaces it from point A(4, – 3, –2) to B (6,1, – 3) then the work done by the force is-





(A) | a + b |

Q.87



Q.94

1

(B)









| a + b |

2

1   | a – b | 2

(D) 2 ( a + b )

angle between 3 a and 4 b will be(A) 60º Q.92

(B) 30º

(C) 0º

(D) 90º

The unit vector which bisect the angle

ˆ (A) k

(C)

(B)

2

(D) None of these

3



(C)

10 3 ˆ) 18 (3 jˆ  4k 13





















be

the

(A) | a  b |  | a | | b |

Q.100 If

vector of a along b is(A)

(D) 4 6

70

















(B) | a  b |  | a || b |

(C) | a  b |  | a || b | (D) | a  b |  | a || b |



ˆ) 18 (3 jˆ  4k

(B) 2 6 (C)

6

If a and b are two vectors, then-

 ˆ , then component If a  4î  6 jˆ and b  3 jˆ  4k 



 ˆ , b  î  3 jˆ  3k ˆ then | a  b | is If a  2î  jˆ  3k

(A)

(î  jˆ)

Q.99

ˆ) (î  jˆ  k

Vector or cross product of two vectors 

Q.98

between î and jˆ is-

Q.93

Question based on

ˆ) 18 (3 jˆ  4k

(B)

ˆ (D) 3î  4k

25



angle

between

vectors

î  2 jˆ  3k ˆ and 3î  2 jˆ  k ˆ , then the value of

sin  is(A)

6/ 7

(C) 1/7

(B)

2 6 7

(D) None of these


8





 



Q.101 If | a  b | = | a . b | then angle between a and 

b is -

Q.109 If adjacent sides of a triangle are represented 

 by vectors a  3î  4 jˆ and b  5î  7 jˆ , then

(A) 0º

(B) 90º

vector area is -

(C) 60º

(D) 45º

(A) 13/2 (C) 41


Q.102 The unit vector perpendicular to vectors ˆ , 2iˆ  jˆ  k ˆ and 3î  jˆ  2k ˆ are Q.110 If î  jˆ  2k

î  jˆ and jˆ  k ˆ is1

(A)

(B)

ˆ) ( î  jˆ  k

(D) None of these

3 1

(C)

3

1

ˆ) ( î  jˆ  k

 



ˆ) ( î  jˆ  k

3



Q.103 If | a . b | = 3 and | a  b | = 4, then the angle

(A) 26

(B) 13

(C) 2 13

(D)

13

ˆ and Q.111 Two constant forces P = 2î  5 jˆ  6k ˆ are acting on a point A (4,–3,– Q=  î  2 jˆ  k





position vectors of vertices of a triangle, then its area is-

between a and b is(A) cos –1 3/4

(B) cos –1 3/5

2). The moment of their resultant about origin

(C) sin –1 4/5

(D) /4

(0, 0, 0) is-





Q.104 If |( a  b )|2 +

  ( a . b )2 =



144 and | a | = 4, then



ˆ (A) 21î  22 jˆ  9k

ˆ) (B)  (21î  22 jˆ  9k

ˆ (C) 21î  22 jˆ  9k

(D) None of these

| b | is equal to (A) 3 (C) 12



(B) 8 (D) 16

Q.112 If a



























Q.107 In a parallelogram PQRS, PQ = a  b and 

PR = a  b , then its vector area is







(C) 2( a × b )



(D) 2( b × a )

Q.108 If the diagonals of a parallelogram are

respectively



a

(D) –74

14

(C) 2 6







ˆ,  î  jˆ  2k

(C) b

(D) Neither a nor b



(D) 38





 





 

(A) | a  b |  | a . b | (C) | a  b |  | a . b | 

a

ˆ,  î  2 jˆ  3k





a and b

between /2 and 3/4, then -

Q.115 If









 





 

lies

(B) | a  b |  | a . b | (D) | a  b |  | a . b | 

ˆ b  î  2 jˆ  k

and



c  3î  jˆ , then unit vector along the direction

of the resultant is-



ˆ, b  iˆ  3 jˆ  4k

(B) 2 14



(B) a

ˆ (A) 3î  5 jˆ  4k

then the area of parallelogram is(A)



(A) both a and b





(B) a × b



(C) 74

Q.114 If angle between vector 

(A) | a |2 – | b |2

(B) 64



(B) a  b

(C) a = 0 or b = 0 (D) None of these





(A) 60 



(A) a || b



Q.113 Vector a  ( b  a ) is perpendicular to-

Q.106 If for vectors a & b , a × b = 0 and a . b = 0,

then-





(B) 1 (D) 2 



then ( a  b ). ( a  c ) equals-

ˆ  î )] equalsˆ ) × ( k Q.105 ( î  jˆ ). [( jˆ  k

(A) 0 (C) –1



ˆ, ˆ , b   î  2 jˆ  4k ˆ & c  î  jˆ  k  2 î  3 jˆ  k

(C)

ˆ 3 î  5 jˆ  4 k 5 2

(B)

ˆ 3 î  5 jˆ  4 k 50

(D) None of these


9

Q.116 If points P(1, –1, 2), Q(2, 0, –1) and

R (0, 2, 1) be any three points, then unit vector perpendicular to the plane PQR isˆ (A) 2î  jˆ  k

(C)

(B)

ˆ 3 î  2 jˆ  k

ˆ 2 î  jˆ  k

then the unit vector perpendicular to both 





(C)



(D) None of these

vectors

(A) 5 6

(B) 6 2

(C) 6 5

(D) 180

(C)

| a × b | equals(B) 8 (D) None of these

(C)



ˆ  k ˆ  î = 0 (B) î  jˆ  jˆ  k

ˆ  k ˆ  î = 3 (D) î  jˆ  jˆ  k 





Q.120 If a . b = a . c and a  b = a  c , a 





(A) b = 0 

(C) b

















0 , then-









  





Q.123

(B) 













(C) a  b

(B) 2 13

13 1 2

3

(D) None of these

its area is (A) 1 unit

(B) 2 unit

(C)

(D)

2 unit

3 2

unit

(B) isosceles (D) None of these

position vector is 2î  jˆ . The moment of F

6

(C) 3

(D)



(2a  3 b)  (5a  7 b ) is equal to-

(A) a  b

(D) None of these

ˆ acts at a point A whose Q.130 A force F  2î  jˆ  k

(D) 0

ˆ )  (î  jˆ  k ˆ ) | is equal to| ( 2î  k

(A) 6

21

2

(B) [ a b c ]

(C) a  b  c Q.122

1

is(A) right angled (C) equilateral



  



(B) 2 21

21

ˆ ,  4î  4 jˆ  6k ˆ be  î  jˆ  8k p.v. of A, B, and C respectively, then  ABC

a  ( b  c )  b  ( c  a )  c  (a  b) equals-

(A) a  b  c



ˆ, Q.129 If î  2 jˆ  3k

(D) None of these

c





Q.121 For any three vectors a , b , c , 





(B) b = c





ˆ then Q.128 If the vertices of any triangle are î , jˆ , k

ˆ . k ˆ =3 (C) î . î + jˆ. jˆ + k





(A)

ˆ . k ˆ = 0 (A) î . î + jˆ. jˆ + k





Q.127 If A (1, –1, 2), B(2, 1, –1), C(3, –1, 2) be any three points, then area of ABC is-

Q.119 Which one of the following is correct-

 

ˆ  3î  2 jˆ  k

ˆ , c  2 jˆ  k ˆ , b  î  k ˆ then  2î  3 jˆ  k

(A)



 

î  2 jˆ  3k ˆ and

represent adjacent sides of a parallelogram, then its area is-



 

(A) 16 (C) 32



a  b and b  c will be-

Q.118 If | a | = 10, | b | = 2 and a . b = 12, then 



the area of the parallelogram with diagonals

ˆ )/ 3 (D) ( î  jˆ  k





(C) 0

Q.126 If a

(B) jˆ

2



(B) 2 | a |2| b |2



ˆ  î k

 





a  b and b  c is-

(A) î





(A) | a |2 | b |2

Q.125 If

  ˆ , b  î  2 jˆ  k ˆ & c  î  2 jˆ  k ˆ, Q.117 Let a  î  jˆk 





a , b {| a  b |2 + ( a . b )2} + | a |2 | b |2 equals-

6

(D) None of these

14

Q.124 For any two vectors



3

about origin isˆ (A) î  2 jˆ  4k

ˆ (B) î  2 jˆ  4k

ˆ (C) î  2 jˆ  4k

ˆ (D) î  2 jˆ  4k



(B) b  a 



(D) 7 a  10 b


10

ˆ passing through A whose Q.131 A force F  3î  k ˆ , then the moment position vector is 2î  jˆ  3k

of the force about point P whose position ˆ isvector is, î  2 jˆ  k

Question based on

ˆ (A)  3î  11 jˆ  9k

ˆ (B) 2î  10 jˆ  8k

ˆ (C) î  3 jˆ  7k

ˆ (D) 4î  3 jˆ  6k

ˆ  î  jˆ  k

are(A) coplanar (B) non- coplanar (C) two are perpendicular to each other (D) none of these Q.139 If the volume of the tetrahedron with edges î  jˆ  k ˆ , î  a jˆ  k ˆ and î  2 jˆ  k ˆ is 6 cubic

Scalar Triple product

ˆ Q.132 If [3î 5 jˆ  3k

units, then a is(A) 1 (B) –1

ˆ ] = 5, then the value of î  k





of

a

(B) 15 (D) 40

ˆ )]. jˆ equals[( î  jˆ)  (î  k

(A) 1 (C) 0

(B) 7

Q.135 If a , b , c are mutually perpendicular unit   

vectors, then [ a b c ] equals(A) 0

(B) ±1

(C) 3

(D) 1

 



Q.141 If a , b , c are any three coplanar unit vectors

then 











(A) a . ( b  c) = 1 (B) a . ( b  c) = 3 











(C) (a  b) . c = 0



Q.142 If vectors a

ˆ ,  2î  3 jˆ  k

Q.136 [ a b c ] will not be zero when





(A) a = b = c



ˆ and b  î  2 jˆ  3k

ˆ are coplanar, then the value of p is c  jˆ  pk

(A) 1 (C) – 1 Q.143 If



(B) 2 (D) – 2

 

a , b , c are three non- zero coplanar 

  



(D) 6







(C) 24

(D) ( c  a ) . b = 1

(B) –1 (D) None of these 



ˆ c  î  jˆ  2k



(A) 10

  ˆ , b  3î  2 jˆ  k ˆ & c  3î  jˆ  2k ˆ Q.133 If a  4î  3 jˆ  k

(A) 60 (C) 30



then a . ( b  c) is equal to -

(B) 2 (D) Not possible

represent three coterminous edges parallelopiped then its volume is-

(D) –17

ˆ , b  î  2 jˆ  k ˆ and  î  jˆ  k 

(A) 1 (C) 3

(C) 2





Q.140 If a

is-

Q.134

ˆ ,  î  jˆ  k ˆ & Q.138 Three vectors î  jˆ  k



 

 

 

(A) a . c = 0

(B) a . c  0

(C) a . c > 0

(D) None of these

 





vectors so that a . b = 0 and b . c = 0, then-

(B) a = b or b = c 

 



(C) a , b , c are coplanar 







 

 

  

(D) a  b or b  c

0 then [ a b c ] equals-



Q.137 The vector a which is collinear with the 

 

Q.144 For any non-zero vector d ; d . a = d . b = d . c =



 vector b  2î  jˆ and a . b = 10 is-

(A) 4î  2 jˆ

(B)  2î  4 jˆ

ˆ (C) 2î  4 jˆ  k

ˆ (D) 4î  2 jˆ  k

(A) 0

(B) 1

(B) – 1

(D) None of these

ˆ Q.145 If [ 2î jˆ  k

(A) –1 (C) 2

ˆ ] = – 4 then  is equal toî  2k

(B) 1 (D) any real number


11

 



Q.146 If a , b , c are coplanar vectors, then which of

the following are non-coplanar vectors









(A) a  b , b  c , c  a 

























vertices of a tetrahedron, then its volume is-









Q.152 If a , b , c , d are position vectors of four 















































(A) (1/2) [a  d b  d c  d]

(B) a  b , b  c , c  a

(B) (1/3) [ a  d b  d c  d]

(C) a  b , b  c , c  a

(C) (1/4) [ a  d b  d c  d]

(D) None of these

(D) (1/6) [a  d b  d c  d]

Q.147 If four points A(1, 2, –1), B(0, 1, m),

C (–1, 2, 1), D(2, 1, 3) are coplanar, then the value of m is(A) 2 (B) 0 (C) 5 (D) – 5

Question based on

Vector triple product 

 ˆ , b  2î  jˆ  k ˆ & c  î  3 jˆ  k ˆ then Q.153 If a  î  2 jˆ  2k







a  ( b  c ) is equal to-

Q.148 A unit vector which is coplanar with vector î  jˆ  2k ˆ and î  2 jˆ  k ˆ and perpendicular to

ˆ (A) 20 î  3 jˆ  7k

î  jˆ  k ˆ is-

ˆ (B) 20 î  3 jˆ  7k

(A)

(C)

( î  jˆ) 2 ˆ  jˆ) (k 2

(B)

(D)

ˆ) ( jˆ  k

ˆ (C) 20 î  3 jˆ  7k

2

(D) None of these

ˆ) (î  jˆ  k

Q.154

3







a  ( b  c ) is coplanar with











Q.149 Four points with position vectors a , b , c , d

(A) [ a b c ] = 0



(D) None of these

  

(B) [ b c d ] = 0 



(C) c and a

  

 



(B) b and c 

are coplanar if-



(A) a and b





(C) [ a – d b – d c – d ] = 0 (D) None of these









is















(A) a  ( b  c)  b . (a  c)

Q.150 If p.v. of vertices A, B, C with respect to









(B) (a  b ) . c  a . ( b  c) 

ˆ vertex O of any tetrahedron are 6 î ,6 jˆ , k

(C) a  ( b  c)  (a  b)  c

respectively, then its volume is-

(D) None of these

(A) 1/3 (C) 3

(B) 1/6 (D) 6

Q.151 If volume of a tetrahedron is 5 units and

vertices are A (2, 1, –1), B(3, 0, 1), C(2, –1, 3) and fourth vertex is on y- axis, then its coordinates are(A) (0, 8, 0) (B) (0, – 7, 0) (C) (0, 8, 0), (0, – 7, 0) (D) None of these



Q.155 For three vectors a , b , c correct statement







Q.156 The value of 



























a  ( b  c)  b  ( c  a )  c  (a  b) is

(A) 0

(B) 1

(C) a  b  c 

  

(D) 2 [ a b c] 





Q.157 If (a  b )  c = a  ( b  c ) , then it is possible

that







(A) a  b (C) a || c









(B) a  c (D) b || c


12







Q.158 For any vectors a , b , c correct statement is









Q.162



(A) a  ( b  c) = (a  b )  c 











 







 

  

Q.163

 

(D) a . ( b  c)  a . b  a . c 

[ a b a  b ] equals



(A) | a  b | 

  

  

  

  

  

  

  

  

ˆ )  ( k ˆ  î )] equals(î  jˆ) .[( jˆ  k

(A) 0 (C) –1







(C) ( b . c) a  (a . c) b (D) (a . c ) b  ( b . c ) a

(C) a . ( b  c )  a . b  a . c





(A) (a . c ) b  (a . b) c (B) (a . b ) c  ( a . c) b

(B) a  b  b  a

Q.159



(a  b)  c equals-









(B) 1 (D) 2

(B) | a  b |2



(C) | a . b |

(D) | a | | b |

Q.160 Which of the following is true statement































(A) (a  b )  c is coplanar with c (B) (a  b )  c is perpendicular to a

(C) (a  b )  c is perpendicular to b (D) (a  b )  c is perpendicular to c ˆ ) equalsQ.161 jˆ  ( jˆ  k

(A) î ˆ (C) k

(B) – î ˆ (D) – k


13

LEVEL- 2 Q.1

If C is mid point of AB and P is any point

Q.7

The mid point of points which divide line joining the points a and b in the ratio 1: 2

(A) PA + PB = PC

and 2 : 1 is-

(B) PA + PB = 2 PC (C) PA + PB + PC = 0 

(C)

(D) PA + PB + 2 PC = 0









Q.8

ˆ  jˆ , then the ratio of the magnitudes of d  k 





(B) 2 : 1

(C) 1 : 3

(D) 1 : 4



If

ˆ, AB = 3î  3k

vector

(B)

(C) 2 3 Q.4





















(C) b and c are like but a and b are unlike

î  2 jˆ  k ˆ

AC =

vectors 



(D) a and c are unlike vectors and so also 



b and c

3

Q.9







(B) square

(C) rectangle

(D) Trapezium

p.

v.

of

vertices

of

a

ABC

ˆ, 3î  6 jˆ  3k

are then

which of the following angles is a right a ngle-

then ABCD is a(A) parallelogram

If

ˆ , 4î  5 jˆ  k ˆ, 2î  4 jˆ  k



d are position vectors of the 

Q.10

(A) A

(B) B

(C) C

(D) None of these







a , b , c are three non zero vectors no two of 





If A, B, P, Q, R be any five points in a plane

them are parallel. If a  b is collinear to c

and forces AP , AQ , AR act at the point A

and b  c is collinear to a , then a  b  c is







equal to-

and forces PB , QB , RB act at the point B,

(B) 3 BA

(C) 3 PQ

(D) 3 PR

(C) c 

Q.11

 ˆ , b  î  3 jˆ  2k ˆ & c  2î  jˆ  5k ˆ If a  î  2 jˆ  3k

 

If | b | = 10, then the vector b which is

are vectors, then the vectors a , b , c are-

ˆ iscollinear with the vector 2 2 î  jˆ  4k

(A) linearly independent

ˆ (A) 4 2 î  2 jˆ  8k

ˆ (B)  4 2 î  2 jˆ  8k

ˆ (C) 4 2 î  2 jˆ  8k

(D) None of these



(D) None of these







(B) b



(A) 3 AB







(A) a

then their resultant is-

Q.6



and c

points A, B, C and D such that a  c  b  d ,

Q.5







(D) 3 2

If a , b , c ,



(B) a and b are unlike vectors and so also a

the length of median AM is6



unlike vectors

represents the sides of any triangle ABC then (A)



a and c are like but b and c are



Q.3

2

If a  5 b  c and a  7 b  2c , then-

vectors ( b  a ) and (d  c) is(A) 1 : 2





a  b

(D) None of these

3

(A)



(B)

a  b



 ˆ , b  2î  3 jˆ , c  3î  5 jˆ  2k ˆ, If a  î  jˆ  k







(A) a  b



Q.2





outside AB, then-

(B) collinear (C) linearly dependent (D) None of these


14

Q.12

If two forces acting at a point are represented

Q.17







If a , b , c be any three unit vectors such t hat 







by n OP and m OQ and their resultant is

3a  4 b  5c = 0 , then-

represented by (m + n) OR , then R is a point

(A) a || b

such that-

(C) a  b





Q.18







(A) m : n = RQ : PR (B) m : n = PR : RQ (C) R is the midpoint of PQ



(B) b || c





(D) None of these



If a , b , c be any three unit vectors such t hat 



a and b are perpendicular to each other and

(D) None of these







2a  3 b   c , then value of  is-

Q.13

ˆ , 2î  3 jˆ  4k ˆ and If 4î  7 jˆ  8k

ˆ 2î  5 jˆ  7k

(A) 1

are the position vectors of the vertices A, B and C respectively of triangle ABC. The position vector of the point where the bisector of angle A meets BC is(A) (B) (C)

2 3

3



















(A) perpendicular

(B) parallel

(C) coincident

(D) None of these







If p , q , r be three mutually perpendicular 







between p and p + q + r is-

ˆ) ( 6î  8 jˆ  6k





vectors of equal magnitude, then the angle

3 2



(D) 13

13

If a  b  c  d and a  c  b  d , then vectors 

Q.20

ˆ 6î  13 jˆ  18k





(C)

a  d and b  c will be-

ˆ) (  3î  4 jˆ  3k

2 ˆ) (D) – ( 6î  8 jˆ  6k 3

Q.14

Q.19

(B) 5

Q.21



(A) cos –1 (1/ 3 )

(B) sin –1 (1/ 3 )

(C) cos –1 (1/3)

(D) sin –1 (1/3)

 

If p , q , r , s are position vectors of points 





(A) PQ and RS bisect each other (B) PQ and PR bisect each other (C) PQ and RS trisect each other



a . b  c

then



P, Q, R, S such that p  q = 2( s  r ), then-

 



If a , b , c are three non- coplanar vectors,  



c  a . b

(A) 0

+

 



 



b.a  c

c . a  b

equals(B) 2

  

(C) 2 [ a b c ]

(D) QS and PR trisect each other

Q.22




If a = (1, 1 – 1), b = (1, – 1, 1), then a unit 



vector c which is perpendicular to a and Q.15

ABCDE is a pentagon. Force AB , AE , DC , ED act at a point. Which force should be

(A) (1/ 3 ) (– 1, 1, 1)

added to this system to make the resultant 2

(B) (1/ 6 ) (2, 1, –1)

AC -

(A) AC Q.16

(B) BC (C) BD

and ABC. Then AA' + BB' + CC' is equal


 



If a , b , c are three non- coplanar vectors and 

 

p , q , r are vectors defined as 

to -

(C) 3 GG'

(C) (1/ 6 ) (2, –1, 1)

(D) AD

If G and G  be centroides of triangles ABC

(A) GG '





coplanar with a and b is given by-



(B) 2 GG' (D)

2 3









ca



p =   , [a b c] 

GG '



b  c

q =   , [a b c] 











r =





a  b  

[a b c]

then



(a  b) . p  ( b  c ) . q  ( c  a ) . r equals-

(A) 0

(B) 1

(C) 2

(D) 3


15

Q.24







If a , b , c be any three non- zero non coplanar vectors and vectors 



b  c







ca





Q.30



(B)

(C) 0

1

 

Q.31



  ˆ and b  jˆ are If vectors c , a  xî  y jˆ  zk

Q.32





(C) angle between a and b is zero 

(A) 0

ˆ (B) zî  xk

ˆ (C)  zî  xk

ˆ (D) zk





 2













(A) (5, 2, 2)

(B) (5/3, 2/3, 2/3)

(C) (2/3, 5/3, 2/3)

(D) (2/3, 2/3, 5/3)

Q.33

 

(A) 1

(B) – 1

(C) 0

(D) –(| a |/| b |)2







Q.34

  



 

  

(A) (a . d) [a b c] (C) ( b . d) [a b c]

 

to

(B) ( c . d) [ a b c] (D) None of these Q.35

(C)

1 75 3 75

(B)

16  a . b

(D)

4  a . b



,



ca



q =   , [a b c]

 2

  



a  b



r =   , where [a b c]



(D) None of these







(A) 3

(B) 2

(C) 1

(D) 0

If a and b are non- parallel unit vectors such 



| a  b | = 3 , then









(2a  5 b) . (3a  b )

(A) 11/2

(B) 0

(C) –11/2

(D) 13/2

If A, B, C, D are four points in space, and



(area of (ABC), then  is equal to -

2 75



| AB × CD + BC × AD + CA × BD | =

ˆ and 2î  jˆ  k

vectors

ˆ , then |p| equals3î  2 jˆ  k

(A)

 

[a b c]

(B)

equals-

  

ˆ is a unit vector and is If pî  q jˆ  r k

perpendicular



b  c

 2

 

 

that

[( a  b )  (a  c)]. d equals 

If p =



a  b are at right angle-





(a  b  c ) . ( p  q  r ) equals-









Let the vectors a and b are at right- angle,





a = b = 2

a , b , c are non- coplanar vectors, then

then what is value of m so that a  m b and

Q.29

 2

 

4  a . b

(C) 2

a . b = 3, then b equals-









 

and b is a vector such that a  b = c and

Q.28



(A) 2 16  a . b



If a = (1, 1, 1), c = (0, 1, –1) are two vectors







If u  a  b and v  a  b , and then | u  v | is equal to-

(D) None of these

Q.27









(B) a and b are in opposite direction

 





(A) a and b are perpendicular

Q.26



system, then c is-

3a  5 b and a  b are perpendicular, then











such that a , c and b form a right- handed

Let a and b two unit vectors. If vectors 









a. b  c

(D) None of these













(D) | c | = 1, | a | = 1

(A) a . b  c







[ p q r ] equals-





(C) | b | = 2, | c | = 2| a |

  

Q.25



(B) | b | = 1, | c | = | a |

p =    , q =    , r =    , then a. b  c a. b  c a. b  c  





(A) | a | = 1, | b | = | c |

a  b







If a  b  c , b  c  a then -

Q.36

(A) 2

(B) 3

(C) 4

(D) 1

  ˆ ) equalsIf a . î = 4, then ( a  jˆ ). ( 2 jˆ  3 k

(A) 0

(B) 2

(C) 12

(D) –12


16



Q.37



















If d  p (a  b)  q( b  c)  r ( c  a ) and [a b c] = 1,

Q.39

then ( p  q  r ) equals







AB . AC + BC . BA + CA . CB equals-



(A) d . (a  b  c)

(B) a  b  c

(C) 1




Q.38



ˆ , a  iˆ  jˆ and let b and b Let b  3 jˆ  4k 1 2 

be component vectors of b 



perpendicular perpendicular to a . If If b1 = 

b 2 is equal to-

(A) – (B) (C) –

If in a right- angled triangle ABC, the hypotenuse AB = p,

parallel and

3 ˆ 3 i + jˆ , then 2 2

Q.40

(A) 2p2

(B) p2/2

(C) p2

(D) 0

The value of x for which the angle between 

 ˆ and b  xiˆ  2x jˆ  k ˆ the vectors a  3iˆ  x jˆ  k

is acute and the angle between b and x-axis lies between /2 and  satisfy(A) x < –1 only (C) x > 1 only

(B) x > 0 (D) x < 0

3 ˆ 3 ˆ i + j 2 2

 3 ˆ 3 ˆ i+ j + 4 k 2 2  3 ˆ 3 ˆ i + j + 4 k 2 2

(D) None of these

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17

LEVEL - 3 Q.1

 ˆ and If the vectors a = (clog2x) î – 6 jˆ + 3 k



Q.6

(A) (– , 0)

(B) (– , –4/3)

(C) (–4/3, 0)

(D) (–4/3, )

a c

(A) 0

If a , b , c are the position vectors of the









 2

ˆ, a2 k

1 a3

a2







(C) a + b = c





























a and b . If the angle between



a1



c1

 





(D) ( a × b ) · ( c × d ) = 0

Q.8









 





c b .

c.c

Q.9

  

(A) [ a b c ]2

(B) [ a b c ]

(C) [ a b c ]3

(D) None of these

  

 

If forces of magnitudes magnitudes 6 and 7 units acting in ˆ and 2 î – 3 jˆ + 6 k ˆ the directions î – 2 jˆ + 2 k

(D) None of these

respectively act on a particle which is





displaced from the point P(2, –1, –3) to Q(3, –1, 1) then the work done by the forces is-

Let the unit vectors a and b be perpendicular to each other and the unit vector 



c

be



inclined at an angle  to both a and b . 



  

(B) b + x a for all scalars x



3  2  2 | a | | b | 4







b . c   a a.c

(D)

a.c





(A) a – c



c3

b.a b b . b.c equals-

equal to

(C) b 

is equal to -

 



c.a



2

(B) 1

a b .



other and r × a = b × a  r  c = 0 then r is 



a.a





 to both   a and b is /6,

If a , b , c are non-coplanar vectors, then 

If a and b are not perpendicular to each 

c2

1  2  2 | a | | b | 4

(C)

(C) ( a × b ) × ( c × d ) = 0 

a3

(A) 0

(B) ( a × c ) · ( b × d ) = 0 

a2

then b1 b 2 b 3

(A) ( a × c ) × ( b × d ) = 0 

(D) –1

vectors such that c is a unit vector

a plane. Then the planes are parallel if 

(C) 2

 ˆ , b = b î + b jˆ + b k ˆ Let a = a1 î + a2 jˆ + a 3 k 1 2 3

Let the pairs a , b and c , d each determines 

(B) 1

 ˆ be three non-zero and c = c1 î + c2 jˆ + c3 k

 2


1  c3

c2



Q.7

(B) | a | = | b | 2 + | c |

(A) a + b + c = 0

Q.5

a jˆ +

then the value of abc is-

  

orthocentre is at the origin, then -

Q.4

î +

a=

non-coplanar vectors and b b 2 1  b 3 =0,

vertices of an equilateral triangle whose

Q.3



vectors

 (0, ), then c belongs

to -

Q.2

the

 ˆ , c = î + c jˆ + c2 k ˆ are three b = î + b jˆ + b2 k

ˆ make on b = (log2x) î + 2 jˆ + (2clog2 x) k

obtuse angle for any x

If





(A) 44 units (C) 7 units



If c = x a + y b + z( a × b ), then(A) x = cos, y = sin, z = cos 2 (B) x = sin, y = cos, z = cos 2  (C) x = y = cos,

z2 =

cos 2

(D) x = y = cos, z2 = –cos 2

Q.10





(B) –4 units (D) –7 units



If a , b , c are three non-zero vectors such that 









 





a + b + c = 0 and m = a  b + b  c + c  a , then

(A) m < 0 (C) m = 0

(B) m > 0 (D) m = 3.

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18

Q.11

If the position positi on vectors of three points A, B, C

Statement type Questions

ˆ , 2 î +3 jˆ – 4 k ˆ and are respectively respectively î + jˆ + k ˆ , then the unit 7 î + 4 jˆ + 9 k unit vector vector perp pe rpen endi dicu cula larr

to the plane of triangle ABC isˆ (A) 31 î –18 jˆ – 9 k

(C)

Q.12

(B)

II. Use the following key to choose the

ˆ 31î  38 jˆ  9 k

appropriate answer.

2486

(A) If both StatementStatement- I Statement- II are true,

(D) None of these

2486

of Statement- I. (B) If Statement-I and Statement-II are true



but

Vectors a and b are inclined at an angle 













(A) 225

(B) 275

(C) 325

(D) 300 















 3

(A) 4 5

(B)

(C) 4 7

(D) None of these











 





acute than | a . b | < | a | | b | Q.18

Statement-1 (A) : ABCDEF is a regular 









then EA is equal to – ( b + c ).



  

Statement-2 (R) : AE = BD = BC + CD Q.19

(B) 2[ a b c ]

  

(C) 3[ a b c ] If the vectors

a and b are mutually 











Q.20

 ˆ and Statement-1 (A) : a = î + p jˆ + 2 k



ˆ are parallel vector. If p b = 2 î + 3 jˆ + q k

is equal to





AB = a + b (Triangle law of addition)

perpendicular, perpendicular, then a × { a × { a × ( a × b )}} 









Statement-1(A) :In ABC, AB + BC + CA = 0 Statement-2 (R) : If OA = a , OB = b then

(D) 0







(A) | a |2 b (C) | a |4 b

(B) | a |3 b

=3/2, q = 4.

(D) None of these

ˆ and Statement-2 (R) : If a = a1 î + a2 jˆ + a3 k





Q.16



hexagon and AB = a , BC = b and CD = c ,

   

(A) [ a b c ]

Q.15





b ) × ( a – b b – c )} is equal ( a + 2 b – c ) {( a – b

to

Statement-1 (A) : If the difference of two

Statement-2 (R) : If angle between a & b is

3



correct

between them is 60o

, then the

length of a diagonal of the parallelogram is-

Q.14

the

unit vectors is again a unit vector then angle

a = 3  –  , b =  +3  . If |  | = |  | = 2 and 

not

(D) If Statement-I is false but Statement-II is true Q. 17

the angle between  and  is

is

(C) If Statement-I is true but Statement-II is false

A parallelogra m is constructed constructe d on the vectors 

Statement-II

explanation of Statement-I

= 120º. If | a | = 1, | b | = 2, then

b )]2 is equal to[( a + 3 b ) × (3 a – b

Q.13

below consists of Statement -I and Statement-

and Statement-II is the correct explanation

ˆ 31î  38 jˆ  9 k



Each of the questions (Q.No.17 to 27) given

ˆ are parallel b = b1 î + b2 jˆ + b3 k

The area of parallelogram parallelogram constructed on the 











vectors a = p + 2 q and b = 2 p + q where 



p and q are unit vectors forming an angle of

30º is(A) 3/2 (C) 0

Q.21

Statement-1 (A) : If

(D) None of these

b1

=

a2 b 2





=

a3 b 3

 

a , b , c are three

coplanar vectors then the vectors 

(B) 1



a1





a × b ,



b × c , c × a are also coplanar. 





Statement-2 (R) : If a , b , c   

are coplanar

vectors then [ a b c ] = 0 .www.iitians pace.com IIT - ian’s PACE ; ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223 / 09 ; .www.iitianspace.com

19

Q.22





Statement-1 (A) : Three points A( a ), B( b ), 











Passage Based Question





C( c ) are collinear if a × b + b × c + c × a = 0

Passage-1 

Statement-2 (R) : Points A , B , C are

  



collinear  AB = t AC , t  R.

c is denoted by [ a b c ] and is defined as   

Q.23



The scalar triple product of three vectors a , b , 











[ a b c ] = a .( b × c ). Three vectors a , b , c   

Let PQ , QR , RS , ST , TU , UP denote the

are coplanar vectors if and only if [ a b c ] = 0.

sides of a regular hexagon.

Volume of the parallelopiped whose three



Statement-1 (A) : PQ ×( RS + ST )  0





  



concurrent edges are a , b , c is |[ a b c ]|



Statement-2 (R) : PQ × RS = 0 and Q.28



PQ × ST = 0

Q.24

Statement-1 (A) : ˆ , î – 2 jˆ + k ˆ & î + jˆ – 2 k ˆ Vectors –  2 î + jˆ + k

ˆ is 546 then  = 2 î + jˆ – 15 k

(A) 2/3 (C) –4

are coplanar for only two values of  .





If the volume of a parallelopiped whose three ˆ , 3 jˆ – k ˆ and concurrent edges are –12 î +  k



Statement-2 (R) : Three vector a , b , c are 





Q.29

coplanar if a . ( b × c ) = 0. Q.25















 









Statement-2 (R) : [ a + b b + c c + a ]

  

Q.26















[a b c]



 





 





b  c 

[a b c]





, b =



ca 

[a b c]



, c =

are reciprocal system of vectors

 

a , b , c .

On the basis of above information, answer

are non coplanar then [ A B C ]  0 Statement-1 (A) : If a , b , c are unit coplanar

  



a  b

Statement-2 (R) : Three vector A , B , C



(D) 2[ a b c ]

the equations a  =





(C) [ a b c ]



a + b – 3 c are also non coplanar.

Q.27

(B) 1



 

2 a – b +3 c , a + b –2 c , 

  

(A) 0



vectors then vectors 

  

Let a , b , c are non coplanar vectors and let

Statement-1(A) : If a , b , c are non coplanar 

  

Passage - 2 :

= [ a b c ] 



  



non coplanar then a + b , b + c , c + a are also non coplanar.

 



If a , b , c , d are four coplanar points then [ b c d ] + [ c a d ] + [ a b d ] is

Statement-1 (A) : Three vector a , b , c are 

(B) –1 (D) –3

the following questions. Q.30

The value of the expression 

 







a . a  + b . b + c . c equals-



vectors then [2 a – b 2 b – c – a ] = 0   

Statement-2 (R) : [ a b c ] = 0 Q.31

(A) 0

(B) 1

(C) 2

(D) 3 











The expression a × a  + b × b + c × c is (A) a unit vector (B) null vector 





 | b | 2  | c | 2 (C)  2   | a |  | b | 2  | c | 2 | a |2

(D) arbitrary vector


20

Q.32

The value of the expression 









Q.34



a  × b + b × c + c × a  is



(A)



 





[a b c] 

(C)



a  b  c

 a  b  c  

[ a b c]

(B)

(D)

three axes, then the tangent of angle becomes 



a  b  c  

[a b c ] 





a  b  c  

 ˆ , b = î – ˆj –2 k ˆ, If a = 2 î +3 jˆ – k

Q.35



ˆ , then c × a equals c = – î + 2 jˆ +2 k

(A) (C)

î  jˆ

ˆ  2 k

3 î  jˆ  2 k ˆ 9

Passage - 3 : 

(B)

(D)



3

(C) 2

(D)

5

 



Q.36

(A) 3

(B) 3 7

(C) 9

(D) 12 



The range of | a – b | will be (A) [1, 7]

(B) [3, 5]

3

(C) [3, 4]

(D) [1, 9]





| a | = 3, | b | = 4, | c | = 5.

Q.37



If a . c = 9 then the value of | a × b | is

ˆ  î  jˆ  2 k

of the two. Also given 

(B) ( 2 )

î  jˆ  2 k ˆ

If each of a , b , c is orthogonal to the sum 

 3   (A) ±   2   

[a b c ]



Q.33



If a makes angles of equal measures with all







Value of | a + b + c | is (A) 12

(B) 5

(C) 5 2

(D) None of these

On the basis of above information, answer the following questions.


21


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans. C

B

D

D

B

D

B

C

C

D

A

A

D

D

C

A

A

A

A

A

Q.No. 21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Ans. A

D

C

C

B

B

C

B

C

D

B

B

C

D

A

C

B

B

C

B

Q.No. 41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

Ans. B

A

B

A

D

C

C

C

A

A

D

A

D

C

B

B

A

A

A

B

Q.No. 61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

Ans. A

A

D

B

D

B

A

B

B

A

C

D

B

C

C

D

B

B

A

B

Q.No. 81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

Ans. A

B

A

B

C

D

D

D

B

D

B

B

B

A

A

C

B

C

99 100 B

B

Q.No. 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 Ans. D

A

B,C

A

D

C

D

A

D

D

B

D

A

D

C

B

C

A

C

B

Q.No. 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 Ans. D

B

B

B

C

D

A

D

C

A

A

D

C

C

B

D

A

B

D

B

Q.No. 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 Ans. C

C

B

A

D

D

C

B,C

C

D

C

D

A

B

B

A

C

D

B

D

Q.No. 161 162 163 D

Ans.

D

B

LEVEL- 2 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans. B

C

A

A

A

A

B

A

A

D

A

B

B

D

B

C

C

C

B

A

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Ans. A

C

D

B

B

B

D

A

A

B

B

A

A

C

C

D

A

C

C

D

Q.No.

LEVEL- 3 Qus.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

C

A

C

C

D

D

C

A

A

A

B

D

C

C

C

A

B

A

C

A

Qus.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

Ans.

A

A

C

A

C

A

A

D

C

D

B

D

B

B

B

A

C


22

IIT – ian’s

P A C E



Q.1

LEVEL –1

Distance between two points

Q.7

3D Cool. Geometry The point which lie on z -axis has the following condition-

The points A(1, –1, – 5), B(3, 1, 3) and

(A) z - coordinate are zero

C(9, 1, –3) are the vertices of-

(B) both x and y coordinate are zero

(A) an equilateral triangle

(C) both y and z coordinate are zero

(B) an isosceles triangle

(D) both x and z coordinate are zero

(C) a right angled triangle (D) none of these

Q.8

The distance of the point (1, 2, 3) from x-axis is

Q.2

Distance of the point (x, y, z) from y-axis is(A) y (C)

(B) y

2

z

2

(D)

x

2

y

2

z

2

x

2

Q.9 Q.3

13

(B)

(C)

10

(D) None of these

If P  (0, 5, 6), Q

5

 (2, 1, 2), R  (a, 3, 4) and

The distance of a point P(x, y, z) from yz

PQ = QR then 'a' equal to-

plane is-

(A) 1

(B) 2

(C) 3

(D) None of these

(A) x

(B) y

(C) z

(D) x + y + z Q.10

Q.4

(A)

Points (1, 2, 3); (3, 5, 7) and (–1, –1, –1) are-

The co-ordinates of the point which are lie

(A) vertices of a equilateral triangle

equally distance from the point (0, 0, 0);

(B) vertices of a right angle triangle

(a, 0, 0) ; (0, b, 0) and (0, 0, c)

(C) vertices of a isosceles triangle

(A) (a/2, b/2, c/2)

(D) collinear

(B) (–a/2, b/2, c/2) Q.11

(C) (–a/2, –b/2, c/2)

If the vertices of points A, B, C of a tetrahedron ABCD are respectively (1, 2, 3) ;

(D) (a/2, –b/2, –c/2)

(–1, 2, 3), (1, –2, 3) and his centroid is (0, 0, Q.5

3/2) then co-ordinate of point D are-

Distance of the point (a, b, c) from z- axis is (A) (C)

a 2  b 2 c

2

a

2

(B) b 2

 c2

(B) (–1, –2, 3)

(C) (–1, –2, –3)

(D) (0, 0, 0)

(D) c Q.12

Q.6

(A) (1, 2, –3)

The point on xy-plane which is equidistant from the points (2, 0, 3), (0, 3, 2), (0, 0, 1) is(A) (2, 3, 0)

(B) (3, 0, 2)

(C) (3, 2, 0)

(D) (2, 3, 1)

The distance of point (1, 2, 3) from coordinate axis are(A) 1, 2 , 3

(B)

5 , 10 , 13

(C)

(D)

13 , 10 , 5

10 , 13 , 5


1

Q.13

The coordinates of the points A and B are

The points (1, 2, 3), (– 1, – 2, – 1), (2, 3, 2)

Q.19

(–2, 2, 3) and (13, –3, 13) respectively. A

and (4, 7, 6) form a-

point P moves so that 3PA = 2 PB, then locus

(A) rectangle

(B) square

of P is-

(C) parallelogram

(D) rhombus

(A) x2 + y2 + z2 + 28x – 12y + 10z – 247 = 0 (B) x2 + y2 + z2 + 28x – 12y + 10z + 247 = 0

If BC, CA and AB are the sides of a triangle ABC whose midpoints are (p, 0, 0), (0, q, 0),

Q.20

(C) x2 + y2 + z2 – 28x + 12y – 10z – 247 = 0 (0, 0, r) then find

(D) None of these

(A) 8 Q.14

A point which lie in yz plane, the sum of

Question based on

 (BC) 2  (CA)2 p 2  q 2  r 2

(AB)2

(B) 6

(C) 5

-

(D) 2

Coordinates of division point

co-ordinate is 3, if distance of point from xz plane is twice the distance of point from xy

Q.15

Find the ratio in which the segment joining

plane, then co-ordinates are-

the points (2, 4, 5), (3, 5, –4) is divided by the

(A) (1, 2, 0)

(B) (0, 1, 2)

yz-plane.

(C) (0, 2, 1)

(D) (2, 0, 1)

(A) 3 : 1

(B) – 2 : 3

(C) – 1 : 3

(D) 1 : 2

A point located in space is moves in such a way that sum of distance from xy and yz

Q.16

Q.21

Q.22

Find the ratio in which the segment joining

plane is equal to distance from zx plane the

(1, 2, –1) and (4, –5, 2) is divided by the

locus of the point are-

plane 2x – 3y + z = 4.

(A) x – y + z = 2

(B) x + y – z = 0

(A) 2 : 1

(B) 3 : 2

(C) x + y – z = 2

(D) x – y + z = 0

(C) 3 : 7

(D) 1 : 2

A (1, 3, 5) and B (– 2, 3, – 4) are two points,

Q.23

If points A (3, 2, –4); B(5,4, –6) and

A point P moves such that PA2 – PB2 = 6c,

C(9, 8,–10) are collinear then B divides AC in

then locus of P is-

the ratio-

(A) x + 3z + 1 – c = 0

(A) 2 : 1

(B) 1 : 2

(C) 2 : 3

(D) 3 : 2

(B) x + 3z – 1 + c = 0 (C) 2x + 3z + 1 – c = 0 (D) 2x + 3z – 1 + c = 0 Q.17

Q.24

If zx plane divides the line joining the points

The locus of the point which moves such that

(1, –1, 5) and (2, 3, 4) in the ratio

its distance from (1, –2, 2) is unity, is-

 equals to-

(A)

x2 +

y2 +

z2 –

2x + 4y + 4z + 8 = 0

(A) 1/3

(B) 3

(B)

x2 +

y2 +

z2 –

2x – 4y – 4z + 8 = 0

(C) –3

(D) –1/3

:1

then

(C) x2 + y2 + z2 + 2x + 4y + 4z + 8 = 0 Q.25

(D) x2 + y2 + z2 – 2x + 4y – 4z + 8 = 0

OABC is a tetrahedron whose vertices are O (0, 0, 0); A (a, 2, 3); B (1, b, 2) and C (2, 1,

Q.18

If distance of any point from z - axis is thrice

c) if its centroid is (1, 2, –1) then distance of

its distance from xy-plane, then its locus is-

point (a, b, c) from origin are-

(A) x2 + y2 – 9z2 = 0 (B) y2 + z2 – 9x2 = 0

(A)

14

(B)

(C)

107 / 14

(D) None of these

(C)

x2 –

9y2 +

z2

= 0 (D)

x2 +

y2 +

z2 =

0

107


2

Q.26

If A(1, 2, –1) and B (–1, 0, 1) are two points

lie on a line meets the point P(2, 7, 1) & Q(3,

externally in the ratio of 1 : 2

10, 11) then coordinates of R is-

(C)

1 3

(B)

1 3

(A) (2, 7, 3)

(3, 4, –3)

(B) (3, 10, 3) (C) (11/5, 38/5, 3)

(1, 4, –1)

(D) None of these

(D) (38/5, 11/5, 3)

The ratio in which the yz-plane divides the

Q.34

join of the points (–2, 4, 7) and (3, –5, 8) is-

Q.28

(A) 2 : 3

(B) 3 : 2

(C) –2 : 3

(D) 4 : –3

A (3, 2, 0), B (5, 3, 2) and C (–9, 6, –3) are

Q.35

A meets BC at D, then its coordinates are(A)

 19 , 57 , 17     8 16 16 

(B)

  19 , 57 , 17     8 16 16 

(C)

 19 , 57 ,  17     8 16 16 

(D)

  19 , 57 , 17     8 16 16 

(C) (2, 3, 4), (4, 8, 6) (D) none of these Q.36

3 lies on(A) XOY plane

7, c) then values of a, b, c are respectively(B) 0, 2, 2

(C) –2, –8, 2

(D) None of these

(B) YOZ plane (C) ZOX plane (D) none of these

The line joining the points (2, –3, 1) and (3, –4, –5) and cuts the plane 2x + y + z =7 in

Q.37

Q.31

(B) (–1, 2, 7)

(C) (1, –2, 7)

(D) (1, –2, –7) are

(A) 1 : 1 (B) 1 : 2 (C) 1 : 3 (external)

A (4, 3, –2), B(3, 0, 1) and C(2, –1 , 3), the

(D) 3 : 1 (external)

The

vertices

of

a

triangle

ABC

length of the median drawn from point 'A' (A) (C) Q.32

The line joining the points (0,0,0) and (1,–2, – 5) is divided by plane x – y + z = 1 in the ratio-

those points, the point are(A) (1, 2, 7)

The point which divides the line joining the points (2, 4, 5) and (3, 5, – 4) in the ratio – 2 :

If origin is the centroid of the triangle ABC

(A) 2, 8, 2

The points trisecting the line segment joining the points (0, 0, 0) and (6, 9, 12) are(A) (2, 3, 4), (4, 6, 8) (B) (3, 4, 2), (6, 8, 4)

with vertices A(a, 1, 3), B(–2, b, –5) and C(4,

Q.30

If three consecutive vertices of a parallelogram are A (1, 2, 3), B (– 1, – 2, – 1) and C (2, 3, 2). Its fourth vertex is(A) (– 4, 5, 3) (B) (4, 7, 6) (C) (3, – 5, 2) (D) (4, 5, 3)

vertices of a triangle ABC. If the bisector of

Q.29

The z-coordinates of a point R is 3, which is

then co-ordinate of points which divide AB

(A) (3, 4, –3)

Q.27

Q.33

1 2 1 3

122

(B)

122

122

(D) None of these

The orthocentre of the triangle with vertices

Question based on

Q.38

Direction cosines and direction Ratio's of a line

Find the d.c's of a line whose direction ratios are 2, 3, –6 (A)

(2, 3, 4), (3, 4, 2) and (4, 2, 3) is(A) (1, 1, 1)

(B) (2, 2, 2)

(C) (3, 3, 3)

(D) None of these

(C)

2 2 2 , , 7 5 7 2 7

,

3

2 , – 4 7

(B) (D)

2 7 3 7

, ,

3

6 , – 7 7 4 7

,

6 7


3

Q.39

The projections of a line segment on x, y and z axes are respectively 3, 4 and 5. Find the

Q.44

y axis then angle makes with the z axis are-

length and direction cosines of the line segment3

(A) 5 3 ; (B) 5 2 ; (C) 5 2 ; (D) 3 2 ; Q.40

5 3 5 5 2 3 5 2 3 3 2

,

4 5 3 3

,

4

,

4

,

3 2

Q.45

3

(B) 45º or 135º

(C) 30º or 150º

(D) 30º or 60º

, ,  be

1 2 1

Q.46

2

(A) 2

(B) 3

(C) 4

(D) None of these

If the direction ratios of a line are 1, –3, 2, then the direction cosines of the line are-

The direction cosines of a line equally

1

(A)

(A) (1, 1, 1) or (–1, –1, –1)

 1  ,  3

 (C)  

1 2

1 3

,

  or 3 

1

   

  or , 2 2 

1

,

1

1

(B) 1

,

3

   

1 3

,

  3 

  , , 2 2 2 

1

14

1

1



1

If the projection of a line on the co-ordinate

Q.47

(D)



1 14

line are(A) 6, –3, 2 (C)

Q.42

7 6

,

7 3

,

(B) 7 2

7

,

3 7

,

2 7

Q.48

3

,

,

14

2 14

,

2 14

3 14

The direction cosine of a line which are (A) 1, 0 , 0

(B) 0, 1, 0

(C) 0, 0 , 1

(D) 1, 1, 1

The co-ordinates of a point P are (3, 12, 4) cosines of OP are-

  

If a line makes angle

(A) 3, 12, 4 with the (B)

 + cos2  + cos 2

equals to(A) –2

(B) –1

(C) 1

(D) 2

(C) (D)

If a line makes angle co-ordinate axis and cos



14

with respect to the origin O, then the direction

(D) none of these

co-ordinate axis then cos2

Q.43

3

,

14

,

14

perpendicular to the yz plane-

axes are 6, –3, 2, then direction cosines of the 6

2

14

1

2

,

14

,

(C)

(D) none of these Q.41

3

,

14

inclined with the coordinate axes are-

(B)

the angles which a line makes

sin2  + sin2  + sin2  =

2

,

If

with the positive directions of the axes, then

1

,

5 2

(A) 60º or 120º

1

, ,

5 2

If a line makes angle 120º and 60º with x and

   with  = 14/15 ]

=1/3 then cos  is equal to ?

(A) 1/5

(B) ± 1/ 5

(C) ± 2/15

(D) None of these

1

,

4

1 3

3 13 3

,

,

, 12

13 13

1 2 1 13

,

,

2 13

4 13

the cos

Q.49

   with the co +   90º] then  equal to-

A line makes angle ordinate axis if (A) 0º

(B) 90º

(C) 180º

(D) None of these


4

Q.50

The length of line segment AB is 14 if its

Q.56

A line located in a space makes equal angle

direction ratio are 2, 3, 6 then its direction

with the co-ordinate axis then the angle made

cosines will be-

by this line with any axis is-

(A) ± 2/7 ± 3/7, ± 6/7

(A) 60º

(B) 45º

(B) ± 2/14, ± 3/14, ± 6/14

(C) cos –1 1/3

(D) cos –1 1/ 3

(C) ± 2/7  3/7, ± 6/7 (D) None of these

Q.57

The angle between the pair of lines with direction ratios 1, 2, 2 and 2 , 3, 6 is-

Q.51

Which

of

the

following

triplets

gives –1

(A) cos

direction cosines of a line? (A) 1, 1, 1 (C) 1, –1, 1

(B) 1, 1, –1 (D)

1 3

,

1 3

,

–1

1

(C) cos

3 Q.58

Question based on

Q.52

If the line through the points (4, 1, 2) and

,

0) is parallel to the line through the

points (2, 1, 1) and (3, 3, –1), find .

Q.53

(A) 3

(B) –3

(C) 2

(D) 4

If the line joining the points (1, 2, 3) and

(B) cos

 20     21 

(D) cos

–1

–1

 19     20   20     19 

If O is origin and P(1, –2, 1) and Q(2, 3 , 4) are other two points then-

Angle between two lines

(5,

 21     20 

Q.59

(A) OP = OQ

(B) OP  OQ

(C) OP || OQ

(D) None of these

The point in which the join of (–9, 4, 5) and (11, 0, –1) is met by the perpendicular from the origin is(A) (2, 1, 2) (B) (2, 2, 1)

(4, 5,

(C) (1, 2, 2)

(D) None of these

7) is perpendicular to the line joining the Q.60

points

Q.54

If vertices of a

are respectively (a, 0,

(–4, 3, –6) and (2, 9, ).

0); (0, b, 0) and (0, 0, c) then  B is equal to-

(A) –15

(B) 20

(A) cos –1

(C) 5/3

(D) 10

( b

2), then A is equal to(B) 60º

(C) 90º

(D) 30º

If co-ordinates of points P, Q, R, S are respectively (1, 2, 3), (4, 5, 7); (–4, 3, –6) and (2, 0, 2) then(A) PQ || RS

(B) PQ  RS

(C) PQ = RS

(D) None of these

(C)

2

 b 2 ) ( b 2  c 2 ) b 2

(B) cos –1

If the coordinates of the vertices of a triangle

(A) 45º

b 2 (a

ABC be A(–1, 3, 2), B(2, 3, 5) and C(3, 5, –

Q.55

ABC

2

 c 2 ) (c 2  a 2 ) b 2

cos –1 (a

2

 b 2 ) (c 2  a 2 )


The co-ordinates of points A, B, C, D are respectively (4, 1, 2); (5, a, 0);(2,1, 1) and (3, 3, – 1), if AB is perpendicular to CD equal to(A) 1/2 (B) –1/2 (C) 3/2

then 'a'

(D) –3/2


5

Q.62

If points (2, 0, –1); (3, 2, –2) and (5, 6,  ) are collinear then

Q.63

Q.64

Q.65

 equal to-

(A) 4

(B) –4

(C) 3

(D) 0

Question based on

Q.70

Projection problems

If P(6, 3, 2); Q(5,1,4); R(3, –4, 7) and S(0, 2, 5) are given points then the projection of PQ on RS is equal to-

The angle between the lines whose direction ratios are 3, 4, 5 and 4, –3, 5 is(A) 30º (B) 45º (C) 60º (D) 90º If the vertices of a right angle isosceles triangles are A(a, 7, 10); B(–1, 6, 6) and C(–

Q.71

(A) 13/7

(B) 13

(C)

(D) 13/ 7

P



If < a, b, c > and < a', b', c' > are the direction ratios of two perpendicular lines, then-

(A)

Q.72

a2

=

b1 b 2

=

c1 c2 Q.73

If A  (k, 1, –1); B  (2k, 0, 2) & C  (2 + 2k, k, 1) if AB  BC, then value of k are(B) 1

1 mn

(C) 2

m

(y2 – y1) +

1 n

(z2 – z1)

[(x2 – x1) + m(y2 – y1) + n(z2 – z1)]

A line makes angle 45º, 60 and 60º with the

(A) 3/2 (C) 1/2

(D) None of these

(A) 0



1

coordinate axis, the projection of line segments on line which joins point (–1, 2, 3) & (–1, 4, 0) are-

(B) a1a2 + b1 b2 + c1c2 = 0 a1

(x2 – x1) +

(D) None of these

(A) a1= a2, b1= b2, c1= c2

(C)

1

(B) (x2 – x1) + m(y2 – y1) + n(z2 – z1)

If direction ratio of two lines are a 1, b1, c1 and only if-

(B) 1/3 (D) 2/3

The projection of point (a, b, c) in yz plane are(A) (0, b, c) (B) (a, 0, c) (C) (a, b, 0)

(D) 3 Q.74

A point P(x, y, z) moves parallel to z-axis.

(D) (a, 0, 0)

The direction cosine of a line are proportional

Which of the three variables x, y, z remain

to 1, 2, 3, the projection of line segment on line which joins point (5, 2, 3) and (–1, 0, 2)-

fixed?

(A) 13

(B) 13/14

(C) 13/ 14

(D) None of these

(A) x and y

(B) y and z

(C) x and z

(D) none of these Q.75

Q.69

y2, z2) are two

m, n then projection of PQ on AB are-

a2, b2, c2 then these lines are parallel if and

Q.68

 (x2,

points if direction cosines of a line AB are ,

(C)

Q.67

(x1, y1, z1) and Q

4, 9, 6) which are right angle on B, then 'a' equal to(A) –1 (B) 0 (C) 2 (D) – 3

(A) a/a' = b/b' = c/c' (B) aa' + bb' + cc' = 0 (C) aa' + bb' + cc' = 1 (D) None of these Q.66

13 /7

If the angle between the line AB and CD is

A point P(x, y, z), moves parallel to yz-plane.

 then

Which of the three variables x, y, z remain

are

fixed?

(A) AB sin 

(B) AB cos 

(C) AB tan 

(D) AB cot 

(A) x

(B) y

(C) z

(D) y and z

projection of line segment AB on CD


6

Q.76

The projections of a line segment on x, y, z axes are 12, 4, 3. The length and the direction

Q.82

(C) 11, < 12/11, 14/11, 3/11 >

Q.83

x  1 

y  2

=

m

=

z  1 n

x 1

(C) (1, 2, –1)

(D) (0, 1, 0)

If the angle between the lines whose direction

Q.79

Q.80

The equation of a line passing through the point (–3, 2, –4) and equally inclined to the axes, are(A) x – 3 = y + 2 = z – 4 (B) x + 3 = y – 2 = z + 4 (C)

x  3

=

y  2

1 2 (D) none of these Q.81

(A) (B) (C) (D)

x  3 1 x  3 1 x  3 7 x  3 7

= = = =

y  2 3 y  2 3

=

y  2

=

7 y  2 7

=

and

7

(C) –10

10



y3 2





y4 2

z4



(D) –7

two

lines

and

1 z 1 2

is-

 2    9   4  (D) cos –1    9  (B) cos –1 

The angle between the lines whose direction ratios are 1, –2, 7 and 3, –2, –1 is (A) 0° (C) 45°

Q.86

(C)

Q.87

(B) 30° (D) 90°

Equation of x-axis is(A)

Question based on

=

2

Q.85

3

The equation of the line passing through the points (3, 2, 4) and (4, 5, 2) is-

z3

A line passing through the point (– 5, 1, 3) and (1, 2, 0) is perpendicular to the line passing through the point (x, 2, 1) and (0, – 4, 6) then x equal to(A) 7/2 (B) –7/2 (C) 1 (D) – 1

z4

=

2k



Q.84

(D) 4

Direction ratios of the line represented by the equation x = ay + b, z = cy + d are(A) (a, 1, c) (B) (a, b – d, c) (C) (c, 1, a) (D) (b, ac, d)



 1    9   3  (C) cos –1    9 

ratios are 2, –1, 2 and a, 3, 5 be 45º, then a = (A) 1 (B) 2 (C) 3

y2

z 6

(A) cos –1 

n) is(B) (1, 1, –1)

3



between

1

line through (1, 2, –1) & ( –1, 0, 1), then (, m,

Q.78

7

x4

is the equation of the

(A) (–1, 0, 1)



(B)

Angle 2

If

10



(A)

(D) None of these

Q.77

y 5



x 1

lines

are at right angles, then 3k 1 5 the value of k will be-

(B) 19, < 12/19, 4/19, 3/19 >

Equation of a line and angle between them

the

x 1

cosines of the line segments are(A) 13, < 12/13, 4/13, 3/13 >

Question based on

If

x 1 x 1

= =

y 1 y 0

=

z 1

=

z 0

(B)

(D)

x 0

=

x 0

=

y 1 y 0

= =

z 1 z 1

Perpendicular distance of a point from a line, foot of the perpendicular

The

co-ordinates

of

the

foot

of

the

z4

perpendicular drawn from the point A (1, 0,

2 z4 2 z4

3) to the join of the point B (4, 7, 1) and C (3, 5, 3) are(A)

6

z4 6

(C)

 5 , 7 , 17     3 3 3   5 , 7 , 17      3 3 3 

(B) (5, 7, 17) (D)

 5 , 7 , 17      3 3 3 


7

Q.88

The length of the perpendicular from point x  6

(1, 2, 3) to the line

3

=

y  7

z7

=

2

2

Q.93

The equation of the plane through the three points (1, 1, 1), (1, –1, 1) and (–7, –3, –5), is-

is-

(A) 3x – 4z + 1 = 0 (B) 3x – 4y + 1 = 0

(A) 5 (C) 7

(B) 6 (D) 8

The perpendicular

distance of the point

(C) 3x + 4y + 1 = 0 (D) None of these

Q.89

(2, 4, –1) from the line z6

9

Q.90

The 5 z3 4

Q.94

3 z6 4 (A)

point y  1

=

2

of

intersection z

=

1

foot

of

the

(B) 2x – 4y + 3z = 29 (C) 2x + 4y – 3z = 29

(D) none of these of

x  1

and

2

lines y  2

=

3

Q.95

The equation of a plane which passes through (2, – 3, 1) and is normal to the line joining the points (3, 4, –1) and (2, – 1, 5) is given by(A) x + 5y – 6z + 19 = 0 (B) x – 5y + 6z – 19 = 0 (C) x + 5y + 6z + 19 = 0 (D) x – 5y – 6z – 19 = 0

Q.96

If O is the origin and A is the point (a, b, c) then the equation of the plane through A and at right angles to OA is-

=

(B) (–1, –1, 1) (D) (–1, 1, –1)

=

y 8

1

=

z3 1

and

x 3

3

=

y  7 2

=

is

(A) a(x – a) – b (y – b) – c (z – c) = 0 (B) a(x + a) + b(y + b) + c (z + c) = 0

30

(C) a (x – a) + b (y – b) + c (z – c) = 0 (D) none of these

(B) 2 30 (D) 3 30



y2 2



x 1

z3

2

1



y2 2

Q.97



z3 3

and

are-

(B) intersecting at 60º (C) skew lines (D) intersecting at right angle

Different forms of the plane

If from a point P(a, b, c) perpendicular PA and PB are drawn to yz and zx planes, then the equation of the plane OAB is(A) bcx + cay + abz = 0 (B) bcx + cay – abz = 0 (C) bcx – cay + abz = 0

(A) parallel lines

Question based on

the

(A) 2x – 4y – 3z = 29

is-

The straight lines

2

of

perpendicular drawn from the origin to a

(B) 5 (D) none of these

(C) 5 30

x 1

co-ordinates

is-

The shortest distance between the lines x  3

The

plane is (2, 4, –3). The equation of the plane

(A) (–1, –1, –1) (C) (1, –1, –1)

Q.92

4

=

Distance between two lines and Intersection point

x  4

Q.91

7

y  3

=

is-

(A) 3 (C) 7 Question based on

x  5

(D) –bcx + cay + abz = 0 Q.98

The equation of a plane which cuts equal intercepts of unit length on the axes, is(A) x + y + z = 0 (B) x + y + z = 1 (C) x + y – z = 1

(D)

x a

+

y a

+

z a

= 1


8

Q.99

The plane ax + by + cz = 1 meets the co-ordinate axes in A, B and C. The centroid

Q.106 The value of k for which the planes 3x – 6y – 2z

= 7 and 2x + y – kz = 5 are perpendicular to

of the triangle is(A) (3a, 3b, 3c) (C)

 3 , 3 , 3     a b c 

 a b c   , ,   3 3 3   1 1 , 1  (D)  ,   3a 3 b 3c 

each other, is(A) 0 (B) 1

(B)

(B) y = 0

(C) z = 0

(D) x + y + z = 0

(D) 3

Q.107 The equation of the plane passing through the

point (–1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0, is(A) 7x – 8y + 3z – 25 = 0

Q.100 The equation of yz-plane is-

(A) x = 0

(C) 2

(B) 7x – 8y + 3z + 25 = 0 (C) –7x + 8y – 3z + 5 = 0 (D) 7x – 8y – 3z + 5 = 0

Q.101 If the length of perpendicular drawn from

origin on a plane is 7 units and its direction

Q.108 The equation of the plane through (1, 2, 3)

ratios are –3, 2, 6, then that plane is(A) –3x + 2y + 6z – 7 = 0

and parallel to the plane 2x + 3y – 4z = 0 is-

(B) –3x + 2y + 6z – 49 = 0 (C) 3x – 2y + 6z + 7 = 0 (D) –3x + 2y – 6z – 49 = 0

(B) 2x + 3y + 4z + 4 = 0 (C) 2x – 3y + 4z + 4 = 0

(A) 2x + 3y + 4z = 4

(D) 2x + 3y – 4z + 4 = 0

Q.102 A plane meets the coordinate axes at A, B and

Q.109 The equation of the plane passing through

C such that the centroid of the triangle is (3,

(1, 1, 1) and (1, – 1, –1) and perpendicular to 2x – y + z + 5 = 0 is(A) 2x + 5y + z – 8 = 0

3, 3) . The equation of the plane is(A) x + y + z = 3 (B) x + y + z = 9

(B) x + y – z – 1 = 0 (C) 2x + 5y + z + 4 = 0

(C) 3x + 3y + 3z = 1 (D) 9x + 9y + 9z = 1

(D) x – y + z – 1 = 0

Q.103 The direction cosines of any normal to the

xz-plane is(A) 1, 0, 0 (C) 1, 1, 0 Question based on

of planes x + 2y + 3z = 4 and 2x + y – z = – 5 & perpendicular to the plane 5x + 3y + 6z + 8 = 0 is(A) 7x – 2y + 3z + 81 = 0

Angle between two planes

the angle between the 2x – y + z = 6 and x + y + 2z = 3 is(A)  / 3

(B)  / 6

(C)  / 2

(D) 0

planes

Q.105 The equation of the plane which is parallel to

y-axis and cuts off intercepts of length 2 and 3 from x-axis and z-axis is(A) 3x + 2z = 1 (B) 3x + 2z = 6 (C) 2x + 3z = 6

(D) 3x + 2z = 0

Intersection of two planes

Q.110 The equation of the plane through intersection

(B) 0, 1, 0 (D) 0, 0, 1

Q.104 Find

Question based on

(B) 23x + 14y – 9z + 48 = 0 (C) 51x + 15y – 50z + 173 = 0 (D) None of these Q.111 The equation of the plane containing the line

of intersection of the planes 2x – y = 0 and y – 3z = 0 and perpendicular to the plane 4x + 5y – 3z – 8 = 0 is(A) 28x –17y + 9z = 0 (B) 28x + 17y + 9z = 0 (C) 28x – 17y – 9z = 0 (D) 7x – 3y + z = 0


9


Q.119 Image point of (1, 3, 4) in the plane

line of intersection of the planes x + y + z = 1

2x – y + z + 3 = 0 is -

and 2x + 3y – z + 4 = 0 and parallel to x-axis is(A) y – 3z – 6 = 0 (B) y – 3z + 6 = 0

(A) (–3, 5, 2) (C) (3, – 5, 3)

(C) y – z – 1 = 0

(B) (3, 5, – 2) (D) none of these

(D) y – z + 1 = 0


Question based on

Bisector of angle between two planes

intersection of the planes x + y + z = 6 and 2x + 3y + 4z + 5 = 0 and the point (1, 1, 1), is(A) 20x + 23y + 26z – 69 = 0

Q.120 The equation of the plane which bisects the

angle between the planes 3x – 6y + 2z + 5 = 0

(B) 20x + 23y + 26z + 69 = 0 (C) 23x + 20y + 26z – 69 = 0 (D) none of these Question based on

and 4x – 12y + 3z – 3 = 0 which contains the origin is(A) 33x – 13y + 32z + 45 = 0 (B) x – 3y + z – 5 = 0

Length & foot of perpendicular & image of the point w.r.t.plane

(C) 33x + 13y + 32z + 45 = 0 (D) 67x – 162y + 47z + 44 = 0

Q.114 Distance of the point (2, 3, 4) from the plane

3x – 6y + 2z + 11 = 0 is(A) 1 (B) 2 (C) 3

Question based on

(D) 0

Q.115 The distance between the planes

x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

Q.121 Equations of the line through (1, 2, 3) and

parallel to the plane 2x + 3y + z + 5 = 0 are (A)

is (A)

Line and Plane

7 2 2

(C)

7 2

(B)

(D)

7

(B)

2 7 2 2

Q.116 If the product of distances of the point

(1, 1, 1) from the origin and the plane x – y + z + k = 0 be 5, then k = (A) –2 (B) –3 (C) 4

(D) 7

(C) (D)

x 1

1 x  1 2 x  1 3

= =

=

x  1

=

y  2

=

1 y  2 3 y  2 2 y  2

=

= =

z3

1 z3 1

z3 1 z3

1 2 1 Q.122 The co-ordinates of the point where the line joining the points (2, –3, 1), (3, –4, –5) cuts the plane 2x + y + z = 7 are(A) (2, 1, 0) (B) (3, 2, 5) (C) (1, –2, 7) (D) None of these

Q.117 The equation of the plane which is parallel to

the plane x – 2y + 2z = 5 and whose distance from the point (1, 2, 3) is 1, is(A) x – 2y + 2z = 3 (B) x – 2y + 2z + 3 = 0 (C) x – 2y + 2z = 6

(D) x – 2y + 2z + 6 = 0

Q.118 The length and foot of the perpendicular from

the point (7, 14, 5) to the plane 2x + 4y – z = 2, are(A)

21 , (1, 2, 8)

(C) 21 3 , (1, 2, 8)

(B) 3 21 , (3, 2, 8) (D) 3 21 , (1, 2, 8)

Q.123 Equations of the line through (1, 1, 1) and perpendicular to the plane 2x + 3y – z – 5 = 0 are-

(A) (B) (C)

x  1

=

2 x  1 2

=

x  1

=

y  1 3 y  1 3 y 1

= =

z  1 1 z 1

1 z  1

= 2 1 1 (D) None of these


10

Q.124 The angle between the line

=

z2 2

  4   406     4   sin –1   406 

Q.125 The

point y  1

x

3

=

y  1 4

and the plane 2x – 3y + z + 4 = 0 is-

(A) cos –1  (C)

x  1

of

= = 1 2 3 is(A) (0, 1, –2)

the line

x  1

=

2

y 1

2

=

(A) 4x – y – 2z + 6 = 0

(D) None of these

(C) 4x – y – 2z – 6 = 0

4

z2 3

, is-

(B) 4x – y + 2z + 6 = 0 (D) none of these

intersection

z2

points (3, 2, 2) and (1, 0, –1) and parallel to

  406 

(B) tan –1

  

equation of the plane plane passing through the Q.129 The equation

of

the

line

& the plane 2x + 3y + z = 0

Q.130 The point where the line

x  1 2

=

y2

3

=

z3 4

meets the plane 2x + 4y – z = 1, is(B) (1, 2, 3)

(C) (–1, 9, –25)

(D)

  1 , 9  25     11 11 11 

(A) (3, –1, 1)

(B) (3, 1, 1)

(C) (1, 1, 3)

(D) (1, 3, 1)

Q.131 The line drawn from (4, –1, 2) to the point Q.126 The equation equation of the plane plane passing through the the origin and perpendicular to the line x = 2y = 3z is(A) 6x + 3y + 2z = 0 (B) x + 2y + 3z = 0 (C) 3x + 2y + z = 0 (D) none of these

(–3, 2, 3) meets a plane at right angles at the point (–10, 5, 4), then the equation of plane is(A) 7x – 3y – z + 89 = 0 (B) 7x + 3y + z + 89 = 0 (C) 7x – 3y + z + 89 = 0 (D) none of these

equation of a line line and a plane plane be Q.127 If the equation x3 2

=

y4 3

=

z5 2

and 4x – 2y – z = 1

respectively, respectively, then(A) line is parallel to the plane (B) line is perpendicular to the plane (C) line lies in the plane (D) none of these

Q.132 The line

x  2 3

=

y  3 4

=

z4 5

is parallel to

the plane(A) 2x + 3y + 4z = 29 (B) 3x + 4y – 5z = 10 (C) 3x + 4y + 5z = 38 (D) x + y + z = 0

Q.128 The equation equation of the plane plane passing through the

lines =

z 5

x  4 1

=

y  3 1

=

z2 2

&

x  3 1

=

y2

4

Q.133 The distance between the line

x  1 3

is-

(A) 11x – y – 3x = 35 (B) 11x + y – 3z = 35

=

y2

2

=

z  1 2

&

the plane 2x + 2y – z = 6 is(A) 9

(B) 1

(C) 2

(D) 3

(C) 11x – y + 3z = 35 (D) none of these 2624522 3 / 09 ; .www.iitianspace.com .www.iitians pace.com IIT - ian’s PACE ; ANDHERI / DADAR / CHEMBUR / THANE ; Tel : 26245223

11

Q.134 The angle between the line x2 a

=

y2 b

=

z2 c

and the plane

ax + by + cz + 6 = 0 is-

  

(A) sin –1 

  2 2 2  a  b  c  1

(B) 45º (C) 60º (D) 90º Q.135 The angle between the line

x  1 2

=

y  2 1

=

z3

2

and the plane

x + y + 4 = 0, is(A) 0º

(B) 30º

(C) 45º

(D) 90º

Q.136 The equation of the plane containing the line

x 1

3

=

y  3 2

=

z2 1

and the point (0, 7, – 7)

is(A) x + y + z = 1

(B) x + y + z = 2

(C) x + y + z = 0

(D) none of these


12

LEVEL- 2 Q.1

The cosines of the angle between any two

Q.7

diagonals of a cube is-

Three lines with direction ratios 1, 1, 2; 3 – 1, 1, –

3 – 1, 4 ; –

3 – 1,

(A) 1/3

(B) 1/2

enclose-

(C) 2/3

(D) 1/ 3

(A) an equilateral equilateral triangle

3 – 1, 4,

(B) an isosceles triangle Q.2

A point moves in such a way that sum of

(C) a right angled triangle

square of its distances from the co-ordinate

(D) a right angled isosceles triangle

axis are 36, then distance of these given point

Q.8

from origin are-

The distance of the point (–1,–5,–10) from the

(A) 6

(B) 2 3

(C) 3 2

line

point x  2

(D) None of these

3

=

of

y  1 4

=

intersection

z2 12

of

and plane x – y + z

= 5 isQ.3

(A) 13

If co-ordinates co-ordina tes of points A and B are (3, 4, 5) and (–1, 3, –7) respectively, then the locus of Q.9

P such that PA2 – PB2 + 2k 2 = 0 is(A) 8x + 2y + 24z = 2k 2 – 9 (C) 8x + 2y – 24z = 2k 2 (D) 8x + 2y – 24z + 9 = 2k 2

Q.10

 1 1 1  If the direction cosines of a line are  , ,  ,  c c c  (A) c > 0

(B) c = ±

(C) 0 < c < 1

(D) c > 2

3

The co-ordinates of points A, B, C, D are (a, 2, 1), (1, –1, 1), (2, –3, 4) and (a + 1, a + 2, a +

are

3) respectively. If AB = 5 and CD = 6, then

vertices

of

a

triangle,

then

its

a=

(A) (1, 4, 3)

(B) (–1, 4, –3)

(C) (1, – 4, 3)

(D) none of these

(A) 2 Q.11

(A) (C)

2 3 2 3

, ,

2 1 3 2 3

,

,

3

1 3

(B)

2 3

,

2 3

,

1 3

2 2 1 (D) – , , 3 3 3

The graph of the equation x2 + y 2 = 0 in three dimensional space is(A) x-axis

(B) y-axis

(C) z-axis

(D) xy-plane

(C) – 2

(D) – 3

The number of straight lines are equally equally axes, is(A) 2

line so directed that the angle made by it with positive direction direction of x-axis is acute, acute, are -

(B) 3

inclined to the three dimensional co-ordinate

A line passes through the points (6, –7, –1) and (2, –3, 1). The direction cosines of the

Q.6

(D) 21

If A(3, 2, – 5), B(– 3, 8, – 5) and C(– 3, 2, 1) circumcentre circumcentre is-

Q.5

(C) 8

then-

(B) 8x + 2y + 24z = 2k 2

Q.4

(B) 10

Q.12

(B) 4

(C) 6

(D) 8

The acute angle between the line joining the point (2, 1, –3), (–3, 1, 7) and a line parallel

x 1

to

3



y 4



z 3 5

through

the

point (–1, 0, 4) is-

  5

  10 

(B) cos 1 

 3    5 10 

(D) cos 1 

(A) cos 1  (C) cos 1 

7

 

  10  1

 1    5 10 


13

Q.13

The

point

x 5

y7



3

of intersection of the lines

1

z2 x 3 , 1  36





y3 2



z6 Q.19

(A)

  21, 

7

The equations of the line passing through the point (1, 2, –4) and perpendicular to the two

5 10  ,  3 3 

(B) (2, 10, 4)

(C) (– 3, 3, 6)

x 8

lines

(D) (5, 7, –2)

3

x  15

(3, 5, –3), (1, 2, 3), (3, 5, 7) respectively, then

(A)

x 1 2

the angle between AB and CD is -

Q.15

 2

(B)

 3

(C)

 4

(D)



(B)

6

2

cosines are given by  + m + n = 0,  + m – n

2

= 0 is(A) /3 Q.16

Q.17

(C)

The angle between two lines whose direction 2

(B) /6

(C) 5/6

(D) 2/3

x 1

2 x 1 3

y  19



 16

y  29 8

  





y2 3 y2 3 y2 2

z  10 7

z 5



and

will be-

5 z4

 

6 z4 8 z4 8


Equation of the plane through (3, 4, –1) which is  ˆ ) +7 = 0 parallel to the plane r . ( 2î  3 jˆ  5k

If the points (1, 1, k) and (–3, 0, 1) be

is-

equidistant from the plane 3x + 4y – 12z + 13

 ˆ ) + 11 = 0 (A) r . ( 2î  3 jˆ  5k

= 0, then k =

 ˆ ) + 11 = 0 (B) r . (3î  4 jˆ  k

(A) 0

(B) 1

(C) 2

(D) None of these

 ˆ ) + 7 = 0 (C) r . (3î  4 jˆ  k  ˆ ) –7 = 0 (D) r . ( 2î  3 jˆ  5k

The equation of the line passing through (1, 2, 3) and parallel to the planes x –y + 2z = Q.21

5 and 3x + y + z = 6, is(A) (B) (C)

x 1

 3 x 1  3 x 1  3

y2 5 y2

z3

 

z 1



0



Q.22

3 4



n,

the

length

of

the



(B) | n |



(D) q/| n |

Equation of the plane through three points ˆ, A, B, C with position vectors – 6î  3 jˆ  2k ˆ , 5î  7 jˆ  3k ˆ is3î  2 jˆ  4k

z 1 4

, then its perpendicular

distance from the origin is (A)

vector



(C) q | n |

perpendicular to the line 3

the

(A) q

4 5 y  2 z 1  5 4

y 1



perpendicular from the origin on the plane is

4

If a plane passes through the point (1, 1, 1) and is x 1



If r . n = q is the equation of a plane normal to




3

If A, B, C, D are the points (2, 3, –1),

(A)

(D) 1

5

4

is

Q.14

(C)

(B)

4

 ˆ ) + 23 = 0 (A) r . (î  jˆ – 7k  ˆ ) = 23 (B) r . (î  jˆ  7k  ˆ ) + 23 = 0 (C) r . (î  jˆ  7k  ˆ ) = 23 (D) r . (î  jˆ  7k

3


14

Q.23



ˆ ) & The lines r = î  jˆ  (2î  k

(C)



ˆ) r = 2î  jˆ (î  jˆ  k

(D)

(A) intersect each other (B) do not intersect

x  2

=

y 3

=

2 x  2 y 3 = = 1 2 1

z4 1 z4 1

 ˆ (C) intersect at r = 3 î  jˆ k

(D) are parallel Q.24

Equation of the plane containing the lines. 

ˆ   ( î  2 jˆ  k ˆ ) and r = î  2 jˆ  k



ˆ   ( î  jˆ  3k ˆ ) isr = î  2 jˆ  k

 ˆ ) = 0 (A) r . (7 î  4 jˆ  k

(B) 7(x – 1) – 4(y – 1) – (z + 3) = 0  ˆ) = 0 (C) r . ( î  2 jˆ  k  ˆ) = 0 (D) r . ( î  jˆ  3k

Q.25

The Cartesian equation of the plane passing through the line of intersection of the planes 

ˆ ) = 1 & r . ( î  jˆ) + 4 = 0 and r . ( 2 î  3 jˆ  4k

 ˆ) + 8 perpendicular to the plane r . ( 2 î  jˆ  k

= 0 is(A) 3x – 4y – 4z = 5 (B) x – 2y + 4z = 3 (C) 5x – 2y – 12z + 47 = 0 (D) 2x + 3y + 4 = 0

Q.26

x  3

If the line

2

=

y  5 k

=

z  1 2k

is parallel to

the plane 6x + 8y + 2z – 4 = 0, then k

Q.27

(A) 1

(B) –1

(C) 2

(D) 3

The equation of a line through (–2, 3, 4) and parallel to the planes 2x + 3y + 4z = 5 and 3x + 4y + 5z = 6 are(A) (B)

x  2 1 x  2 2

= =

y  3 2 y  3 3

= =

z4

1 z4 1


15

LEVEL- 3 Q.1

A plane is such that the foot of perpendicular

If the foot of perpendicular from the point

drawn from the origin to it is (2, –1, 1). The

(1, –5, –10) to the plane x – y + z = 5 is

distance of (1, 2, 3) from the plane is-

(a, b, c) then a + b + c =

(A)

3 2

(B)

(C) 2 Q.2

Q.7

3

(A) 10

(B) –10

2

(C) 11

(D) –11

(D) None of these

 both with axes. A possible value of  is    (A) 0,  (B) 0,   4  2 A line makes an angle

(C)

 ,   4 2 

(D)

Q.8

the point (2, –1, 3) measured in the direction

x and y-

 ,   3 6 

The distance of the plane x + 2y – z = 2 from with d.r.’s 2, 2, 1 is-

Q.9

(A) 1

(B) 2

(C) 3

(D)

5 6

A variable plane makes with coordinate planes a tetrahedron of unit volume. The

Q.3

locus of the centroid of the tetrahedron is-

If the plane x + y + z = 1 is rotated through 90º about its line of intersection with the

(B) xyz =

(C) x + y + z = 6

(D) x3 + y3 + z3 = 3

pl a ne x – 2y + 3z = 0, the new position of the plane

3

(A) xyz = 6

32

is(A) x – 5y + 4z = 1

(B) x – 5y + 4z = – 1

(C) x – 8y + 7z = 2

(D) x – 8y + 7z =– 2

Q.10

An equation of the plane passing through the origin

and

containing

the

lines

whose

direction Q.4

The shortest distance between the lines

cosines are proportional to 1, –2, 2 & 2, 3, –1



is-



(A) x – 2y + 2z = 0

(B) 2x + 3y – z = 0

(C) x + 5y – 3z = 0

(D) 4x – 5y – 7z = 0

ˆ ) +  (2 î + 3 jˆ + 4 k ˆ ) and r = – ( î + jˆ + k ˆ ) isr = – î +  (3 î + 4 jˆ + 5 k

(A) 1

(B)

1 2

(C)

1 3

(D)

1 6

Q.11





Q.5









The angle between a diagonal of unit cube

r = b + ( c × a ) will intersect if

and an edge is-

(A) a × c = b × c (B) a . c = b . c

(A)

cos –1

(C) sin –1

Q.6





The lines r = a +  ( b × c ) and

1 3 1 3

(B)

cos –1

(D) tan –1





1













(C) b × a = c × a





 

(D) None of these

3 1 3

If A = (0, 1, –2), B = (2, –1, 0), C = (1, 2, 3), then

Q.12

If  denotes the acute angle between the line 

ˆ )   ( î  jˆ  k ˆ ) and r  ( î  2 jˆ  k

the

plane



ˆ ) = 4, then sin  + 2 cos  = r .( 2 î  jˆ  k

a bisector of angle BAC has direction ratios(A) 1, 1, 1

(B) 1, 1, –1

(A) 1 / 2

(B) 1

(C) 0, –1, 1

(D) None of these

(C)

(D) 1  2

2


16

Q.13

Direction ratios of the line x – y + z – 5 = 0

Q.19

= x – 3y – 6 are-

from the origin and meets the axes in A, B

(A) 3, 1, –2 3

(C)

14

,

(B) 2, –4, 1

1 14

,

A variable plane is at a constant distance p

2 14

(D)

2 41

,

and C. The locus of the centroid of the

4 41

,

1

tetrahedron OABC is-

41

(A) x –2 + y –2 + z –2 = 16p –2 (B) x –2 + y –2 + z –2 = 16p –1

Q.14

The

distance 

ˆ)  line r  ( î  jˆ  2k

between

the

ˆ) &  (2 î  5 jˆ  3k

the

(D) None of these



Q.20

ˆ ) =5 is plane r .( 2 î  jˆ  3k 5

(A)

14 7

(C)

Q.15

14

(B)

(D)

(C) x –2 + y –2 + z –2 = 16

ay pass through one line, if 6

(A) a + b + c = 0

14

(B) a + b + c = 1

8

(C) a2 + b2 + c2 = 1

14

(D) a2 + b2 + c2 + 2abc = 1

Volume of the tetrahedron included between

Q.21

origin meets the co-ordinates axes in A, B, C.

coordinate planes is-

Through these points planes are drawn

(B) 6

(C) 18

parallel to co-ordinate planes. Then locus of

(D) 12

the point of intersection is-

,

If A(3, –4, 7), B(0, 2, 5), C(6, 3, 2) and D(5, 1,

(A)

4) are four given points (Projection of AB on CD ) : (projection of CD on AB ) is-

Q.17

(A) 3 : 7

(B) 7 : 3

(C) 4 : 5

(D) 5 : 6

The points on the line distant

x  1 1

Q.22

=

y  3 3

=

(D) (2, 6, –4), (3, –4, –5) Distance of the point (0, 1, 2) from the plane 2x – y + z = 3 measured parallel to the line 1

1  1

(A) 0 (C)

z

is equal to(B) 3 3

3

y

2



1 z

2



1 p

2

(D) None of these

(B) x2 + y2 + z2 = p2

(D)

1 x

1

1

y

z

   p

The equation of the planes passing through 4z = 0 and x + 3y + 6 = 0 whose distance from the origin is 1, are-

(14) from the point in which the line

=

1

2

(A) x – 2y – 2z – 3 = 0, 2x + y – 2z + 3 = 0 (B) x – 2y + 2z – 3 = 0, 2x + y + 2z + 3 = 0 (C) x + 2y – 2z – 3 = 0, 2x – y – 2z + 3 = 0 (D) None of these

(C) (0, 0, 0), (2, 6, –4)

y



the line of intersection of the planes 3x – y –

(B) (0, 0, 0), (3, –4, –5)

=

x

2

z2

(A) (0, 0, 0), (2, –4, 6)

x

1

(C) x + y + z = p

meets the plane 3x + 4y + 5z – 5 = 0 are-

Q.18

A variable plane at a constant distance p from

the plane 2x – 3y – z – 6 = 0 and the (A) 3 Q.16

The planes x = cy + bz, y = az + cx, z = bx +

Q.23

The

lines

x  b  c





x a d

 y  b







ya



z  b  c





z a d



and

are coplanar and

then equation to the plane in which they lie, is(A) x + y + z = 0

(B) x – y + z = 0

(C) x – 2y + z = 0

(D) x + y – 2z = 0


17

Q.24

If P1 and

P2 are

the

lengths

of

the

Q.28



perpendiculars from the points (2, 3, 4) and







 

always intersects.

(A) P2 – 23P + 7 = 0 (B) 7P2 – 23P + 16 = 0 Q.29

Statement- 1 (A) : If lines x = ay + b, z = 3y

+ 4 and x = 2y + 6, z = ay + d are

ratio in which the line segment joining

(2, 4, 5) and (3, 5, –4) is divided by the yz-

perpendicular to each other then a = 1/5

plane is 2 : 3.

Statement- 2 (R) : If two lines with d.rs a 1, b1,

II.The

c1 and a2, b2, c2 are perpendicular then

line joining (x 1, y1, z 1) and (x2, y2, z 2) is

a1a2 + b1 b2 + c1c2 = 0

divided by xy-plane in the ratio –z 1 : z2. Which of the statement is true?

(A) both I and II

(B) only I

(C) only II

(D) neither I nor II

Q.30

Statement- 1 (A) : The line of intersection of

the planes 2x + 3y + z = 10 and x + 3y + 2z = ˆ 5 is parallel to vector î  jˆ  k

Statement type questions

Statement- 2 (R) : The line of intersection of

Each of the questions given below consists of

two

Statement -I and Statement- II. Use the following



r .n 2

key to choose the appropriate answer. (A) If both Statement- I Statement- II are true, and Statement- II is the correct explanation of Statement- I. (B) If Statement- I and Statement-II are true but Statement-II is not the correct explanation of Statement- I (C) If Statement- I is true but Statement- II is false (D) If Statement- I is false but Statement- II is true. Q. 26 Statement-1 (A) : The angle between the rays

of with d.r's (4, –3, 5) a nd (3, 4, 5) is /3. Statement-2 (R) The angle between the rays

whose d.c's are 1, m1, n 1 and 2, m2, n 2 is given by  whose cos  = 12 + m1 m2 + n1n2 Q.27



Statement- 2 (R) : Two coplanar lines

equation-

I. The



then (c  a).{ b  d}  0

2z + 11 = 0, then P 1 and P2 are the roots of the

Q.25



and r  c  d intersects at a point

(1, 1, 4) respectively from the plane 3x – 6y +

(C) P2 – 17P + 16 = 0 (D) P2 – 16P + 7 = 0



Statement- 1 (A) : If the lines r  a  b

Statement 1 (A) : A line makes 60º with x-

axis and 30º with y-axis then it makes 90° with z-plane. Statement 2 (R) :

If

a

ray

makes

angles

, , 

with

x-axis, y-axis and z-axis respectively then sin2  + sin2 + sin2  = 1

Q.31

non

parallel



planes

  2 is always parallel to

List-I (P) The points (–1, 0, 7) (3, 2, –k) and (5, 3, –2)



r .n1  1 and 

n1  n 2

List-II (1) 22/7

are collinear then k = (Q) The length of the (2) 1 projection of the line segment joining the points (–1, 0, 3) and (2, 5, 1) on the line whose d.r's are 6, 2, 3 is (R) The distance of the point (3) –1 (1, –2, 8) from the plane 2x – 3y + 6z = 63 is (S) The distance between the (4) 1/6 parallel planes 2x – 2y + z +3 = 0, 4x – 4y + 2z + 5 = 0 Correct match for List-I from List-II is P Q R S (A) 1 4 5 3 (B) 3 1 2 4 (C) 2 5 1 2 (D) 4 2 3 1


18

Passage based questions Passage-1

Consider the line

x  1 2

=

y

1

=

z  1 2

and the

point C(–1, 1, 2). Let the point D be the image of C in the line. Q.32

The distance of C from the line is (A) (C)

Q.33

5 3 4

3

(B) 5

(D)

2

5

3 5

5

3

The distance of the origin from the plane through C and the line is 1 2 (A) (B) 5 5 3 4 (C) (D) 5 5

Q.34

The distance of D from the origin is (A)

15

(B)

21

(C)

26

(D)

30


19

ANSWER KEY LEVEL- 1 Qus.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

A

D

A

A

A

C

B

A

D

D

C

D

A

C

D

B

D

A

C

A

Qus.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

Ans.

B

C

B

A

B

A

A

A

C

C

A

C

C

B

A

B

C

B

C

B

Qus.

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

Ans.

B

B

C

B

A A,C

A

D

B

A

D

A

A

C

D

D

C

B

C

A

Qus.

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

Ans.

D

B

C

B

B

C

D

A

A

A

B

C

A

C

B

A

B

D

A

B

Qus.

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99 100

Ans.

B

A

D

B

D

C

A

C

D

A

D

D

A

C

A

C

B

B

D

A

Qus. 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 Ans.

B

B

B

A

B

A

B

D

B

C

A

B

A

A

A

C

C

D

A

D

Qus. 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 Ans.

A

C

B

C

D

A

A

D

D

A

A

B

D

D

C

C

LEVEL- 2 Qus.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

A

C

A

B

A

C

A

A

B

D

B

A

A

A

D

B

A

C

A

A

Qus.

21

22

23

24

25

26

27

Ans.

D

A

B

A

C

B

C

LEVEL- 3 Qus.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Ans.

B

C

D

D

B

D

D

C

B

D

B

C

A

D

B

B

C

C

A

D

Qus.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

Ans.

A

A

C

B

C

B

C

C

D

A

B

D

A

C


20

IIT – ian’s

P A C E



LEVEL –1 Q.7

Expansion of Determinants

Determinants The minors of the elements of the first row in 2 1 4 the determinant

a 1 1

Q.1

1

1

1

 1 = 4, then the value a is -

1

1

1

If (A) 1

(B) –1

(C) –2

If

x

y

4

2

= 7 and

(A) x = – 3, y = –

The value of (A) 12

Q.4

3

y

x

(C) x =

5i 4i

(B) 17

sec x

sin x

tan x

0

1

0

tan x

cot x

sec x

Q.6

2

1 xy

5

x2 +

y3

Q.9

(D) 24

xy +

(D)

1

2

3

4

–2,

y3

–3 and 4 in

are-

(A) 4, 3, 2, 1 (C) 4, –3, –2, 1

c3


If A = (aij) is a 4 × 4 matrix and cij is the co-

equals(A) 0 (C) 1

(B) –4, 3, 2, –1 (D) –4, –3, –2, –1

If

(B) – 1 (D) Det. (A)

cofactor of x 1 2 1

2x

x 1

x

(A) 0 (C) 1

1

Minors & Cofactor and their properties

The cofactors of 1,

c 2 and A2, B2, C2 are

expression a11c11+ a12c12+ a13c13 + a14c14

1 is -

x3 –

= a 2 b 2

factor of the element a ij in Det (A), then the

(B) x2 – xy + y2 y2



(A) –  (C) 

0

3 x3

If

c1

a1A2 + b1B2 + c1C2 is equal to-

Q.10

0

2

respectively cofactors of a 2, b2, c2 then

is equal to -

1

1

(B) 7, 11, 2 (D) 7, 2, 11

a 3 b 3

, y = 3

(C) 14

(A) x + y

Question based on

Q.8

(B) – 1 (D) None of these

The value of

(C)

5

 3i is 5i

(A) 0 (C) 1

Q.5

= 4, then -

5 (B) x = – , y = – 3 2 2

2

1

 3 are-

(D) 0

5

5

(C) x = 3, y =

Q.3

2

2

(A) 2, 7, 11 (C) 11, 2, 7 a1 b1

Q.2

4

Question based on

Q.11

2x

in

the

determinant

x  1 is zero, then x equals to0

(B) 2 (D) –1

Some basic properties

a1

ma1 b1

The value of the determinant a 2

ma 2 b 2

a3

ma 3 b 3

is (A) 0

(B) ma1a2a3

(C) ma1 b2a2

(D) mb1 b2 b3


1

a

Q.12

If



= b c

0 c

p 2a

0 a , then

a b

pb pc

to(A) p (C) p3

c

a is equal

Q.19

If Dr =

The value of the determinant 1 / b 1 ca is 1 / c 1 ab

Q.14

n (n  1) / 2

2r  1

y

n2

z

n(3n  1) / 2

Q.20

Q.16

If each row of a determinant of third order of value  is multiplied by 3, then the value of new determinant is (A)  (B) 27  (C) 21  (D) 54 

Q.21

The sum of infinite series 1 2 1/ 2 2 1/ 4 2 + + + ........ is 6 4 2 4 2/3 4

If

ax

ax

ax

ax

a  x = 0, then value of x

ax

ax

ax

(A) 0, a

(B) 0, – a

(C) a, – a

(D) 0, 3a a

b

2

2

The value of the determinant a

a2

ma  nx

x

The value of b

mb  ny

y is-

c

mc  nz

z

The value of

Q.22

Q.18

The

2a

3a  2 b

3a

6a  3 b 10a  6 b  3c

ka kb kc

 a2 k 2  b 2 k 2  c 2 k 2

the

If (a  1) 2

c2

( b  1) 2

2

( b  1)

2

(A) 1 (C) 4

a  b

of

ca

ab

a 2 b 2

(c  1) 2 = k a (c  1)

2

1

c2

b

c ,

1

1

then k is equal to(B) 2 (D) 0 a b  c

a  b  c 4a  3 b  2c is

Q.23

1 is 1

(A) k (a + b) (b + c) (c + a) (B) k abc (a 2 + b2 + c2) (C) k (a – b) (b – c) ( c – a) (D) k (a + b – c) (b + c – a) (c + a – b)

a3

c  a b 3 isc3

a  b

(A) (a – b) (b – c) (c – a) (B) abc (a – b) (b – c) (c – a) (C) – (a + b + c)2 (a – b) (b – c) (c – a) (D) None of these

determinant

1

The value of b c

(B) b3 (D) a3 + b3 + c3

value

b 2

(a  1)

a

equal to (A) a3 (C) c3

c 2 is -

(B) (a – b) (b – c) (c – a) (a + b + c) (C) (a – b) (b – c) (c – a) (ab + bc + ca) (D) None of these

(B) 0 (D)  a

b

c

(A) abc (a – b) (b – c) (c – a)

(A) a + b + c (B) x + y + z (C) m(a + b + c) + n(x + y + z) (D) 0

Q.17

r

r 1

are-

(B) 1/abc (D) None of these

(A) –10 (C) 10

 D

ax

bc

Q.15

n

, then

is equal to 1 1 (A) n(n + 1)(2n + 1) (B) n2(n + 1)2 6 4 (C) 0 (D) None of these

(B) p2 (D) 2p

equal to (A) abc (C) 0

x

3r  2

a b

1 / a 1 bc

Q.13

r

0 0

Q.24

If

x

is

real

x 1

x2

x

x2

x3

x   = 0 then  

x3

x4

x

(A) A.P. (C) H.P.

number

such

that

 are in



2

Q.25

The determinant

1

1

1

a

b

c

a 2  bc b 2  ca

Q.26

(B) 1

(C) – 1

(D) None of these

m m

C1

1

1

m 1

m2

m1

C2

C1

m2

C2

Find

the

1

4

20

1

2

5

1

2x

5x

the

=0

x 3

x  2a

x3

x4

x  2 b equals -

x4

x5

x  2c

(B) 0

i

(C) 2a

1  i (where i =

 1 ) equals -

a  b

3a

4a  3 b

6a

9a  6 b 11a  9 b  6c

value

( x  2) 2

(x  1) 2

( x  1) 2

x2

x2

(x  1) 2

(A) 0

6x (B) 3 (D) None of these

Symmetric and skew symmetric Determinants

If A + B + C =  , then sin (A  B  C) sin B

5a  4 b  3c where a = i,

(B) 8x2

the

determinant

x2 (x  1) 2 is(x  2) 2

(C) 8

tan A equals-

0

 tan A

0

(B) 2sinB tanA cosC (D) None of these

The value of an odd order skew symmetric determinant is(A) perfect square (B) negative (C) ±1 (D) 0

Q.37

The value of b  a ca

a  b 0 c  b

a c b  c is0

(A) 0 (B) abc (C) (a – b) (b – c) (c – a) (D) None of these

a  b  c

of

cos C

Q.36

(B) 7 – 4i (D) 4 – 7i

a

The

3

0

b =   c =  then  is equal to(A) i (B) – 2 (C)  (D) – i Q.31

3

= 0 then x =

3

The value of an even order skew symmetric determinant is(A) 0 (B) perfect square (C) ±1 (D) None of these

(D) 2x

i

(A) 7 + 4i (C) 4 + 7i

=

3

6

6 3 x

(D) 4

Q.35

1 i 1 i

i

(C) – 20

(A) 0 (C) 1

x2

1 i

If

(B) – 2

 sin B cos (A  B)

equation

If a, b, c are in A.P., then the value of

1 i 1 i

Q.30

in

(B) –1, 0 (D) 1, 2

(A) 1

Q.29

x

Q.34

2

(A) –1, 2 (C) 2, 0 Q.28

Question based on

C2

of

(A) 20

Q.33

=

7581 7591

(A) 6 (C) 0

(B) m(m – 1) (D) 0

value

7579 7589

3 x

C1 =

(A) m(m + 1) (C) 1 Q.27

c 2  ab

equal to (A) 0

1

is

Q.32

(D) –8

Question based on

Crammer's Rule

Q.38

The equations x + 2y + 3z = 1, 2x + y + 3z = 2 and 5x + 5y + 9z = 4 have(A) unique solution (B) many solutions (C) inconsistent (D) None of these

Q.39

The existence of unique solution of the system x + y + z = b, 2x + 3y – z = 6, 5x – y + az = 10 depends on(A) b only (B) a only (C) a and b (D) neither a nor b


3

Q.40

Given the system of equations px + y + z = 1, x + py + z = p, x + y + pz = p 2, then for what value of p does this system have no solution (A) –2 (B) –1 (C) 1 (D) 0

Q.41

The value of k for which the set of equations 3x + ky – 2z = 0, x + ky + 3z = 0 and 2x + 3y – 4z = 0 has a non – trivial solution is(A) 15 (B) 16 (C) 31/2 (D) 33/2


4

LEVEL – 2 a 2  b 2

c

c

Q.1

b

a

c

2

Q.6

2

(D) 0

33

43

3

3

4

5

3.2

9 , then

p

Q.7

ca

a  b

If the determinant b  c

c  a

a  b

2

a

(C) 92

(D) None of these ca

ca

ca

a  b =



a b

c

b

a then

a  b b  c

c

c

(B) 2

(C) 3

(D) 4

Q.8

Q.4

If

c

x

y

z and

p

q

r

 =

(C) 1

(D) 2

the first column consists of sum of two terms,

a b

each element of the second column consists of sum of three terms and each element of third column consists of sum of four terms, then it can be decomposed into determinants, where n has the value-

y b

2

(B) 0

In a third order determinant each element of

(A) 1 a b

c , then the

value of m is-

 is equal to(A) 1

c

a b c

(B) 1

a  b b  c

b

is expressible as m a b

(A) 0

If b  c

(B) 25 (D) None of these b  c

(A) – 1

Q.3

25 10

b  c c  a a  b

 3.2  1 2 3.3  3.3  1 is equal to3.4 2  3.4  1

3

35

3

(A) 0 (C) 625

(C) 4abc 2

If D p = p 2

b

(B) 2abc

3

8

D1 + D2 + D3 + D4 + D5 is equal to-

(A) abc

3

15

 a2

c2

b

p

is equal to-

a

a

b

Q.2

c

=

(B) 9

(C) 16

n

(D) 24

q

x

a p then

z

c

Q.9

r

1 is equal to-

For any

ABC,

sin 2 A

cot A 1

2

cot B 1 is-

2

cot C 1

sin B

(A) 22

(B) 2

(C) – 2

(D) None of these

the value of determinant

sin C

(A) 0 (B) 1

ax by

Q.5

(C) sin A sin B sin C

cz

The determinant x 2

y2

z 2 is equal to-

1

1

1

1

(A)

(C)

1

1

a

b

c

x2

y2

z2

1

1

1

x

y

z

2

2

2

a

b

c

a

(B)

x

b y

yz zx

c

(D) sin A + sin B + sin C

Q.10

If Sr =

x

6r 2  1

y

3

4r

z xy

2r

 2nr

n ( n  1) n 2 ( 2n  3)

z

then

n (n  1) 3

n

 S does not depends on r

r 1

(D) None of these

(A) x

(B) y

(C) n

(D) all of these


5

Q.11

If a, b, c are non-zero real numbers, then

Q.16

equal to a and – 1

b 2 c 2 bc b  c c2a 2 2 2

a b

Q.12

ca

c  a is equal to

ab a  b

(A) abc

(B) a2 b2c2

(C) ab + bc + ca

(D) None of these

x p

q

p

x

q is equal to -

p

q

x

Q.17

[ y]

[z ]

[x]

[ y]  1

[z ]

[x]

[ y]

[ z]  1

(B) [y]

(C) [z]

(D) None of these

The 5 5

value

of

C0

5

C3

C1

5

C4

1 is-

C2

5

C5

1

the

determinant

14

(C) (x – p) (x – q) (x – p – q)

(A) 0

(B) – (6!)

(D) (x + p) (x + q) (x + p + q)

(C) 80

(D) None of these

If x is a positive integer then the value of (x  1)! ( x  2)!

x!

a b

Q.18

If  = b

determinant ( x  1)! (x  2)! ( x  3)! is-

c

( x  2)! ( x  3)! ( x  4)!

c

(A) ca  b 2

(B) 2 (x) !. (x + 1) !. (x + 2) !

c a , then 2 is equal to-

a b

bc  a 2

(A) (2x) !. (x + 1) !. (x + 2) !. (x + 3) !

<

is equal to-

(A) [x]

5

(B) (x – p) (x – q) (x + p + q)

 x < 0, 0  y < 1, 1  z

[x ]  1

2, then

(A) (x + p) (x + q) (x – p – q)

Q.13

If [a] denotes the greatest integer less than or

ca  b 2

ab  c 2

ab  c 2 bc  a 2

ab  c 2 bc  a 2

ca  b 2

(C) (2x) !. (x + 3) ! bc  a 2

(D) None of these

(B) ca  b 2 Q.14

The determinant cos (  ) sin 

 cos 

 sin (  ) cos  sin 

cos 2 sin  is-

a 2  bc b 2  ca

c 2  ab

c 2  ab

a 2  bc

a 2  bc b 2  ca

(D) None of these

 (C) independent of  (B) independent of

Q.19

(D) independent of both

Q.15

ab  c 2 bc  a 2 ca  b 2

c 2  ab

(A) 0

ab  c 2

ab  c 2 bc  a 2

(C) b 2  ca

cos 

ca  b 2

If ax + by + cz = 1, bx + cy + az = 0 = cx + ay +

 and  bz, then

13  3

2 5

15  26

5

10 equals-

65  3

15

5

5

x

y

z

a b

z

x

y

c

y

z

x b

c

a b is equal toc

(A) 0

(B) 1

(C) – 1

(D) 2

a

(A) 0 (B) 5 3 ( 6 – 5) (C) 5 3 (5 – 6 ) (D) None of these


6

LEVEL – 3

Q.1

xn

x n 2

x n 3

If y n

y n2

y n 3

zn

z n 2

z n 3

Q.6

The values  and  for which the system of equations x + y + z = 6, x + 2y + 3z = 10 and x + 2y + z =  have unique solution are(A)   3,   R (B)   3,   10 (C)   3,   10 (D)   3,   10

Q.7

The system of linear equations x + y + z = 2, 2x + y – z = 3, 3x + 2y + kz = 4 has a unique solution if(A) k   (B) –1 < k > 1 (C) –2 < k < 2 (D) k = 0

 1 1 1     , then n =  x y z 

= (x–y) (y–z) (z–x)  (A) 1 (C) 2 Q.2

If

(B) –1 (D) –2

   are

the roots of x 3 + ax2 + b = 0,

   then the value of    is equals to   (A) – a3 (C) a3

Q.8

If

If A, B and C are the angles of a triangle and 1 1 1 1  sin A

1  sin B

sin A  sin 2 A

1  sin C

Q.9

3x  3 = Ax – 12,

2x  1 2x  1

cos 

Q.10

c

r

7(8 )

16 Σ D r is r 1

449

450

451

2(416  1) , 4(8

3x 3  81

2

3

f (1). f (3) + f (3). f (5) + f (5). f (1) is equal to(A) f (1) (B) f (3) (C) f (1) + f (3) (D) f (1) + f (5)

216  1 16

2x 2  18

1

Q.11

D r  b 3(4 r )

446 447 is-

If f (x) = x  5 2x 2  50 4x 3  500 then

1 is-

(D) none of these 2 r

445

x 3

 sin(  ) 1 (A) independent of  (B) independent of  (C) independent of  and 

a

The value of

443

(A) 441 × 446 × 451 (B) 0 (C) –1 (D) 1

cos(   )

value of

3x

441 442

The value of the determinant cos  1  sin 

Let

 3x  1  2x  3

sin B  sin 2 B sin C  sin 2 C

sin 

Q.5

2

x2

then the value of A is(A) 12 (B) 24 (C) –12 (D) –24

 0 ,

then the triangle ABC is(A) isosceles (B) equilateral (C) right angled isosceles (D) none of these Q.4

x 1

2x 2 x

Q.3

x

x2

(B) a3 –3b (D) a2 – 3b

then

the

If the system of equations, x + 2y – 3z =1, (k + 3)z = 3,(2k + 1)x + z = 0 is inconsistent, then the value of k is(A) –3 (B) 1/2 (C) 0 (D) 2

 1) 1 a b

equals to-

(A) 0 (B) a + b + c (C) ab + bc + ca (D) none of these

Q.12

In a ABC, if 1 c

a

1 b

c

sin2 A

+ (A) 9/4 (C) 1

sin2

B+

 0 then

sin2 C

is equal to(B) 4/9 (D) 3 3


7

Q.13

The equation x + 2y + 3z = 1, 2x + y + 3z = 2, 5x + 5y + 9z = 4 have(A) unique solution (B) infinitely many solutions (C) inconsistent (D) None of these 1

Q.14

1

cos ( nx ) cos ( n  1) x sin ( nx )

sin ( n  1) x

dependent(A) on x (C) both on x and n

1 cos ( n  2) x is

not

sin (n  2) x

(B) on n (D) None of these


8


1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

Ans.

D

B

C

C

C

A

B

B

D

C

A

B

C

B

A

D

A

C

C

D

C

Q.No.

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

Ans.

C

C

A

A

C

A

B

C

A

D

C

C

A

B

D

A

A

B

A

D

LEVEL- 2 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

Ans.

C

A

B

B

B

D

B

D

A

D

D

B

B

B

B

C

D

A

B

LEVEL- 3 Q.No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Ans.

B

C

A

A

A

A

A

B

B

B

A

A

A

B


9

IIT – ian’s

P A C E



Q.1

LEVEL –1

Theory till scalar multiplication of matrices

Q.7

3 4

of A has-

Q.2

(B) 4 elements

(C) 12 elements

(D) 7 elements

(A) 

In the following, upper triangular matrix is-

1 (A) 0 3

0

0

0

 0  3

0 (C)  0

2

3

2

5 (B) 0 0

0

 3  1

0 0

2 (D) 0 0

 4

2

4

Q.8

3

1



3

1 2 (C)   1 0   Q.4

3

(D) 



7 8

3



6 6



12

The scalar matr ix is3

 4 0



4

0 2

(B) 

3



0

(D) None of these

For any square matr ix A = [ai j], a i j = 0, when

i  j, then A is-

 Q.10

6

(A) unit matrix

(B) scalar matrix

(C) diagonal matrix

(D) none of these

A row matrix has only(A) one element

3B| equals-

(D) one row and one column (B) –53

(C) 53

(D) –77

(C) one column with one or more rows

Q.11

A matrix A = (a ij) m x n is said to be a square

If a matrix B is obtained by multiplying each

matrix if-

element of a matrix A of order 2 × 2 by 3,

(A) m = n

(B) m  n

then relation between A and B is-

(C) m  n

(D) m < n

(A) A = 3B

(B) 3A = B

(C) 9A = B

(D) A = 9B

 If  x

 2

7

   1 –  3   6  2 1

2



4 

5 = 4   y 

0

 then

3

(B) one row with one or more columns

5

Q.6

2 4

6 2 (C)   8 6

(C) 

3

(D) 

2

5 2 2 3  If A =  and B =    , then |2A – 1 0 5  1 (A) 77

Q.5

2

4 4

(B) 

4 0

 3  0

3 (B)  2

1 and A – B =   9 0

 3

 1 2

In the following, singular matrix is-

2 (A)  1

4

1

(A) 

Q.9 Q.3

7 8

If A + B = 

the value of A is-

If A is a matrix of order 3 × 4, then each row (A) 3 elements

Matrices

  4 then4  5

(A) x = 1, y = – 2

(B) x = – 1, y = 2

(C) x = 1, y = 2

(D) x = – 1, y = – 2

Q.12

In the following, diagonal matrix is-

0 4

(A) 

3

 0

1 0 0 (C)   0 0 1

1 0

0

3 0

0

(B) 

(D) 

0



2 0



4


1

Q.13

If every row of a matrix A contains p elements and its column contains q elements,

Q.18

then the order of A is(A) p × p

(B) q × q

(C) p × q

(D) q × p

 sin 2  If A =  2 cosec 

sec 2   1/ 2

 and 

 cos 2   tan 2  B =  , then A + B is equal 1 / 2   – cot 2  to

Q.14

3x  y 

 x zx  z

3 =  3y  w  4

If 

2

 , then7

(A) x = 3, y = 7, z = 1, w = 14 (B) x = 3, y = – 5, x = – 1, w = – 4

 1  1 (A)   – 1 1  

1 1 (B)   1 1

1 0 (C)   0 1

(D) 

 1 0    0  1

(C) x = 3, y = 6, z = 2, w = 7 (D) None of these Q.19

Q.15

Q.16

1  If A = 2 0

2

3

3

4 , then 2A =

4 5

 4 6

1  (C)  2 0

2

3

3

4

3

 3  19 (A)   10 29 

6

2  (A) 2 0

6



10 12

1  (B) 4 0

2

3

2  (D) 4 0

4

6

6

8

 3 10



5

6



Q.20

10 12

2 (C)  4

Q.17



29

(D) None of these

 16  10 0 + 2B =    6 8  0

If 

19 

0

 , then B is equal

0

 8  5   3 4   8 5  (C)    3  4

2

(A) 

, 2

2 0   4  3

 8  3

5

(B) 



4

8  5 (D)   3  4

(B) 

Q.21

2

(D) None of these



3

1 5  If A =  0 7 11 8



29

 3  10

(B) 

to

then A =

2 2   4  3

19 

(C) 

6 8

6 4  0 If 2A + B =  and A – B =   6  11 6

(A) 

1  3 and B =    , then 4 2  7

5

2A – 3B is equal to-



5

3  2

If A = 

1 0

If X = 

a

2 and 3X –   1 0

3

1 =  2 0

3

 ,

1

then the value of a is7

 9  , then the value of tr A is9 

(A) 17

(B) 25

(C) 3

(D) 12

Question based on

Q.22

(A) –2

(B) 0

(C) 2

(D) 1

Multiplication of matrices

If A and B are matrices of order m × n and n × n respectively, then which of the following are defined(A) AB, BA

(B) AB, A2

(C) A2, B2

(D) AB, B2


2

Q.23

3

1

2 If A =   and A + k I = 8A, then k 7 5  

Q.30

equals

Q.24

(A) 4

(B) 8

(C) 1/4

(D) 1/16

and (A) 3 × 3

(B) 1 × 3

(C) 1 × 1

(D) 3 × 1

0 1

Q.31

Q.26

Q.27

1 2

If A = 

 1 1  , B =   1  1 , then2   (B) AB = 2I

(C) BA = 0

(D) B2 = I

Q.32

0

0  i   , then 0

1

If A =   and B =  i 1 0    (A) AB = BA (C) AB = – BA

0  [x 1 2] 1 1

1 1

(A) 1/3

(B) – 1/3 (C) 0

  0

0 1 1

x  1 = 0 is   1 

Q.33

(B) AB = B2 (D) None of these

 0 0    1 1   1 0   (C)   0 1 

2

 then AB

0

 2  4 (B)   4 4  2 4 (D)    4 4

 2  4  3 2

(C) 

 cos   sin   cos 2 (A)   sin 2 If A = 

sin  

 , then A2 equals-

cos  

sin 2 

cos 2  sin 2 (B)    cos 2  sin 2 cos 2   sin 2 cos 2  1 0 (C)  (D)     cos 2 sin 2  0 1 For matrices A and B, AB = 0, then(A) A = 0 or B = 0 (B) A = 0 and B = 0

 0  1  1 (D)   0

(A) 

If A, B are two matrices such that

(A) I

 1  2   and B =   , then AB 1    2 5  2 

equals (D) 1

3  2 

 5  2

If A = 

equals-

Q.29

10

The root of the equation

1  2 A + B =  , A – B = 2 4 

Q.28



7 , then

(D) None of these

1

(A) AB = 0

5

 tan  / 2  tan  / 2 1  1 tan  / 2   1  equals1 0    (A) zero matrix (B) sec2 .I2 (C) I2

Q.25

 1  and B =  2  2  3 2

(A) AB and BA both exist (B) AB exists but not BA (C) BA exists but not AB (D) both AB and BA do not exist

If A,B,C are matrices of order 1 × 3, 3 × 3 3 × 1 respectively, the order of ABC will be-

 1 If A =  3

Q.34

(B) 

0  i  0 , B =  0 1

If A =  i

1 



0  1 



0 

1

1 0  , C =  , 0 0  1

then which of the following statement is true-

Q.35

(A) AB = BA

(B) AB = – BA

(C) A2 = BC

(D) A2 = B + C

 2

If A =  0

1

2  and f(x) = 2x – 3x, then f(A)

3

equals-

14 1  (A)    0  9 14  1 (C)   0 9 

 14 1   0 9  14  1 (D)    0  9 (B) 

(C) It is not necessary that A = 0 or B = 0 (D) All above statements are wrong


3

Q.36

 2 0 1 If A = 2 1 3 then A2 – 5A + 6I = 1  1 0  1  1  5  1 1  3      (A)   1  1 4  (B)   1  1  10  3  10 4   5 4 4 

Q.37

If order of A + B is n × n, then the order of AB is (A) n × n (C) m × n

Q.38

(B) n × m (D) not defined

a  , then  a 

If A = [a b], B = [–b –a] and C =  true statement is (A) A = – B (C) AC = BC

Q.39

2

2

 , then correct

1

(A) AB = BA

(B) AAT = A2

(C) AB = B2

(D) None of these

cos   sin   , then AAT equalscos   cos 2  sin 2   cos 2  sin 2 (A)  (B)  2    sin  cos 2   sin 2 cos 2  If A=   sin 

1 0 (C)   0 1 Q.45

(B) A + B = A – B (D) CA = CB

   If   is square root of I2, then ,  and      will satisfy the relation (A) 1+ 2 +  = 0 (B) 1– 2 +  = 0 (C) 1+ 2 –  = 0 (D) –1+ 2 +  = 0

 1 and B =   4  1

3 1

If A = 

statement is -

Q.44

(D) I

(C) 0

Q.43

Q.46

a  b

If A = 

0 0

0

(D) 



0

b

, then |A +AT| equals  a

(A) 4(a2 – b2)

(B) 2(a2 – b2)

(C) a2 – b2

(D) 4 ab

For suitable matrices A, B; the false statement is(A) (AB)T =ATBT (B) (AT)T= A (C) (A – B)T = AT – BT (D) (AT) –1 = (A –1 )T

Question based on

Q.40

Transpose of a Matrix

1 If A =   1

2

3 4  and B =    , then 2  2  2

Q.47

(AB)T is-

11  2 (A)    5  6 7 1   0  8

(C)  Q.41

3 4

3  , B = 2  5 1 4

4



1 , then (AB) 3

equals-

 11 5     2  6 7 0  (D)   1  8 (B) 

16 22 (A)   23 31

16 23 (B)   22 31

22 31 (C)   16 30

(D) 

If A and B are matrices of order m × n and n × m respectively, then the order of matrix BT (AT)T is (A) m × n (B) m × m (C) n × n (D) Not defined

Q.42

2 If A =  3

Q.48

3 y

If A = 

(B) A + B + C

(C) – (A + B+ C)

(D) (A + B + C)T

Q.49

16 



20

x

 and A = AT, then -

0

(A) x = 0, y = 3

(B) x + y = 3

(C) x = y

(D) x = – y

If A, B, C, are three matrices, then AT + BT + CT is (A) zero matrix

23  31

 cos   sin 

If A =  (A) I

sin  

 , then AA equals -

cos  

(B) A

(C) A

(D) 0


4

Q.50

If A is a matrix of order 3 × 4, then both AB T

Q.56

If A is symmetric matrix and B is a skew-

and BTA are defined if order of B is -

symmetric matrix, then for n

(A) 3 × 3

(B) 4 × 4

statement is -

(C) 4 × 3

(D) 3 × 4

(A) An is symmetric



N, false

(B) An is symmetric only when n is even Question based on

Q.51

Symmetric & Skew symmetric Matrices

(C) Bn is skew symmetric when n is odd (D) Bn is symmetric when n is even

5  7 0   Matrix  5 0 11  is a 7  11 0 

Q.57

If A is a square matrix, then A– A is (A) unit matrix (B) null matrix

(A) diagonal matrix

(C) A

(B) upper triangular matrix

(D) a skew symmetric matrix

(C) skew-symmetric matrix Q.58

(D) symmetric matrix

Let A be a square matrix. Then which of the following is not a symmetric matrix -

Q.52

If A and B are square matrices of same order,

(A) A + A

then which of the following is skew-

(B) AA

symmetric-

(C) AA

(A) (C)

Q.53

AA 2 AT

T

 BT 2

(B)

(D)

A

T

B

T

(D) A – A

2 B  BT

Question based on

2

If A is symmetric as well as skew symmetric

Q.59

matrix, then -

Q.55

3 0

For any 2 × 2 matrix A , A(adj A) = 

(A) A is a diagonal matrix

then |A| equals-

(B) A is a null matrix

(A) 0

(B) 3

(C) A is a unit matrix

(C) 6

(D) 9

(D) A is a triangular matrix Q.54

Adjoint of a Matrix

(A) orthogonal matrix

1 If A = 2 0

(B) symmetric matrix

A) is-

(C) skew- symmetric matrix

(A) |A|2

(B) – 2A

(D) triangular matrix

(C) 2A

(D) A2

If A – A = 0, then A is -

x

y

If   is symmetric matrix, then u v

Q.60

Q.61

0

,

3

2

3

3

4 , then the value of adj (adj

0

 cos x  sin x

If A = 

  2

sin x 

1 & A. adj A= k   cos x  0

(A) x + v = 0

then k equals-

(B) x – v = 0

(A) sin x cos x

(B) 1

(C) y + u = 0

(C) 2

(D) 3

0

,

1

(D) y – u = 0


5

Q.62

1 If A = 0 2 9 (A) 0 0 0  (C)  0 9

Q.63

2 3 1

3

0 0

  9 9  0  0

9 0 0 0 9 0

14  (C)  4 22

 1 , then A (adj A) equals 2 9 (B) – 0 0

  9

0

Q.68

Q.69

If A and B are square matrices of same orders, then adj (AB) equals(B) adj B . adj A

(C) adj A + adj B

(D) adj A – adj B Question based on

Q.64

Q.65

 1 2 3   If A =  4 0  1 , then (adj A) 23 =  3 1 5  (A) 13

(B) – 13

(C) 5

(D) – 5

Q.70

(A) 2 |A|

(B) 2 |A| I

(C) zero matrix

(D) Unit matrix

4 3

2

Q.67

4 3

(B) 

2



3

(D) None of these

1 3 5 If A = 3 5 1 then adj A equals 5 1 3  14  4  22 (A)   4  22 14   22 14  4  22   14 4   (B)  4 22  14   22  14 4 

(B) |A|n (D) |A|n+2

The adjoint of symmetric matrix is(A) symmetric matrix (B) skew-symmetric matrix (C) diagonal matrix (D) None of these Inverse of a Matrix

4 1

7

The inverse matrix of 

2 1

If A = 

 is -

2

 2  1   7 4   2 1  (D)    7  4 (B) 

3

4 , B =  3 2

6

1 , C =  3 0

0

 , then

1

invertible matrices are(A) A and B (B) B and C (C) A and C (D) All

 , then adj (adj A) is equal to-

3

 3  2   3 4  4 2 (C) 6   3 3 (A) 

If A is a square matrix of order n, then the

 2  7   1 4   2 7  (C)    1  4

(adj AT) – (adj A) T equals-

If A = 

14

(A) 

Q.71 Q.66

  4 

22 14

value of |adj A| is(A) |A|n–1 (C) |A|n+1

(D) None of these

(A) adj A . adj B

22

(D) None of these

0 0 9 0

4

Q.72

Q.73

  1  Matrix  3 0   1 1 (A)  = – 15 (B)  = – 17 (C)  = – 16 (D)  = – 18

4



1 is not invertible if 2

1 2  1 , B =  0 3  5 

If A = 

0

 and X is a matrix

2

such that A = BX, then X equals 1  2 4 1 2 4  (A) (B)     2  3 5 2 3  5

2 4   3  5

(C) 

(D) None of these


6

Q.74

2 0 0 If A = 0 2 0 , then value of A –1 is0 0 2 0  1 0 0 1 / 2 0     (A) 0 1 0 (B)  0 1 / 2 0  0 0 1  0 0 1 / 2  2 0 0    (C)  0  2 0  (D) None of these  0 0  2

Q.75

For any square matrix A, which statement is wrong(A) (adj A) –1 = adj (A –1) (B) (AT) –1 = (A –1)T (C) (A3) –1 = (A –1)3 (D) None of these

Q.76

If A = 

Q.77

Which of the following matrix is inverse of itself 1 1 1 1 0 0      (A) 1 1 1 (B) 0 1 0

 1 2  , then A –1 = 4 1     2  1  2 1  1 (A) (B)     7  4  1 4 1 1  1 2 1  1  2 (C) (D)     1  9 4 1 7 4

1 1  (C)  0 1

1 1

0 1 0 0

 0 1

0 0  (D) 1 0

0 1

1 0



1 1 1 0


7

LEVEL- 2 Q.1

Q.2

1 1 

If A =

1

(A) A. A = A and

n  and n  N, then A is equal to1

(A) 2nA

(B) 2n–1 A

(C) nA

(D) None of these

 cos   sin 

If E () = 

 cos n   sin n   cos n  sin n

sin   cos 



cos n

(C) A. A = A+ and

(A) E (0°)

(B) E ()

(C) E ( + )

(D) E ( – )

 cos n  n  sin 

(A)n = 

(A) nk

(B) n + k

(C) nk

(D) k n

3  4 If A =   , then for every positive 1 1   

sin n  



cos n  

(D) A. A = A+ and

If A = [aij] is scalar matrix of order n × n

 cos n  sin n

(A)n = 

Q.8

1 2

If M = 

sin n 

2

2  and M – M – 2 = 0, then 

3

equals (A) –2

(B) 2

1  2n 4n  1  2n  4n  (A)  (B)    1  2n  1  2n   n  n

(C) –4

(D) 4

  1  2n  4n



cos n

integer n, An is equal to -

1  2n (C)   n Q.5

sin n 

(A)n = 

 then E() E() is

such that a ij = k for all i, then |A| equals -

Q.4



cos n  

(B) A. A = A and

equal to-

Q.3

sin n  

(A)n = 

Q.9

(D) None of these

If A is any skew- symmetric matrix of odd

orders, then |A| equals -

If D = diag (d1, d2,.....,dn), then D n equals -

(A) –1

(B) 0

(C) 1

(D) None of these

(A) D (B) diag (d1n–1, d2n–1, ....dnn–1)

Q.10

AB = B and BA = A, then A 2 + B2 is equal

(C) diag (d1n, d2n, .....,dnn)

to-

(D) None of these

Q.6

Q.7

If A and B are square matrices such that

(A) 2AB

(B) 2BA

(C) A + B

(D) AB

cos   sin   If A = sin  cos    0 0

0

(A) adj A = A

(B) adj A = A –1

all i = 1,2,....,n; then D –1 is equal to-

(C) A –1 = –A

(D) None of these

(A) D

 cos   sin 

If A= 

 0 , then 1

sin  

 , then which of

cos  

following statement is true –

Q.11

If D = diag (d1, d 2, d 3, ...,dn), where di  0 for

(B) In (C) diag (d1 –1, d2 –1, ....,dn –1) (D) None of these


8

Q.12

If A be a matrix such that inverse of 7A is the

 1 2   , then A equals  4  7

matrix 

1 4

2

(A) 

1 2

4 / 7

(B) 

4

(C) 

Q.13

 1 2 / 7

 1



1/ 7 

 1 4 / 7

 1

2 / 7

(D) 



1/ 7 

If A is invertible matrix, then det (A –1)

equals-

Q.14

(A) det A

(B) 1/det A

(C) 1

(D) None of these

If A and B are non-zero square matrices of

the same order such that AB = 0, then (A) adj A = 0 or adj B = 0 (B) adj A = 0 and adj B = 0 (C) |A| = 0 or |B| = 0 (D) None of these

Q.15

Q.16

If A =

0  1 

1

and (aI2 + bA)2 = A, then  0

(A) a = b = 2

(B) a = b = 1/ 2

(C) a = b = 3

(D) a = b = 1/ 3

a 2  0 c  b    If A =  c 0 a  and B =  ab   ac  b  a 0  

ab b

2

bc

ac 



bc  , c2 



then AB is equal to (A) A

Q.17

(B) B

 1  tan  If   1   tan  a =  b

(C) I

tan   1  tan  1     b  , then a

(D) 0 1

(A) a = 1, b = 1 (B) a = cos 2 , b = sin 2 (C) a = sin 2 , b = cos 2  (D) None of these


9

LEVEL- 3 Q.1

Q.2

Q.3

1 3 If A=   and A2 – kA – 5 I2 = O, then the 3 4  

(A)

value of k is-

(C) 

(A) 3

(B) 5

(C) 7

(D) –7

cos   sin  0   Let F() =  sin  cos  0 , where   R.  0 0 1 Then (F()) –1 is equal to (A) F (– ) (B) F ( –1) (C) F (2) (D) none of these

Q.7

20

(B)

(D)

2 5

1 5 2 5

 a i b i  If Ai =  i and if |a|<1, |b|<1, then det  i i 1  b a 



(Ai) is equal to(A)

(C)

a2

1  a2 a2

1  a 

2

 

b 2

1  b 2 b 2 2

1  b 

(B)

(D)

a 2  b 2 (1  a 2 ) (1  b 2 ) a 1 a



b 1  b

If a matrix A is such that 3A3 + 2A2 + 5A + I = 0, then A –1 is equal to(A) – (3 A2 +2 A + 5) (B) (3 A2 +2 A + 5) (C) 3 A2 – 2 A – 5)

Q.4

21

Q.8

(D) none of these

1 For the matrix A = 1 2

1

5A, then

0

  0

2 1 , which of the 1

1  1 1  If A = 0 2  3 and B = (adj A), and C = 2 1 0  adjB C

is equal to-

(A) 5

(B) 25

(C) –1

(D) 1

following is correct (A) A3 + 3A2 – I = O (B) A3 – 3A2 – I = O (C) A3 + 2A2 – I = O (D) A3 – A2 + I = O

Q.5

1 If 3 2

2 1 3

Q.9

 4  2  y   0  6 2 , then (x, y,     1  z   1 2 

3  x 

 2  1

Q.6

(B) (4, –2, –2)

(C) (4, 2, 2)

(D) (–4, –2, –2)

1 If matrix A = 3 0

0 4 6

 a11 1  denoted by A  a 21 a 31 value of a 23 is equal to-

 1  5 and its inverse is  7  a12 a13   a 22 a 23 , then the  a 32 a 33 

 i   i i     i  and Q =  0 0  , then  i i  0   

PQ is equal to-

z) is equal to(A) (–4, 2, 2)

 i 0  If P =  0  i  i i 

Q.10

  2  (A)  1  1 

  1  1 

 2  2    (B)   1 1   1 1   

 2 (C)   1

2   1 

 1  (D)  0 0 

2 

0 0 



1 0 0 1 

If A, B are symmetric matrices of the same order then (AB – BA) is : (A) symmetric matrix (B) skew symmetric matrix (C) null matrix (D) unit matrix


10

Each of the questions given below consists of

Q.14

Statement I : Trace of matrix

 a11  A = a 21 a 31

Statement – I and Statement – II. Use the following Key to choose the appropriate answer.

(A) If both Statement- I and Statement- II

a 12

a13 

a 22

a 23  is equal to a 11 + a22 +



a 33 

a 32

a33

are true, and Statement - II is the correct

Statement II : Trace of a matrix is equal to

explanation of Statement– I.

sum of its diagonal elements.

(B) If both Statement - I and Statement - II are true but Statement - II is not the

Q.15

 0 q  p   r  p

correct explanation of Statement – I. (C) If Statement - I is true but Statement -II is false. (D) If Statement - I is false but Statement - II



0 r  q

q  r  = 0



0

symmetric matrix of odd order is zero.

5 0  0

Statement I :

0 3 0

0

 0 is a diagonal 2

Q.16

Statement I : The order of the matrix A is

4 × 5 and that of B is 3 × 4. Then the matrix AB is not possible.

matrix.

Statement II : AB is defined if number of

Statement II : A square matrix A = (a ij) is a

columns of A = number of rows of B.

diagonal matrix if aij = 0  i  j. Q.12

p  q p  r 

Statement II : The determinant of a skew

is true.

Q.11

Statement I : The determinant of matrix

Q.17

1 2  9

Statement I : The determinant of matrix

0 x y  x 0 p    y  p 0    z  q  r

Statement I : The inverse of the matrix

z

  is a perfect square. r   0

q

3

5



6 10 does not exist.  8

7 

1 2  9

Statement II : The matrix

Statement II : The determinant of a skew

symmetric matrix of even order is a perfect

1

singular.

square.

3

5

 is  7 

6 10 8

5

[  2 6 10 = 0, since R 2 = 2R 1] 9

Q.13

3

8

7

Statement I : The inverse of the matrix

 1  3 0  does not exist. 1 2  Statement II : |A|  0 1 4 1 1 4 1 [ |A| = 2 3 0 = 0  5 2 1  A = 2 0

4

0 1

2

0

1

2

= –10 – 2 = – 12  0 ].

Q.18

a  Statement I : If A = 0 0 1 a  A –1 =  0  0 

0

1

0

0

0

b

0 , then

0



c



0 b

0

 

1 c 

Statement II : The inverse of a diagonal

matrix is a diagonal matrix.


11

PARAGRAPH from Comprehension based problems in Mathematics

Let A and B are two matrices of same order

1  3  i.e. where A = 2 K 4 2 Q.19

Q.20

 5 ; B = 1

2 4  3

1

3

2

4

3



5

If A is singular matrix then tr(A + B) is equal to (A) 5

(B) 3

(C) 4

(D) 6

If K = 2 then tr (AB) + tr (BA) is equal to (A) 66 (C) 84

Q.21

2

(B) 42 (D) 63

If C = A – B and tr (C) = 0 then K is equal to(A) 5 (B) –5 (C) 7 (D) –7


12

ALGEBRA (Practice Sheet).pdf

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