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IIT MATHEMATICS SET-II
INDEX 1.
COMPLEX NUMBERS ......................................................................................
2
2.
FUNCTIONS .......................................................................................................
44
3.
LIMIT CONTINUITY AND DIFFERENTIABILITY.........................................
4.
APPLICATION OF DERIVATIVE .....................................................................
80
120
IIT-MATHEMATICS-SET-II
1
COMPLEX NUMBERS
2
COMPLEX NUMBERS
INTRODUCTION
If x2 + 1 = 0 , then x 1 .
1 is represented as i. This is taken as unit of imaginaries. If x 2 x 1 0 , then x
1i 3 1 1 4 or x . 2 2
Here roots of this equation are of the form x + iy, where x and y are real numbers. Roots of this form are called complex roots. Any number of the form x + iy (where x and y are real numbers) is called a complex number. A complex number x + iy is also defined as an ordered pair of real numbers x and y and may bewrittenas(x, y). If z = x + iy, then x is called the real part of complex number and y is called the imaginary part of the complex number z. ‘x’ is denoted as Re(z) and ‘y’ is denoted as Im(z).
ALGEBRAIC OPERATIONS WITH COMPLEX NUMBERS (i)
Addition: (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i (y1 + y2)
(ii) Subtraction: (x1 + iy1) – (x2 + iy2) = (x1 – x2) + i(y1–y2) (iii) Multiplication: (x1 + iy1) (x2 + iy2) = (x1x2 – y1y2) + i (x1y2 + x2y1) x iy
x x y y
i(x y x y )
1 1 1 2 1 2 2 1 1 2 (iv) Division: x iy x 2 y 2 x 2 y 2 2 2 2 2 2 2
(v) Equality: x1 + iy1 = x2 + iy2 if and only if x1 = x2 & y1 = y2 . (vi) The complex number do not posses the property of order i.e., (x1 + iy1) < or > (x2 + iy2) is not defined.
ARGAND PLANE AND GEOMETRICAL REPRESENTATION OF COMPLEX NUMBERS (a) Let O be the origin and OX and OY be the x-axis and y-axis respectively. Corresponding to each complex number x + iy there will be one and only one point P(x, y) in the xy – plane. Thus each complex number x + iy can be represented by a point P(x, y) of the xy-plane and conversely corresponding to each point P(x, y) of the xyplanetherewillbeauniquecomplexnumber x + iy. The xy-plane is called the Argand Plane, Complex Plane or Gaussian Plane, xaxis is called the real axis and y-axis is called the imaginary axis.
3
IIT-MATHEMATICS-SET-II Z = x + iy P(x, y)
Y
y x L
O
X’
X
Y’
(b) Each complex number z can be represented by a vector OP , where P is the point representing the complex number z.
Thus z OP y P x
Note:
(i)
Any other vector AB which has the same magnitude, direction and sense as
that of OP but has a different initial point, also represents the complex number z. (ii) Complex numbers can be considered as vectors in case of sum, difference and modulus of complex numbers.
CONJUGATE OF A COMPLEX NUMBER The complex numbers z = x + iy and z x iy, where x and y are real numbers, i 1 and y 0 are said to be complex conjugate of each other. (Here the complex conjugate is obtained by just changing the sign of i). Note that, sum = (x + iy) + (x – iy) = 2x, which is real and product = (x + iy) (x – iy) = x2 + y2 which is real. Properties of Conjugate: (i) z is the mirror image of z along real axis. (ii) z = z z is purely real (iii) z = – z z is purely imaginary (iv) Re (z) = Re ( z ) =
(v) Im (z) =
z z 2
z z 2i
4
COMPLEX NUMBERS (vi) z1 z2 z1 z2 (vii) z1 z 2 z1 z 2 (viii) z1z 2 z1 z 2 z1 z1 (z 2 0) (ix) z z 2 2
(x)
z1 z 2 z1 z 2 2 Re (z1 z 2 ) 2 Re (z1 z 2 )
(xi) z n (z) n Imaginary axis A (z)
O
Real axis B (z)
MODULUS OF A COMPLEX NUMBER Distance of a complex number z from origin is called the modulus of the complex number z and it is denoted by |z| . Therefore if z = x + iy, then |z| = x 2 y 2 . Im. axis
|z| y x
Re-axis
Properties of Modulus (i)
|z| = 0 z = 0
(ii)
Re z | z |, Imz | z |
(iii) zz | z |2 5
IIT-MATHEMATICS-SET-II (iv) | z1z 2 | | z1 | | z 2 |
(v)
z1 | z1 | z 2 | z 2 | (z 2 0) (vi)
| z1 z 2 | | z1 | | z 2 | | z1 z 2 ... z n | | z1 | | z 2 | ... | z n | , n N ,
(vii)
| z1 z 2 | || z1 | | z 2 || (Equality holds when arg z1 = arg z2).
(viii)
| z1 z 2 |2 (z1 z 2 ) (z1 z2 ) | z1 |2 | z 2 |2 z1 z2 z 2 z1
(ix)
z 1
(x)
|z|n = |zn|, n N
z | z |2
ARGUMENT OF A COMPLEX NUMBER y z (x, y) y
x
O
x
We have cos
and sin
x x2 y2
y 2
x y2
... (1)
... (2)
Value of , – , satisfying equations (1) and (2) simultaneously, is called the principal argument of z. Method of calculating principal argument: first calculate tan 1
y . x
Now , , or 2 becomes the principal argument of z according as point P(z = x + iy) lies in Ist, IInd, IIIrd or IVth quadrant respectively. Note: Whenever we have to calculate the argument of a complex number, it is obvious that we have to calculate the principal argument.
6
COMPLEX NUMBERS Properties of Arguments (i)
arg (z1 z2) = arg (z1) + arg (z2)
z1 (ii) arg z a r g z 1 a r g z 2 2 (iii) arg (–z) = arg (z) (iv) arg (iy) =
if y > 0 2
=
(v)
if y < 0 2
arg (z z)
2
1 (vi) arg ( z) arg (z) arg z
(vii) arg (z) = 0 or z is purely real. (viii) arg(z) = / 2 z is purely imaginary.. Note: Here arg(z) means general argument of z. Polar form : z = r cos ir sin re i , where r = |z|, and = principal argument of z.
DE-MOVIER’S THEOREM If n is any interger, then (cos i sin )n cos n i sin n. Writing the Binomial expansion of (cos i sin )n , n N and equating the real and the imaginary parts , we get cos n cos n n C2 cos n 2 sin 2 n C 4 cos n 4 sin 4 ... sin n n C1 cos n 1 sin n C3 cos n 3 sin 3 n C 5 cos n 5 sin 5 ...
If n = p/q, where p and q are integers (q > 0) and p, q have no common factor, then (cos i sin ) n has q distinct values, one of which is cos n isin n . If zn = r (cos sin ), n N, then 2k 2k , k = 0, 1, 2, ..., n – 1. z1/ n r1/ n cos i sin n n
7
IIT-MATHEMATICS-SET-II
CUBE ROOTS OF UNITY Roots of x3 – 1 = 0 are called the cube roots of unity Now x3 –1 = 0 (x – 1) (x2 + x + 1) = 0 Therefore, x = 1,
1 i 3 1 i 3 . 2 2
If second root be represented by , then third root will be 2 . 2 Cube roots of unity are 1, , . 1 is real cube root of unity and and 2 are nonreal cube roots of unity.
Cube roots of unity can be taken as vertices of an equilateral triangle ABC inscribed in a circle of radius 1 and centre at origin. y B A(1)
x
O
C
Properties of Cube Roots of Unity (1) 1 2 0 (2) 3 1 (3) 1 n 2n 3 (if n is multiple of 3) (4) 1 n 2n 0 (if n is not a multiple of 3).
The nth Roots of Unity Let x be an nth root of unity. Then xn = 1 = cos 2k i sin 2k (where k is an integer) x cos
Let cos
2k 2k i sin , k = 0, 1, 2, ... n – 1 n n
2 2 i sin . Then the n, nth roots of unity are t n n
(t = 0, 1, 2, ... , n – 1), i.e., the nth roots of unity are 1, , 2 , ..., n 1. 8
COMPLEX NUMBERS Sum of the Roots 2 n 1 1 + ....
1 n 0 1
Thus the sum of the roots of unity is zero.
Product of the Roots
2
1.. ...
n 1
n (n 1) 2
2 2 cos i sin n n
n 1 n 2
cos{(n 1)} isin{(n 1)}
1, n is even 1, n is odd 2
A 2( )
A 2( 2/n 2/n A1(1) 2/n O n
An
Note : The points represented by n, nth roots of unity are located at the vertices of a regular polygon of n sides inscribed in a unit cirlce having centre at the origin, one vertex being on the positive real axis.
GEOMETRICAL REPRESENTATION OF SUM AND DIFFERENCE OF TWO COMPLEX NUMBERS Let z1 = x1 + iy1 and z2 = x2 + iy2 be two complex numbers. Let z1 and z2 be represented by the points P(x1 , y1) and Q(x2, y2) in the Argand plane. Let O be the origin. Join OP and OQ. Produce QO backwards to Q such that OQ = OQ. Then, the co-ordinates of Q will be (–x2 , – y2) and therefore, Q will represent the complex number –z2 = –x2 + i (–y2). Complete the parallelogram POQ R and let its diagonals intersect at the point H. x x 2 y1 y 2 Since H is the middle point of PQ , its co-ordinates are 1 , 2 2
Now, if (x, y) be the co-ordinates of R, then since H is the middle point of OR, the coordinates of H are .
9
IIT-MATHEMATICS-SET-II x y , 2 2
x x1 x 2 y y y2 and 1 2 2 2 2
x = x1 – x2 and y = y1 – y2. Thus, the co-ordinates of R are (x1 – x2, y1 – y2). Y Q(z2)
S P(z1)
O
X
X’ H
R(z1 – z2)
Q’(–z2) Y’
Hence, the point R represents the complex number (x1 – x2) + i (y1 – y2) = [(x1 + iy1) – (x2 + iy2)] = z1 – z2 Clearly, |z1| = OP, |z2| = | – z2 | = OQ and |z1 – z2| = OR = PQ So if P(z1) and Q(z2) be two points then |z1 – z2| is the distance between P and Q. Similarly, the point S represents the complex number z1 + z2. Since in a triangle sum of two sides is greater than the 3rd side and difference of two sides is less than the 3rd side, we have |z1| + |z2| |z1 + z2|, equality holds when arg z1 = arg z2 ||z1| – |z2|| |z1 – z2|, equality holds when arg z1 = arg z2 .
GEOMETRICAL REPRESENTATION OF PRODUCT AND QUOTIENT OF TWO COMPLEX NUMBERS Let z1 and z2 be two complex numbers. Let z1 = r1 (cos 1 i sin 1 ) and z2 = r2 (cos 2 i sin 2 ) . Let z1 and z2 be represented by the points P and Q respectively. Let O be the origin. We join OP and OQ. Then OP = r1 ; POX 1; OQ r2 and QOX 2 Now z1z2 = r1 (cos 1 i sin 1 ).r2 (cos 2 i sin 2 ) = r1r2 [cos( 1 2 ) i sin (1 2 )]. 10
COMPLEX NUMBERS = r(cos i sin ), where r = r1 r2 and 1 2 Thus z1z2 is a complex number whose modulus is r1r2 and argument is 1 2 . Take a point A on the real axis such that OA = 1. Join PA. Now, construct a triangle OQR similar to the triangle OAP From similar OAP and OQR , we have OR OP OR r1 or OQ OA r2 1 or, OR = r1 r2.
and, XOR XOP POR 1 2 Y R(z1.z2)
r2
Q(z2) P(z1) r1
O
A
X
z1 Thus, the point R represents z1z2. Similalry we can represent z as a complex number having 2
modulus r1/r2 and argument 1 2 .
Rotation Theorem (Coni Method)
If z0, z1 and z2 be the affixes of the vertices of a triangle described in counter-clockwise sense, then z 2 z0 z z 1 0 ei | z 2 z 0 | | z1 z 0 |
or,
z1 z 0 z z z z 2 0 ei(2 ) 2 0 e i | z1 z 0 | | z 2 z 0 | | z 2 z0 |
Imaginary axis (2–)
z2
z1
z0 Real axis O
11
IIT-MATHEMATICS-SET-II Condition for Four Points to be concyclic If points A, B, C and D are concyclic, then ABD = ACD Using rotation theorem (z1 z 2 ) (z 4 z 2 ) i In ABD | z z | | z z | e ... (1) 1 2 4 2
In ACD
(z1 z 3 ) z 4 z 3 i e | z1 z 3 | | z 4 z 3 |
... (2)
From (1) and (2)
(z1 z 2 ) (z 4 z 3 ) (z1 z 2 ) (z 4 z 3 ) (z1 z 3 ) (z 4 z 2 ) (z1 z 3 ) (z 4 z 2 ) = a positive real number So if z1, z2 z3 and z4 are such that A(z1)
D(z4) C(z3)
B(z2)
(z1 z 2 ) (z 4 z3 ) (z1 z3 ) (z 4 z 2 ) is a positive real, then these four points are concyclic and vice-versa.
GENERAL EQUATION OF A LINE Equation of straight line through z1 and z2 is given by
z z z1 z z1 or z1 z 2 z1 z2 z1 z2
z
I
z1 z2
I 0. I
z(z2 z1 ) z1 z2 z1 z1 z(z 2 z1 ) z1z 2 z1 z1 z(z2 z1 ) z(z1 z 2 ) z1z 2 z1 z2 0 Here z1z 2 z1 z2 is a purely imaginary number as z1z 2 z1 z2 2i lm (z1z 2 ) . Let z1z 2 z1 z2 ib, b R
z(z2 z1 ) z(z1 z 2 ) ib 0
z i (z1 z2 ) zi(z 2 z1 ) b 0 Let a = i (z2 – z1) a i (z1 z2 ) za za b 0
12
COMPLEX NUMBERS This is the general equation of a line in the complex plane. Slope of a given line: Let the given line be za za b 0. Replacing z by x + iy, we get (x iy)a (x iy)a b 0 (a a )x iy (a a) b 0.
It’s slope is
aa 2 Re(a) Re(a) 2 i(a a) 2i lm(a) lm(a)
Equation of a line parallel to the line za za b 0 is za za 0 (where is a real number). Equation of a line perpendicular to the line za za b 0is za za i 0 (where is a real number)
Equation of Perpendicualr Bisector
Consider a line segment joining A(z1) and B(z2). Let the line L be it’s perpendicualr bisector. If P(z) be any point on the ‘L’, we have PA = PB |z – z1| = |z – z2| |z – z1|2 = |z – z2|2 (z – z ) ( z – z1 ) = (z – z ) (z z2 ) 1 2 zz zz1 z1 z z1 z1 zz zz2 z 2 z z 2 z2 z(z z ) z (z z ) | z |2 | z |2 0 1 2 1 2 2 1 P(z) B(z2)
L B(z1)
Distance of a given point from a given line
Let the given line be za za b 0, and the given point be zc Say zc = xc + iyc . Replacing z by x + iy, in the given equation, Distance of (xc, yc) from this line is, | x c (a a) iyc (a a) b | (a a) 2 (a a) 2
13
| z c a zca b | 2|a|
| z c a zc a b | 4(Re(a)) 2 4(im(a)) 2
IIT-MATHEMATICS-SET-II zc
D
za + za + b = 0 arg (z – z0) = represents a line passing through z0 with slope tan (making angle with the positive direction of x-axis).
(a) Slope of the line segment joining two points If A and B represent complex numbers z1 and z2 in the Argand plane, then the complex slope of AB is defined as
z1 z 2 . z1 z2
Slope of AB =
a coeff . of z a coeff . of z
Thus, the slope of the line az az b 0 is
a a
(c) if 1 and 2 are the complex slopes of two lines on the Argand plane, then the lines are (i)
parallel, if and only if 1 2 .
(ii) perpendicular, if and only if 1 2 0 .
CIRCLE (a) If z0 be a fixed point on the complex plane such that z is a moving point always at a distance r from z0, then z lie on a circle whose equation is |z – z0| = r. (b) The general equation of circles is zz b 0 is zz az az b 0, where b is a real number. The centre of this circler is ‘–a’ and its radius is aa b . (c) The equation of the circle described on the line segment joining z1 and z2 as diameter is (z – z1) (z z2 ) + (z z2 ) (z z1 ) 0
z z1 (d) arg z z , (, ], 2
(e)
represents an arc of a circle through z1 and z2, 0, .
z z1 , 1 represents a circle having the following properties. z z2
(f) C, P and Q are collinear, where C is the center of the circle and z1 and z2 represents the points P and Q. 14
COMPLEX NUMBERS (g) CP. CQ = r2, r being the radius of the cirlce. (h) the circle has a diameter AB, where A and B divide PQ in : 1 internally and externally respectivley.
15.
CONIC SECTIONS (i)
Parabola
Equation of parabola, whose focus is at z0 and the line az az b 0 is the directrix is
| z z0 |
| az az b | 2|a |
(ii) Ellipse Ellipse is locus of a point whose sum of distances from two fixed points z1 and z2 is always a constant. |z – z1| + |z – z2| = a. Here a > |z1 – z2| If a = |z1 – z2|, it represents line segment between z1 and z2. (iii) Hyperbola Equation of hyperbola, having foci at z1 and z2 is given by |z – z1| – |z – z2| = a where a is a positive real number & a < |z1 – z2|.
15
IIT-MATHEMATICS-SETII
ASSIGNMENT
16
COMPLEX NUMBERS
WORKED OUT ILLUSTRATIONS
ILLUSTRATION : 01 6
Let Z k k 0,1,2,...............6 be the roots of the equation z 1 z 0 , then Re z k 7
7
k 0
is
equal to
(a) 3 - 2i
(c)
(b) 0
7 2
(d) 3 2i
Solution : Let z k x k iy k , we have z k 17 z 7 k 0
z k 17
| z k 1 | 7 | z k |7
| z k 1 || z k |
| x k iy k 1 | 2 | x k iy k | 2
xk 12 y k 2 x 2 k y 2 k
2 x k 1 0 or x k
z 7 k
6
6
k 0
k 0
Thus, Re z k x k
1 2
7 2
ILLUSTRATION : 02 If m and x are two real numbers, then e
(a) cos x i sin x
(b)
m 2
2 mi cot 1 x xi 1
(c)1
Solution : 1 Let cot 1 x , then cot x or tan . x
We have e 2i cot 17
1
x
e 2i cos2 i sin 2
xi 1
(d)
m
is equal to
m 1 2
IIT-MATHEMATICS-SETII 1 tan 2
2 tan 1 1 / .x 2 2 / x i i = 1 tan 2 1 tan 2 1 1 / x 2 1 1/ x 2
=
x2 1 x2 1
2i cot e
2ix x2 1
1 x m
2 mi cot e
x i 2 x i ix i 2 x i x i x i ix i
ix 1 ix 1
1 x ix
1 ix 1
ix 1 ix 1
m
m
1
ILLUSTRATION : 03 If 1 and is a nth root of unity, then value of 1 4 9 2 163 ...... n 2 n 1 is (a) n
(b)
n2 1
(c) n 2 1 2n
(d)None of these
Solution : We have, for x ¹1,
1 x x 2 x 3 ...... x n x n 1 1 / x 1. Differentiating w.r.t. we get 2
1 2 x 3x ...... nx
n 1
n 1x n x 1
x n1 1 x 12 ……….(1)
Multiplying both the sides by x, we get x 2 x 2 3x 3 ....... nx n
n 1x n 1 x n 2 x x 1 x 12
………(2)
Differentiating again w.r.t.x we get
1 2 2 x 3 2 x 2 ........ n 2 x n1 2 n 1 x n 2n 3x n1 1 2x n 2 x x 1 x 12 x 13
Putting x and using n 1, we get
1 4 9 2 ....... n 2 n1
18
COMPLEX NUMBERS =
n 12 2n 3 1 2 2 1 12 13
=
n 12 2n 3 1 2 1 12 12
=
n 12 1 2n 3 1 2 12
n 1 =
2
2
2n 3 2 n 1 1
12
n 2 n 2 2n n 2 1 2n = 12 12
ILLUSTRATION : 04 Im z1 If z1 and z1 represent adjacent vertices of a regular polygon of n sides and if 2 1 , Re z 1
then n is equal to (a) 8
(b) 16
(c) 18
(d) 24
Solution : Since z1 and z1 are the adjacent vertices of a regular polygon of n sides, we have z1oz1
2 and | z1 || z1 | . n
Thus, z1 z1e 2i / n Let z1 r cos i sin re i
z1 re i
since z1 z1e 2i / n
re i re i e 2i / n re 2i / n i
2 or n n
Therefore z1 r cos i sin = r cos i sin n n
19
IIT-MATHEMATICS-SETII r sin n Im z1 2 1 Now, = 2 1 Re z1 r cos n
tan 2 1 = tan n 8
n=8
ILLUSTRATION : 05 If | z i Re( z ) || z Im( z ) |, then z lies on (a) Re(z) = 2
(b) Im(z) = 2
(c) Re(z) + Im(z)= 2 (d) None of these
Solution : Let z x iy , then | z i Re( z) || z Im( z ) | Implies | x iy ix | x iy y | 2 2 x 2 y x x y y 2 or x 2 y 2 or x y .
Thus, z lies on Re(z) = Im(z ) .
ILLUSTRATION : 06 If
is
a
complex
cube
root
of
unity,
then
value
of
expression
2 cos 1 1 ...... 10 10 2 900
(a) -1
(b) 0
(c) 1
(d)
Solution : 2
We have k k k 2 k 2 3 = k 2 k 1 1 k 2 k 1 10
k k k 1
=
2
10
k 2 k 1 k 1
10 x 11 x 21 10 x 11 10 385 55 10 450 6 2
20
COMPLEX NUMBERS
10 2 450 Thus, . cos k k 900 cos 900 0 k 1
ILLUSTRATION : 07 The roots z1 ,z2 ,z3 of the equation x 3 3 px 2 3qx r 0 p,q,r are complex correspond to points A, B and C. Then triangle ABC is equilateral if (a) p q 2 (b) p 2 3q
(c) p 2 q
(d) q 2 3 p
Solution : We have z1 z 2 z3 3 p, z2 z3 z3 z1 z1 z2 3q and z1 z2 z3 r Triangle A(z1), B(z2), and C(z3) is an equilateral triangle if and only if 1 1 1 0 z2 z3 z3 z1 z1 z 2
z3 z1 z1 z2 z2 z3 z2 z3 z2 z3 z3 z1 0
z12 z 22 z32 z2 z3 z3` z1 z3 z1 z1 z2
z1 z2 z3
3 p
2
2
3 z 2 z 3 z 3 z1 z 1 z 2
3 3q
p2 p
ILLUSTRATION : 08 The system of equation | z 1 i | 2 and |z| = 3 has (a) no solution
(b) one solution
(c) two solutions
(d) infinite number of solutions.
Solution : The given system of equations represent the system of circles (x+1)2 + (y-1)2 = 2 and and The distance between their centers is and difference between their radii is 3 2 2 . Therefore, the first circle lies within the second circle. Therefore the given system of given has no solution.
21
IIT-MATHEMATICS-SETII
ILLUSTRATION : 09 If x iy
3 , then 4x x 2 y 2 reduces to cos i sin 2
(a) 2
(b) 3
(c) 4
(d) 5
Solution :
x iy 1 1 cos 2 i sin 2 2 x y x iy 3
x 1 y 1 cos 2 , 2 sin 2 2 x y 3 x y 3
x 2 y 1 2 2 2 2 3 x y 9 x y
2
2
x2 y 2
2
4x
x
1 3 4x 1 0 x y2
2
y
2 2
3 x2 y
2 2
1 0 3
2
4x - x2 - y2 = 3
ILLUSTRATION : 10 If a,b,c, p,q,r are three complex numbers such that value of (a) 0
Solution :
a b c p q r 1 i and 0 then p q r a b c
p2 q 2 r 2 is a2 b2 c 2 (b) -1
p q r 1 i, a b c
(c) 2i
(d) -2i 2
2 p q r 1 i 2i a b c
p 2 q 2 r 2 2abc a b c p 2 q2 r 2 qr rp pq 2 2i 2 2 2 2 2i a b c pqr p q r a b2 c 2 bc ca ab
p2 q 2 r 2 2 2 2 2i a b c 22
COMPLEX NUMBERS
SECTION-A SINGLE ANSWER TYPE QUESTIONS 1.
If iz 3 z 2 z i 0 then the value of |z| is (A) 1
2.
(B) 2
(B) parabola
2
then :
2
(C)Re(z)=1
(D) Im(z) = 1
(C)|z| >5
(D)None of these
| z | 2 | z | 1 2 then the locus of z is log If 3 2 | z | (B) |z|<5
If w is a complex cube root of unity, then the value of (A) 1
6.
(D) ellipse
log 1 z 2 log 1 z
(B) Im (z)>1
(A) |z|=5
5.
(C) line
Let z 2 be a complex numbers such that (A) Re(z) >1
4.
(D) >1
If the complex numbers z1 , z 2 , z 3 are in A.P. then they lie on (A) circle
3.
(C) <1
(B) –1
a b c 2 a b c 2 is c a b 2 b c a 2
(C) 2
(D) 0
The value of the expression
2.1 1 2 32 2 2 43 3 2 ..... n 1n n 2 where w is an imaginary cube roots of unity is : nn 1 (A) 2
7.
2
1 1 1 (B) z z ..... z (C)0 1 2 n
(B) p 2 3q
Given that the real points of to z
23
(D) None of the above
(D) None of these
The Origin and the roots of the equation z 2 pz q 0 form an equilateral triangle if (A) p 2 q
9.
2
nn 1 (C) n 2
If | z1 || z 2 | ...... | z n | 1, hen the value of | z1 z 2 z 3 ........ z n | is
(A) n 8.
2
nn 1 (B) n 2
5 12i 5 12i 5 12i 5 12i
:
5 12i and
(C) q 2 3 p
(D) q 2 p
5 12i are negative, then the number reduces
IIT-MATHEMATICS-SETII (A) 10.
3 i 2
3 (B) i 2
2 (C) 3 i 5
(D) None of these
The value of z satisfying the equation log z log z 2 ...... log z n 0 is 4m
4m
4m
4m
(A) cos n n 1 i sin n n 1 , m 1, 2,...... (B) cos n n 1 i sin n n 1 , m 1,2,.... (C) sin 11.
4m 4m i cos , m 1,2,... n n
The complex numbers sin x i cos 2 x and cos x i sin 2 x are conjugate to each other for : (A) x = np
12.
(D) 0
1 2
(B) x n
5z 1
(C) x = 0
(D) No value of x
2z 2 3z1
If 7 z is a purely imaginary number then 2z 3z = 2 2 1 (A) 5/7
(B) 1
(C) 7/5
(D) None of these az
13.
bz
1 2 If z1 and z 2 are two distinct points in an Argand plane. If a | z1 | b | z 2 | then bz az is a point on 2 1
the (A) line segment [-2, 2] of the real axis (C) unit circle |z| =1
14.
(B) the imaginary axis (D) the line are z= tan-12
q1
q2
q3
If q1 , q 2 , q3 are the roots of the equation x 3 64 0 then the value of the determinant q 2
q3
q1
q1
q2
q3
is : (A) 1 15.
(D) None of these
(B) 4k+2
(C) 4k+3
(D) 4k
If a,b,c are real and a + ib = (c + id)1/3 then 4 a 2 b 2 is equal to (A)
17.
(C) 16
Let Z1 and Z2 be the nth roots of unity which subtend a right angle at the origin. Then n must be of the form. (A) 4k+1
16.
(B) 4
a c b d
(B)
a b c d
(C)
c d a b
(D) None of these
3 3 1, , 2 are the cube roots of unity, then a b 3 a b 2 a 2 b
(A) a 3 b 3
(B) 3 a 3 b 3
(C) a 3 b 3
(D) a 3 b 3 3ab
24
COMPLEX NUMBERS 18.
1 i 2 i 3i
(A)
1 2
(B)
1 2
(C)1
(D)-1
(C) 3
(D) 4
81
19.
1 1 3 is The value of 2 2
(A) 1 20.
(B) 2
If 1 2i is a root of the equation x 2 bx c 0 where b and c are real then (b,c) is given by: (A) (2,-5)
21.
(B) (-3,1)
n n 1 2
2
2
n n 1 n 2
__
(C)
(B) 3
(C) 4
(D) 6
(B) concyclic (D) the vertices of a triangle 2
2
1 2 1 1 27 If x x 1 0, then the value of x x 2 .... x 27 is x x x 2
(B) 72
(C) 45
If | i | 1 , i 0 for i 1,2,3,....n and | 11 2 2 ...... n n | is
(D) 54
1 2 ....... n 1 then t he value of
(A) equal to 1 (B) less than 1 (C) greater than 1 (D) none of the above If p, q are 2 real numbers lying between 0 and 1 such that z1 p i , z 2 1 qi,& z 3 0 form an equilateral triangle, then (p, q) =
(A) 2 3 , 2 3 27.
(D) None of these
If the roots of z 1n i z 1n are plotted in the argand plane they are
(A) 27.1
26.
(B) 2 3, 2 3
(C) 3 5 ,3 5
(D) None of these
If a,b,c and u , v, w are complex numbers representing the vertices of two triangles such that c = (1-r)a+rb & w = (1-r)u + rv where r then the two triangles are (A) similar
25
__
2
25.
The equation z z 2 3i z 2 3i z 4 =0 represents a circle of radius :
(A) on a parabola (C) collinear
24.
2
n n 1 n 2
(B)
(A) 2 23.
(D) (3,1)
The value of the expression 1.2 . 2 2 + 2.3 3 2 ...... + n 1n n 2 , where w is an imaginary cube root of unity is (A)
22.
(C) (-2, 5)
(B) congruent
(C) of same area
(D) none of these
IIT-MATHEMATICS-SETII 28.
If z is a complex number, then 3z 1 3 z 2 represents A) y - axis
B) a circle
C) x - axis
D) a line parallel to y - axis
n
29.
zi , n integral, then w lies on the unit circle for If w 1 iz A) only even n
30.
B) only odd n
Two of the three values of 11 / 3 are cis π π A) cos i sin 3 3
31.
B) cos
1
5π 5π i sin 3 3
1
2 i 2 2 i 2 A)
C) -1
D) 1
x 1
w
w2
w
x w2
1
1
xw
B) x = w
D) all n
5 and cis . The third value is 3 3
If w is a cube root of unity then a root of
A) x = 1
32.
C) only positive n
w
2
C) x = w2
0
is
D) x = 0
8 5
B)
25 8
C)
5 8
D)
8 25
8
33.
1 cos i sin 8 8 1 cos i sin 8 8
A) 1 + i 34.
B) 0
D) -1
C) i
D) none of these
1 sin 1 z 1 , where z is non real, can be angle of a triangle if i A) Re(z) = 1, Im (z) = 2 C) Re(z) + Im (z) = 0
36.
C) 1
If the fourth roots of unity are z1 , z 2 , z 3 , z 4 then z12 z 22 z 32 z 24 is equal to A) 1
35.
B) 1 - i
B) Re(z) = 1, -1 Im (z) 1 D) none of these
If z = -1, then principal value of is equal to A)
B)
C)
D) 26
COMPLEX NUMBERS 37.
If a and b are two distinct complex numbers such that and Re () > 0, Im(b) < 0, then
may be
A) zero B) purely imaginary C) real and positive D) real and negative 38.
If z is a complex number such that z 0 and Re z = 0 then A ) Re z2 = 0
39.
1 1 i 2
3 2
B)
45.
46.
If x
D) none of these
3 2
C) 0
D) 1
C) 2
D) 1
C)
D) 0
C) 10
D) 2
1 1 2 cos , then x 5 5 x 10 x
B) 32
If z1 , z 2 , z3 are the vertices of an equilateral triangle inscribed in the circle |z| = 2 and if then z1 1 i 3 , A) z2 2, z3 1 i 3
B) z2 2, z3 1 i 3
C) z2 2, z3 1 i 3
D) z2 1 i 3, z3 1 i 3
If i tan 1 z, z x iy and is constant, then locus of z is A) x 2 y 2 2x cot 2 1
B) x 2 y 2 cot 2 1 x
C) x 2 y 2 2 y tan 2 1
D) x 2 y 2 2x sin 2 1
If x = 2 + 5i (where i2 = -1), then the value of x 3 5x 2 33x 19 B) 8
C) 10
D) 12
C) Im (z) > 0
D) Im (z) < 0
|z - i| < |z+i| represents the region A) Re (z) > 0
27
1 1 i 2
a b c 1, and then cos cos cos b c a
B) i
A) 6 47.
C)
If i = and n is a positive integer, then
A) 0 44.
1 1 i 2
B) 3
A) 1
43.
D) none of these
Number of solutions of system of equations Re(z2) = 0, |z| = 2 is i n i n 1 i n 2 i n 3 A) 4
42.
B)
If a = cis , b = cis , c = cis
A) 41.
C) Im z2 = 0
If z2 = -i, then z is equal to A)
40.
B) Re z2 = Im z2
B) Re (z) < 0
IIT-MATHEMATICS-SETII 48.
n1
n
n2
For positive integers n1 and n2 , the value of the expression 1 i 1 1 i 3 i i 5 1 i7
n2
is a real number if and only if i 2 1 A) n1 n2 1 49.
B) 3
4
C) 2 2 ,
(B) 170
D)
(C) 197
(B) 4
2,
2
(D) -188
(C) 0
(D) None of these
If is an nth root of unity, then 1 2 3 2 ...... n n 1 equals n 1
(B)
n (C) 1 2
n 1
(D) None of these
The value of x 4 9x 3 35x 2 x 4 for x 5 2 4 is (A) 0
55.
4
If m,n,p,q are consecutive integers, then the value of i m i n i p i q is
(A)
54.
D) 6
Let 2z = 7 i 3 . Then the expression (z2 - 7z + 1)2 = 2(z2 - 7z) reduces to …..
(A) 1 53.
C) 4
2,
B)
(A) 145 52.
D) n1 0, n2 0
Complex numbers 8 + 5i, -3+i and -2 - 3i represent the point A, B and C respectively, then the modulus and argument of the complex number representing the centroid of the triangle ABC are A) 2,
51.
C) n1 n2
The equation zz 2 3i z 2 3i z 4 0 represents a circle of radius A) 2
50.
B) n1 n2 1
(B) –160
(C) 160
(D) -164
The equation of the right bisector of the line joining the points z1 & z 2 is : __
(A) z
__
1 z1 z2 2
(B) z z
1 z1 z2 2
(C) ( z z1 )( z z1 ) = ( z z2 )( z z2 ) (D) None of the above 56.
Common roots of the equations z 3 2z 2 2z 1 0 and z 1985 z 100 1 0 are (A) w, w2
57.
(C)-1, w, w2
(D) -w, -w2
If , and are the cube roots of p, p<0, then for any x, y and z, the values of (A) w, w2
58.
(B) 1, w, w2
The
(B) -w, -w2 value
(C) 1, -1 of
x y z are x y z
(D) none of these t he
expression 28
COMPLEX NUMBERS 1 1 1 1 2
1 1 2 2 2
1 1 1 1 3 3 2 ....... n n 2 where w is
an imaginary cube root of unity is (A)
59.
n n2 2 3
(C)
3
(D) None of these
4m (C) n n 1
m (D) n 1
are the nth roots of unity, then 1 a1 1 a2 ....... 1 an1 is (B) 1
(C) n
(D) -n
If is an imaginary cube root of unity, then the value of
1 1 1 is 1 2w 2 w 1 w
(A) –2 62.
2m (B) n n 1
I f 1, a1, a2, a3…….an1
(A) 0
61.
3
n n 2 1
If cos i sin cos 2 i sin 2 ..... cos n i sin n 1 then the value of q is: (A) 4m
60.
(B)
n n2 2
(B) –1
(C) 1
(D) 0
(C) a pair of st. lines
(D) none of the above
If z 2 z | z | | z |2 0 then the locus of z is: (A) a circle
(B) a straight line
n
63.
1 bi If a1 ib1 a2 ib2 ..... an ibn A iB, then tan a is i 1 i
(A) B/A
64.
29
(B) 25i
(C) 25 1 i
(B) 4,15
(C) 0,
The maximum value of |z| when z satisfies the condition z (A)
67.
(D) tan-1 (A/B)
(D)100 1 i
If z be a complex number such that | z 5 | 7, then the minimum and maximum values of | z 2 | are (A) 2,10
66.
(C) tan-1 (B/A)
The sequences S i 2i 2 3i 3 ....... upto 100terms simplifies to (A) 50 1 i
65.
(B) tan (B/A)
3 1
(B) 3
If | z | 2 1 then | z 2 2 z cos | is
(D) 0, 10 2 2 is z
(C) 3 1
(D) 2 3
IIT-MATHEMATICS-SETII (A) less than 1
68.
(B)
The complex numbers z1, z2 and z3 satisfying (A) of area zero (C) right angled isosceles
69.
If =
(D)None of these
2 1
z1 z3 1 i 3 are the vertices of a triangle which is z2 z3 2
(B) Equilateral (D)obtuse-angled isosceles
2 , then the 10th term of the series1 cos i sin cos i sin ..... is 6
(A)-i 70.
(C)
2 1
(B) i
(C)
1 i 3 2
(D)
1 i 3 2
If 1 and 2 are the complex cube roots of unity, then1n 2n equals. 1 (A) 1 2
(B) 212
1 (C) 1
(D) 212
2
13
71.
n n 1 The value of i i , where i 1 equals i 1
(A) i 72.
1 2 (B) | z | 3
The value of x,y which satisfy the equation (A) x =0, y =1
74.
(C)-i
(D) 0
The area of the triangle with vertices affixes at z, iz, (z(1+i)) is 1 2 (A) | z | 4
73.
(B) i –1
(B) x =1, y=0
1 2 (D) | z | 2
(C) | z |2
1 i x 2i 2 3i y i i 3i
3i
(C) x=3, y=-1 n
are (D) x =-1, y=3
n1
n2
n2
For positive integers n1, n2 the value of the expression 1 i 1 1 i 3 1 i 5 1 i 7 where
i 1 is a real numbers if and only if
75.
(A) n1 n2
(B) n1 n2 1
(C) n1 n2
(D) for all integral value of n
a ib tan i log is equal to a ib
(A) ab
(B)
2ab 2 a b2
(C)
a2 b2 2ab
(D)
2ab a b2 2
30
COMPLEX NUMBERS 76. If 1,, 2 , ……….. n 1 are nth roots of unity.. The value of 3 3 2 3 3 ………. 3 n 1 is (A) n 77.
78.
(B)0
(C)
3n 1 2
(D)
3n 1 2
If z1 , z2 , z3 be vertices of an equilateral triangle occurring in the anticlock wise sense, then: (A) z12 z22 z32 2 z1 z2 z2 z3 z3 z1
1 1 1 (B) z z z z z z 0 1 2 2 3 3 1
(C) z1 z1 2 z1 0
(D) None of the above 3
The roots of the cubic equation z 3 , 0 represent the vertices of a triangle of sides of length: (A)
1 | | 3
(B) 3 | |
(C) 3 | |
(D)
1 | | 3
79.
The centre of a square ABCD is at z =0. The affix of the vertices A is Z, then the affix of the centrioid of the triangle ABC is z (A) z1 cos i sin (B) 1 cos i sin 3 z1 (C) z1 cos i sin (D) cos i sin 2 2 3 2 2
80.
If S = i n i n where i 1 and n is an integer, then the total number of possible distinct values of S is A) 1
81.
If
C) a parabola
D) none of these
B) |z| > 1
C) |z| < 1
D) none of these
B) 2 2
C) 2 2 1
D) 2 2 2
The area of the triangle whose vertices represents the complex numbers z, -iz and z+iz is A) |z|2
31
B) a circle
If |z - 2 - 2i| = 1 then the minimum value of |z| is A) 2 2 1
84.
D) more than 3
z 1 is purely imaginary then z 1
A) |z| = 1 83.
C) 3
The points representing the complex number z for which |z+3|2 - |z - 3|2 = 6 lie on A) a straight line
82.
B) 2
B)
1 2 |z| 2
C) 2 |z|2
D) none of these
IIT-MATHEMATICS-SETII 85.
If z 0 is a complex number such that arg(z) = A) Im (z2) = 0
86.
89.
D) none of these
B)
C)
2
D)
2
1 1 1 If z1 , z 2 , z3 are complex numbers such that | z1 || z2 || z3 | z z z 1, then | z1 z2 z3 | is 1 2 3 A) equal to 1
88.
C) Re (z) = Im (z2)
If arg (z) < 0, then arg(-z)-arg(z) = A) -
87.
B) Re (z2) = 0
, then 4
B) less than 1
D) greater than 3
D) equal to 3
The complex numbers z1 , z 2 , z3 satisfying
z1 z3 1 i 3 are the vertices of a triangle, which is z2 z3 2
A) of area zero C) equilateral
B) right angled isosceles D) obtuse angled isosceles
For what values of x and y, the complex numbers 9y2 - 4 - 10xi and 8 y 2 20i 7 are conjugate to each other A) x = -2, y = 2
90.
B) x = - 2, y = -1
C) x = 2, y = 2
D) x = 2, y = -2
The point represented by the complex number 2 – i is rotated about origin through an angle
of in 2
clockwise direction. The new position of the point is A) 1 + 2i 91.
D) -1 + 2i
B) Re z < 0
C) Re z > 3
D) Re z > 2
z and w are two non-zero complex numbers such that |z| = |w| and Arg z + Arg w = p, then z = A) w
93.
C) 2 + i
If |z - 4| < |z - 2|, its solution is given by A) Re z > 0
92.
B) -1-2i
B) w
C) w
D) -w
The locus of the centre of a circle which touches the circles | z z1 | a and | z z2 | b externally ( z , z1 , z 2 are complex numbers) will be A) an ellipse
94.
B) a hyperbola
C) a circle
D) none of these
If z and w are two non zero complex numbers such that |zw|=1 and Arg(z)-Arg(w) = z , z1 , z 2 , then zw is equal to
A) - 1
B) 1
C) -i
D) i 32
COMPLEX NUMBERS 95.
Let be the roots of the equation z 2 az b 0, z being complex. Further, assume that the origin, z1 and z2 form an equilateral triangle, then A) a2 = 4b
96.
B) a2 = b
The complex number z is such that |z | = 1, z -1 and w
| z 1|2 A) | z 1|2 97.
C) a2 = 2b
B) -1
C)
D) a2 = 3b
z 1 , then real part of w is z 1
3 | z |2
D) 0
If z1 , z2 and z3 are any three complex numbers then the fourth vertex of the parallelogram, whose three vertices taken in order are z1 , z2 , z3 is A) z1 z2 z3
98.
C)
n
1 z1 z2 z3 3
D)
1 z1 z2 z3 3
D)
2
n
If n is a positive integer, then 1 i 1 i is equal to A)
99.
B) z1 z2 z3
2
n 2
cos
n 4
B)
2
n 2
sin
n 4
C)
2
n
n 2
cos
n 4
n 2
sin
n 4
n
If w ( 1) be a cube root of unity and 1 w2 1 w4 , then the least positive integral value of n is A) 2
B) 3
C) 5
D) 1
100. If A z1 , B z2 be two points such that | z1 z2 | | z1 z2 | and iz1 kz2 ; k R, then an angle between AB and AB’; B’ being reflection of B in the origin, is 1 2 k A) tan 2 k 1
1 2 k B) tan 2 1 k
C) 2 tan 1 k
D) 2 tan 1 k
C) Im (z) < 0
D) Im (z) > 0
101. If log1/6 | z 1| log1/ 6 | z 1|, then A) Re (z) < 0
B) Re (z) > 0
102. If z C and | z 4 | 3, then the least value of |z + 1| is A) -6 103. If z A)
B) 3
C) 0
D) 2
4 2, then the greatest value of |z| is z
5
B) 5 1
C)
5 1
104. If |z-5i| £ 3, then |maximum amp(z) - minimum amp (z)| is equal to 33
D) none of these
IIT-MATHEMATICS-SETII 1 3 1 3 A) sin cos 5 5
B)
1 3 C) 2 cos 5
3 cos 1 2 5 1 3 D) cos 5
1 1 1 105. 1 z1 , z 2 ...........zn 1 , are nth roots of unity the value of 3 z 3 z ....... 3 z is 1 2 n 1 n.3n 1 1 a) n 3 1 2
n.3n 1 1 B) n 3 1
n.3n 1 1 C) n 3 1
D) none of these
34
COMPLEX NUMBERS
KEY 1
2
3
4
5
6
7
8
9
10
11
12 13
14
15
A
C
A
C
B
C
B
A
A
AB
D
B
A
D
D
16
17
18 19
20
21
22 23
24
25
26
27 28
29
30
C
B
C
A
C
B
B
C
D
B
B
A
D
D
C
31
32
33 34
35
36
37 38
39
40
41
42 43
44
45
D
D
D
B
B
B
B
C
C
D
A
D
A
A
A
46
47
48 49
50
51
52 53
54
55
56
57 58
59
60
C
C
D
C
B
B
C
B
B
C
A
A
A
C
C
61
62
63 64
65
66
67 68
69
70
71
72 73
74
75
D
C
C
D
C
A
A
A
B
D
D
B
35
A
B
C
IIT-MATHEMATICS-SETII
KEY 76
77
78 79
80
81
82 83
84
85
86
87 88
89
90
C
C
B
D
C
A
A
B
B
B
A
A
B
91
92
93 94
95
96
97 98
99 100 101 102 103 104 105
B
B
B
D
D
A
B
C
A
C
C
A
C
C
B
A
A
36
COMPLEX NUMBERS
SECTION-B MORE THAN ONE ANSWER TYPE QUESTIONS
1.
The nonzero real value of x for which A) 2
2.
C)- 2
B) 1 1
D) none of these
1
If z1 = a i , a 0 and z2 = 1 bi , b ¹ 0 such that z1 = z 2 then A) a = 1, b = 1
3.
(1 ix ) (1 2ix ) is purely real is 1 ix
B) a = -1, b = 1
C) a = 1, b = -1
D) none of these
If z1, z2,z3 z4 are roots of the equation a0z4 + a1z3 + a2z2 + a3z + a4 = 0, where a0, a1, a2, a3 and a4 are real, then A) z1 , z 2 , z 3 , z 4 are also roots of the equation B) z1 is equal to at least of z1 , z 2 , z 3 , z 4 C) z1 , z 2 , z 3 , z 4 are also root of the equation D) none of these
4.
If a is a complex constant such that az2 + z + = 0 has a real root then A) + = 1 C) + = -1
5.
B) + = 0 D) the absolute value of the real root is 1
If amp (z1, z2) = 0 and çz1ç = çz2ç = 1 then A) z1 + z2 = 0 B) z1 z2 = 1
C) z1 = D) none of these
z 6.
If z is a nonzero complex number then z z is equal to
A) 7.
2
z z
B) 1
C) z
If w is a nonreal cube root of unity then the value of 1 . (2 - ) (2 - 2) + 2 . (3 - ) (3 - 2) + ... + (n - 1) (n - ) (n - 2) is
37
D) none of these
IIT-MATHEMATICS-SETII A) real 8.
n 2 (n 1) 2 B) - n +1 4
D) amp z = tan-1 2
B) Re(z) + 2Im(z) = 0 C) = Ö5
B) nonreal, whose real and imagniary parts are eqaul D) none of these
If z1, z2 are two complex numbers then B) z1 z2 z1 z2
C) z1 z2 z1 . z2
D) z1 z2 z1 z2
Let z1, z2 be two complex numbers represented by points on the circle |z| = 1 and z| = 2 respectively then
A) max 2 z1 z2 = 4 B) min z1 z2 = 1 16.
D) none of these
zi If z is different from ± i and z = 1 then z i is
A) z1 z2 z1 z2 15.
D) none of these
C) 1 - 2w2
B) 1 - 2w
A) purely real C) purely imagniary 14.
C) 3
1 3i If z = 1 i then
A) Re(z) = 2Im(z)
13.
B) -1
Let x be a nonreal complex number satisfying (x - 1)3 + 8 = 0 then x is A) 1 + 2 w
12.
B) -1 if n is not a multiple of 3 D) none of these
The value of a4n-1 + a4n-2 + a4n-3 n N and is nonreal fourth root of unity, is A) 0
11.
B) (-1)n-1 when n is not a multiple of 3 D) 0 when n is not a multiple of 3
The value of a-n + a-2n, n N and a is a nonreal cube root of unity, is A) 3 if n is a multiple of 3 C) 2 if n is a multiple of 3
10.
D) none of these
If z is a complex number satisfying z + z-1 = 1 then zn + z-n, n N, has the value A) 2(-1)n when n is a multiple of 3 C) (-1)n+1 when n is multiple of 3
9.
n ( n 1)2 - n C) 2
1 C) z2 z 3
D) none of these
1
ABCD is a square, vertices being taken in the anticlockwise sense. If A represents the complex number z and the intersection of the diagonals is the origin then A) B represents the complex number iz C) B represents the complex number i
B) D represents the complex number i z D) D represents the complex number - iz
38
COMPLEX NUMBERS 17. If z ( z ) z ( z ) = 0, where a is a complex constant, then z is represented by a point on A) a string line 18.
B) a circle
C) a parabola
D) none of these
If z1, z2, z3, z4 are the four complex numbers represented by the vertices of a quadrilateral taken in order z4 z1 such that z1- z4 = z2 - z3 and amp z z 2 then the quadrilateral is a 2
A) rhombus 19.
1
B) square
C) rectangle
D) a cyclic quadrilateral
If z0, z1 represent points P, Q on the locus z 1 = 1 and the line segment PQ subtends an angle /2 at the point z = 1 then z1 is equal to i B) z 1 0
A) 1 + i (z0 - 1) 20.
B) z1 + z2 = z3 + z4
z2 z4 C) amp z z 2 1 3
z1 z2 D) amp z z 2 3 4
B) z0 z + z0 z = 12
C) z0
D) none of these
z
+ z0 z = 0
1 1 If z1 - z2 and z1 z2 z z then 1 2
Let z = 1
A) z1 z2
39
D) none of these
z2 If amp 2 z 3i = 0 and z0 = 3 + 4i then
A) at least one of z1, z2 is unimodular C) z1 . z2 is unimodular
25.
C) z1 z2 = 1
If z1, z2, z3, z4 are represented by the vertices of a rhombus taken in the anticlockwise order then
A) z0 z + z0 z = 12
24.
C) z1 z2 + z2 z3 + z3 z1 = 0 D) none of these
B) z1 z2 = 1
A) z1 - z2 + z3 - z4= 0
23.
B) z1 z2 z3 = 1
Let A, B, C be three collinear points which are such that AB. AC = 1 and the points are represented in the Argand plane the complex numbers 0, z1, z2 respectively. Then A) z1z2 = 1
22.
D) i (z0 - 1)
If |z1| = |z2| = |z3| = 1 and z1, z2, z3 are represented by the vertices of an equilateral triangle then A) z1 + z2 + z3 = 0
21.
C) 1 - i (z0 - 1)
2
2
3 i . 1 3i 1 i
B) both z1, z2 are unimodular D) none of these
, z 1 3i . 2
1 i
3 i
. Then
B) amp z1 + amp z2 = 0 C) 3 |z1| = |z2|D) 3 amp z1 + amp z2 = 0
IIT-MATHEMATICS-SETII 26.
If z1 z2 z1 z2 then A) amp z1 amp z2
2
B) amp z1 amp z2
C) z1/z2 is purely real 27.
2
2
If z1 z2 z1 z2
D) z1/z2 is purely imaginary 2
then
z1 A) z is purely real
z1 B) z is purely imaginary 2
C) z1 z2 z 2 z1 0
z1 D) amp z 2 2
2
28.
z1 = a + ib, z2 = c + id are complex numbers given that çz1ç = çz2ç = 1 and R (z1 ) = 0, then a pair of complex numbers w1 = a + ic and w2 = b + id satisfies (a,b,c,d ÎR A) çw1ç = 1
29.
B) çw2ç = 1
B) modulus = 6
Which of the following are correct for any two complex numbers z1 and z2?
If z = x + iy, then the equation A) ½
33.
D) arg z = tan-1 (18)
B) çz1 + z2ç = çz1ç + çz2ç D) çarg z1 - arg z2 ç = p/3
A) çz1 z2ç = çz1ç çz2ç C) çz1 + z2ç = çz1ç + çz2ç
32.
3 C) arg z = tan-1 4
If the vertices of an equilateral triangle are situated at z = 0, z = z1 and z = z2, then which of the following are true? A) çz1ç = çz2ç C) çz1 - z2ç = çz1ç
31.
D) All above
The modulus and the principal argument of the complex number z = +4i are A) modulus = 13
30.
C) Re (w1 ) = 0
B) arg (z1 z2) = arg (z1). arg (z2) D) çz1 - z2ç ³ çz1ç - çz2ç
2z i z 1 = m represents a circle when m =
B) 1
C) 2
D) 3
If the points z1, z2, z3 are the affixes of vertices of an equilateral triangle, then 1 1 1 A) z z z z z z = 0 1 2 2 3 3 1 2
2
B) z12 z22 z32 z1 z2 z2 z3 z3 z1 2
C) z1 z2 z 2 z3 z3 z1 = 0 D) z13 z2 3 z33 3z1 z2 z3 = 0 34.
If the imaginary part of the complex number (z - 1) (cos - i sin ) + (z - 1)-1 (cos + i sin ) is zero, 40
COMPLEX NUMBERS then A) z - 1= 1 35.
B) arg (z - 1) =
C) arg (z) =
D) z = 1
If P (x) and Q(x) be complex polynomials and let f(x) = P(x3) + xQ(x3). Suppose f(x) is divisible by x2 + x + 1, then A) P(x) is divisible by (x - 1) but Q(x) is not divisible by x - 1 B) Q(x) is divisible by (x - 1) but P(x) is not divisible by x - 1 C) both P(X) and Q(x) divisible by x - 1 D) f(x) is divisible by x - 1
36.
2
If z is the affix of a moving point in argand plane then the equation z 2 z 2 2 z z z = 0 represents a A) straight line
37.
B) conic
D) parabola
Equation of the line in argand plane joining two points with affixes z1 and z2 must be A) z = tz1 + (1 - t)z2, t R
C)
38.
C) byperbola
z
z
z1 z2
z1 1 =0 z2 1
z z1 B) arg z z is purely real 2 1
1 D) z z1 z z2 z z2 z z1
The inequality sin çzç > 0 represent A) a circle whose centre is origin and whose radius is p B) a parabola whose vertex is (0, 0) C) an annular region between two concentric circles centred at (0, 0) and having radii 2p and 3p D) an ellipse of semi-axes p and 2p
39.
40.
If z0 =
1 i 2 22 2n then the value of the product 1 z0 1 z0 1 z0 ... 1 z0 must be 2
B) (1 + i) if n > 1
C) (1 + i) if n = 1
D) 0
If x, y a, b are real numbers such that (x + iy)1/5 = a + ib, and P = x/a - y/b. then B) (a + b) is a factor of P D) a - ib is a factor of P
If z1, z2,z3 be the affixes of vertices of an equilateral triangle and z0 be the affix of the circum centre then A) z0 = z1 + z2 + z3
41
A) 2n 1
A) (a - b) is a factor of P C) a + ib is a factor of p 41.
B) çz0 - z1ç = çz0 - z2ç = çz0 - z3
IIT-MATHEMATICS-SETII C) z02 z1 z2 z2 z3 z3 z1 42.
D) z0 =
z1 z2 z3 3
If x0, x1, x2 ....xn-1 be n, nth roots of unity where x0 = 1 then A) x1 + x2 + ...+ xn-1 = -1 C) (1 - x1) (1 - x2) ... (1 - xn-1) = n
B) x0 x1 x2 + ...............xn-1 = 1 D) x1 + x2 +x3.... xn-1 = 1.
42
COMPLEX NUMBERS
KEY 1
2
A,C
C
16
10
11
A,B A,C,D B,C A,B A,B A,B B,C
B
B,C A,C C
A,B A,B,C
17
18 19
20
22 23
24
25
26
29
A,D
B
C,D A,C
A,B B,C A,C B
C
A,D A,D B,C,DABCD
31
32
33 34
35
36
37 38
39
40
41
D
D
D
B
B
B
C
D
A
43
3
4
B
5
6
21
7
8
C
9
12 13
27 28
14
15
30
IIT MATHEMATICS
2
FUNCTIONS
44
FUNCTION
NUMBER SYSTEM
(i)
Natural Numbers
The set of numbers {1, 2, 3, 4, ..... } are called natural numbers, and is denoted by N. i.e., N = {1, 2, 3, 4, ..... } (ii) Integers The set of numbers {....., –3, –2, –1, 0, 1, 2, 3, .....} are called integers and the set is denoted by I or Z. where we represent; (a) Positive integers by I+ = {1, 2, 3, 4, .....} = Natural numbers. (b) Negative integers by I– = {....., –4, –3, –2, –1} (c) Non-negative integers = {0, 1, 2, 3, 4, ....} = Whole numbers (d) Non-positive integers = {....., –3, –2, –1, 0} (iii) Rational Numbers a , where a and b are integers, b 0 are called rational numbers b and their set is denoted by Q.
All the numbers of the form
i.e., Q
Note:
a such that a, b I and b 0 and HCF of a, b is 1. b
(1) Every integer is a rational number as it could be written as Q =
a (where b = 1) b
(2) All recurring decimals are rational numbers. 1 e.g., Q 0.3333.... 3
(iv) Irrational Numbers Those values which neither terminate nor could be expressed as recurring decimals are irrational numbers. i.e., it can not be expressed as
a form, and are denoted by Qc (i.e., compleb
ment of Q). e.g.,
2 , q 2,
1 3 2 1 , , , 3, 1 3, , ... etc. 2 2 2 3
(v) Real Numbers 45
IIT MATHEMATICS The set which contain both rational and irrational are called real number and is denoted by R. i.e., R = Q Q c
Note:
5 3 7 1 1 1 , , , , , , ....., 2 , 3 , , .....} 6 4 9 3 7 5
R = {..... –2, –1, 0, 1, 2, 3, ....., As from above definitions;
N I Q R , it could be shown that real numbers can be expressed on number line with respect to origin as;
–3
–2 2
–1
0
1
2
2
3
INTERVALS The set of numbers between any two real numbers is called interval. The following are the types of interval. (a) Closed Interval: [a, b] = {x : a x b} (b) Open Interval: (a, b) or ]a, b[ = {x : a < x < b} (c) Semi open or semi closed interval: [a, b[ or [a, b) = {x: a x < b} ]a, b] or (a, b] = {x: a < x b}
THE ABSOLUTE VALUE OF A REAL NUMBER The absolute value (or modulus) of a real number x (written |x|) is a non negative real number that satisfies the conditions. | x | = x if x 0 | x | = – x if x < 0 Example: | 2 | = 2, | –5 | = 5, | 0 | = 0 From the definition it follows that the relationship x |x| holds for any x. The properties of absolute values are (1) the inequality | x | means that – x ; if > 0 (2) the inequality | x | means that x or x – . (3) | x y| |x| + |y|; (4) | x y| | | x | – | y ||; 46
FUNCTION (5) |xy| = | x | | y |; x |x| = (y 0). y |y|
(6)
INEQUALITIES The following are some very useful points to remember: •
a b either a < b or a = b
•
a < b and b < c a < c
•
a < b –a > –b i.e., inequality sign reverses if both sides are multiplied by a negative number
•
a < b and c < d a + c < b + d and a – d < b – c. c R
•
a < b ma < mb if m > 0 and ma > mb if m < 0
•
0 < a < b ar < br if r > 0 and ar > br if r < 0
•
1 a 2 for a > 0 and equality holds for a = 1 a
•
1 a – 2 for a < 0 and equality holds for a = –1 a
DEFINITION OF FUNCTION Let A and B be two non-empty sets. Then a function ‘f ’ from set A to set B is a rule which associates elements of set A to elements of set B such that (i)
All elements of set A are associated to element in set B.
(ii) An element of set A are associated to a unique element in set B. Terms such as “map” (or mapping), “correspondence” are used as synonyms for function. If f f is a function from a set A to set B, then we write f : A B or A B. which is read as f is a function from A to B or f maps A to B. Example 1: Let A = {2, 4, 6, 8} and B = {s, t, u, v, w} be two sets and let f1, f2, f3 and f4 be rules associating elements of A to elements of B as shown in the following figures. 2 4 6 8
47
f1
s t u v w
f2 2 4 6 8
s t u v w
IIT MATHEMATICS f3
2 4 6 8
s t u v w
2 4 6 8
f4
s t u v w
Now see that f1 is not function from set A to set B, since there is an element 6 A which is not associated to any element of B, but f2 and f3 are the function from A to B, because under f2 and f3 each elements in A is associated to a unique element in B. But f4 is not function from A to B because an elements 8 A is associated to two elements u and w in B. Domain : Set A is called domain of f i.e. Set of those elements from which functions is to be defined. Co-Domain : Here set B is called co-domain of function. Range : Set of images of each element in A, is called range of f. Note: Range Co-domain
SOME ELEMENTARY FUNCTIONS General Exponential Function If a > 0, a 1 then the function defined by f(x) = ax, x R is called an Exponential Function with base a. Y –x
y=2
–x
y = 4 y = 10
–x
x
y = 10x y = 4 y = 2x
Domain : R Range : R+
a>1
Nature : one-one 0
O
X
Logarithmic Function If a > 0, a 1, then the function y = loga x, x R+(set of positive real numbers) is called the logarithmic Function with base a.
48
FUNCTION Y y = log2x y = log4x y = log10x
Domain : R + Range : R
X
O
Nature: one-one
y = log1/10x y = log1/4x y = log1/2x
Rational Function The function which can be written as the quotient of two polynomial function is said to be a rational function. If
P(x) = a0 + a1x + a2x2 + . . . + anxn Q (x) = b0 + b1x + b2x2 + . . . + bmxm
be two polynomial functions then a function f defined by P( x )
f(x) = Q( x ) is a rational function of x , where Q(x) 0 Examples: f(x) =
7x 4 x 2 2 x 2 4x 3
is a rational function which is defined for all real values of x except 1 and
3. Constant Function Let c be a fixed real number. The function f : R R (function f from R to R) is said to be a constant function if f(x) = c for every x R Clearly, domain of f = R and range of f = {c} Identity Function A map f : R R is said to be an identity function, iff f(x) = x, x R. The identity function is sometimes also called the function x Domain of the identity function = R Range of the identity function = R.
49
IIT MATHEMATICS y y = x, x > 0
y = –x, x < 0
x
x
O
y
Modulus Function f(x) = |x| =
x , x 0 x, x 0
Domain : R, Range : [0, ) Graph is symmetrical with respect to y-axis. y y = 1, x > 0
x
x
O
y = –1, x > 0
y Signum Function 1, x 0 |x| f (x) = , x 0 , or f(x) = 1, x 0 0, x 0 x
Domain : R; Range: {–1, 0, 1} Greatest Integer Function A function is said to be greatest integer function if it is of the form of f(x) = [x] = integer equal to or less than x. Examples: [3.7] = 3, [–3 2] = – 4, [5] = 5 etc.
50
FUNCTION y 3 2 1 -1
-2
01
x
2 3
-1 -2 -3
Properties of Greatest Integer Function (i)
x – 1 < [x] x
(ii) [x] + 1 > x (iii) If f(x) = [x + n], where n I then f (x) = n + [x] (iv) x = [x] + {x} where [.] and {.} denotes the integral and fractional part of x respectively 0; x I
(v) [x] + [– x] = 1; x I 0; x I
(vi) {x} + {– x} = 1; x I
Fractional Part of x f(x) = x – [x], x R i.e., f(x) = {x} x 1, x [ 1,0) x, x [0,1) = x 1, x [1,2) 0, xZ
y (0, 1)
-2
-1
0 1
2
3
x
Domain : R, Range : [0, 1), Nature : Many one into This is a many one function with period 1.
51
IIT MATHEMATICS
ALGEBRA OF FUNCTIONS Given function f : D1 R and g : D2 R, we describe function f + g, f – g, fg and f/g as follows f + g : D R is a function defined by (f + g)(x) = f(x) + g(x), f – g : D R is a function defined by (f – g) (x) = f(x) – g(x) fg : D R is a function defined by (fg) (x) = f(x) g(x)
f f f ( x) : C R is a function defined by ( ) (x) = g( x ) , g(x) 0, g g where C = {x D : g (x) 0} and D D1 D2
COMPOSITE FUNCTION Consider two functions f : X Y,,
g :Y Z
one can define h : X Z such that h(x) = g{f(x)} Domain of gof (x) i.e. g{f(x)} = {x : x Dom f, f(x) Dom g} Domain of fog (x) i.e f g(x) = {x : x Dom g, g(x) Dom f)
X
Y
f
x
f(x)
g
Z g (f(x))
h h = gof
EVEN AND ODD FUNCTION Let f : D R be a real function such that – x D as x D. Then f is called an even function if f(–x) = f(x) for every x D and an odd function if f(- x) = – f(x) x D. Graph of an even function is symmetrical about y-axis i.e., in I and II (or III and IV) quadrants always, whereas graph of an odd function is always symmetrical in diagonally opposite quadrants.
PERIODIC FUNCTION 52
FUNCTION Definition: A function f(x) is said to be periodic function if, there exists a positive real number T, such that, f(x + T) = f(x), x Domain. Then, f(x) is periodic with period T, where T is least positive value. A function is said to be periodic function if its each value is repeated after a definite interval. Here the least positive value of T is called the fundamental period of the function. Clearly f(x) = f(x + T) = f(x + 2T) = f(x + 3T) = . . . For example, sinx, cosx and tanx are periodic functions with fundamental period 2 , 2 and respectively..
Properties of Periodic Function: (i)
If f(x) is periodic with period T , then (a) c. f(x) is periodic with period T. (b) f(x + c) is periodic with period T. (c) f(x) c is periodic with period T.. where c is any constant not equal to zero.
(ii) If f(x) is periodic with period T, then, T
k f(cx + d) has period | c | , i.e., period is affected only by coefficient of x where; k, c, d constant, c, k 0 (iii) If f(x), g(x) are periodic functions with periods T1, T2 respectively then; we have h(x) = af(x) bg (x) has period as, (1) LCM of {T1, T2}; if (x) and g(x) can not be interchanged by adding a least positive number less than the LCM of {T1, T2}. (2) k; if f(x) and g(x) can be interchanged by adding a least positive number k (k < LCM of {T1, T2}).
CLASSIFICATION OF FUNCTION The following are the different kinds of function 1.
One-One Function (Injection): If each element in the domain of a function has a distinct image in the co-domain the function is said to be one-one function and is also known as Injective mapping. e.g. f : R R+ given by y = ex g : R R, g(x) = 3x – 7 are one - one functions.
53
IIT MATHEMATICS or, f : A B is one - one a b f(a) f(b) for all a, b A f(a) = f(b) a = b for all a, b A
2.
Onto Function (Surjection): Let , f : X Y be a function. If each element in the co-domain Y has at least one pre-image in the domain X i.e. Range f = Co domain, then f is called onto. Onto function are also called surjective and if function be both one-one and onto then function is called bijective. or, f : A B is a surjection iff for each b B a A such that f(a) = b . e.g. If f : R+ R is defined by y = log2x, then f(x) is Onto function.
3.
Into Function: If there exist one or more than one element in the Co-domain Y which is not an image of any element in the domain X. Then f is into. In other words f : A B is an into function if it is not an onto function. e.g. Let f : R R is defined by y = x2 + 1, then f(x) is an into function. But when f : R R+ is defined by y = x2 + 1, then f(x) is not onto function.
4.
Many-One Function: If there are two or more than two elements of domain having the same image then f(x) is called Many - One mapping. e.g. f : R R+ g : R R+
f(x) = x2 + 4 g(x) = x8 + x4 +x2 + 4
Both functions are many one Note: (i) = f(y).
f : A B is a many - one function if there exist x, y A such that x y but f(x)
e.g y = sin x, y = cos x, y = tan x, y = x2, y = x4, . . . . . are many one functions. (ii) Every even function is Many - One (iii) Every periodic function is Many - One
INVERSE OF A FUNCTION Let f : X Y be a function defined by y = f(x) such that f is both one–one and onto, then there exists a unique function g : Y X such that for each y Y, g(y) = x y = f(x). The function g so defined is called the inverse of f and in general denoted by f –1.
54
FUNCTION Further, if g is the inverse of f, then f is the inverse of g and the two functions f and g are said to be the inverse of each other. For the inverse of a function to exist, the function must be one– one and onto. Some standard functions are given below along with their inverse:
FUNCTION INVERSE FUNCTION 1.
f : [0, ) [0, ) f – 1 : [0, ) [0, ) defined by f(x) = x2 defined by f – 1 (x) = x
2.
3.
f : , [–1, 1] 2 2
f – 1: [–1, 1] , 2 2
defined by f(x) = sinx
defined by f – 1 (x) = sin–1x
f : [0, ] [–1, 1] f – 1 : [–1, 1] [0, ] defined by f(x) = cosx
defined by f – 1 (x) = cos–1x
GRAPHICAL TRANSFORMATIONS Few graphical transformations, which are pivotal in understading the pictorial representation of a function are given below. Students are advised to go through them and understand. F
Drawing the graph of y = |f(x)| from the known graph of y = f(x) |f(x)| = f(x) if f(x) 0 and |f(x)| = – f(x) if f(x) < 0. It means that the graph of f(x) and |f(x)| would coincide if f(x) 0 and the portions where f(x) < 0 would get inverted in the upward direction.
The above figure would make the procedure clear. F
Drawing the graph of y = f(|x|) from the known graph of y = f(x) It is clear that, f(|x|) = f(x), x 0 and f(|x|) = f(–x), x < 0. Thus f(|x|) would be an even function. Graphs of f(|x|) and f(x) would be identical in the first and the fourth quadrants (as x 0) and as such the graph of f(|x|) would be symmetrical about the y–axis (as (|x|) is even).
The figure would make the procedure clear. F
55
Drawing the graph of |y| = f(x) from the known graph of y = f(x)
IIT MATHEMATICS Clearly |y| 0. If f(x) < 0, graph of |y| = f(x) would not exist. And if f(x) 0, |y| = f(x) would y
y = |f(x)| x O y = f(x)
56
FUNCTION g i v e y = f(x). Hence graph of |y| = f(x) would exist only in the regions where f(x) 0 and will be reflected about x–axis only in those regions. Regions where f(x) < 0 will be neglected. y
|y| = f(x) x
O y = f(x)
Full lines show the graph of |y| = f(x) and dotted lines depict the corresponding graph of y = f(x). F
Drawing the graph of y = f(x + a), a R from the known graph of y = f(x) y = f(x)
y = f(x + a), a > 0
y = f(x + a), a < 0 x0 - |a|
x0
x0 + |a|
Let us take any point x0 domain of f(x), and set x + a = x0 or x = x0 – a. a > 0
x < x0, and a < 0 x > x0. That mean x0 and x0 – a would give us same abscissa for f(x) and f(x + a) respectively.
As such for a > 0, graph of f(x + a) can be obtained simply by translating the graph of f(x) in the negative x–direction through a distance ‘a’ units. If a < 0, graph of f(x + a) can be obtained by translating the graph of f(x) in the positive x–direction through a distance a units. F
Drawing the graph of y = a f(x) from the known graph of y = f(x) y = a f(x), a > 1
y = f(x) y = af(x), 0 < a < 1 x
It is clear that the corresponding points (points with same x co–ordinates) would have their ordinates in the ratio of 1 : a. F
57
Drawing the graph of y = f(ax) from the known graph of y = f(x)
IIT MATHEMATICS y y = f(x) y = f(ax), 1 < a
y = f(ax), 0 < a <1 x
Let us take any point x0 domain of f(x). Let ax = x0 or x =
x0 a
Clearly if 0 < a < 1 then x > x0 and f(x) will stretch by 1/a units against y–axis, and if a > 1, x < x0, then f(x) will compress by ‘a’ units against y–axis. F
Drawing the graph of y = f–1 (x) from the known graph of y = f(x) For drawing the graph of y = f–1(x) we have to first of all find the interval in which the function is bijective (invertible). Then take the reflection of y = f(x) (within the invertible region) about the line –1 y = x. The reflected part would give us the graph of y = f (x). e.g. let us draw the graph of y = sin–1 x. We know that y = f(x) = sin x is invertible
if f : , 1,1 the inverse mapping would be f–1 : [–1, 1] , 2 2 2 2 Y
(0, /2)
(1, / 2) y = sin–1 x y=x (/2, 1)
(–/2, 0)
(0, 1)
y = sinx O(1, 0)
/ 2, 1)
(/2, 0)
X
(0, – 1)
58
FUNCTION
59
IIT MATHEMATICS
ASSIGNMENT
60
FUNCTION
WORKED OUT ILLUSTRATIONS ILLUSTRATION : 01 A and B are two sets having 3 and 4 elements respectively and having 2 elements in common. The number of relations which can be defined from A to B is (a) 25
(b)210-1
(c) 212 1
(d) None of these
Solution : The number of elements in A x B is 12. Hence the number of subsets of A x B is 212.
ILLUSTRATION : 02
The period of the function f x cos 2 3 x tan 4 x is (a)
3
(b)
4
(c)
6
(d)
Solution : f x
1 1 cos 6 x tan 4 x. The period of cos6x is 2 and the period of tan 4 x is 2 6 3
. Hence the period of f is 1.c.m. of and 4 3 4
ILLUSTRATION : 03
x 1 The domain of the function f x sin log 3 is 3 (a) 1,9
(b) 1,9
(c) 9,1
(d) 3,9
Solution :
x The function f is defined only if 1 log 3 1 . This inequality is possible only if i . e . 3 1 x 3 1 x 9. 3 3
ILLUSTRATION : 04
61
IIT MATHEMATICS The domain of the function f x (a) (1, 4)
(b) (-2, 4)
log 0.3 x 1 x 2 2x 8
is
(c) (2,4)
(d) None of these
Solution : Since for , 0 a 1 , log a x 0 for x 1 so log 0.3 x 1 0 for x 2. Also x 2 2 x 8 0 if and only if x 2,4 Hence the domain of the given function is (2,4)
ILLUSTRATION : 05 1 x 1 x 2 The domain of definition of f x log 0.4 is x 5 x 36 (a) x : x 0, x 6
(b) x : x 0, x 1, x 6
(c) x : x 1, x 6
(d) x : x 1, x 6
Solution : x 1 x 1 tobe defined. We must have 0 1, which is true if x 1. x 5 x 5
For log 0.4
Morever,
1 is defined for x 6,6 . Hence the domain of f is x 36 2
ILLUSTRATION : 06 The function defined by is (a) both one-one and onto (c) onto but not one- one Solution :
(b) neither one-one nor onto (d) one- one but not onto
Since i.e. or so is onto. More ever the function is one-one on so if then which implies that which implies that The real solution of the last equation is given by . Hence is one-one.
ILLUSTRATION : 07 Part of the domain of the function lying in the interval is 1 5 (a) , ,6 6 3 3
1 5 (b) , ,6 6 3 3
1 (c) , 6 6
(d)None of these
Solution : The function
f is meaningful only if cos x
1 0, 6 35 x 6 x20 2
or
62
FUNCTION cos x
1 0, 6 35 x 6 x 2 0 i.e. 2
1 1 cos x , 6 x 1 6 x 0 or cos x , 6 x 1 6 x ) 2 2
1 5 These inequalities are satisfied if x , , 6 . 6 3 3
ILLUSTRATION : 08 1 Given f x | x | x and g x
1 x | x | then
(a) domf and dom g = (c)f and g have the same domain
(b) domf = and dom g (d) domf = and dom g =
Solution :
domf x :| x | x and domg x : x | x | Thus domf = R- (the set of negative real numbers) and domg = .
ILLUSTRATION : 09 Which of the following functions is not onto (a) f : R R, f x 3x 4
(b) f : R R , f x x 2 2
(c) f : R R , f x x
(d) None of these
Solution : y4 The function is onto as for R, f y . The function R+®R+, f x x is onto as for R+, 3
R®R+, f y 2 y, f : f x x 2 2 is not onto as 1 Î R+ has no pre-image.
ILLUSTRATION : 10 Which of the following functions is non periodic (a) f x x x (c) f x
8 8 1 cos x 1 cos x
1 if x is a rational number (b) f x 0 if x is an irrational number (d) cos x
Solution : (d) The function in (a) is periodic with period 1 and the function in (b) is also periodic since
4 f x r f x for every rational r. The function in (c) is equal to and thus has | sin x | period . 63
IIT MATHEMATICS All are periodic. In ‘b’ there is no period .
SECTION-A SINGLE ANSWER TYPE QUESTIONS 1.
The domain of the function f(x) = A) [1, )
2.
x 1 6 x is
B) (-, 6)
C) [1, 6]
1 The domain of definition of the function y = log (1 x) x 2 is 10 A) (-3, -2) excluding –2.5 C) [-2, 1) excluding 0
3.
B) [0, 1] excluding 0.5 D) none of these
Which o the following functions is an even function ?
a x ax A) f(x) = x a a x 4.
a x 1 B) f(x) = x a 1
6.
D)f(x)= log2 x x 2 1
x2 . Then x3
B) f is one-one D)one-to-one but not onto
e | x| e x Let f:R®R be a function defined by f(x) = x . Then e ex A) f is both one-one and onto
B) f is one-one but not onto
C) f is onto but not one-one
D) f is neither one-one nor onto
Which of the following functions have inverse defined on their ranges ? A) f(x) = x2, x ÎR
7.
a x 1 C) f(x) = x x a 1
Let A = R – {3}, B = R – {1}. Let f:A®B be defined by f(x) = A) f is bijective C) f is onto
5.
D) none of these
B) f(x) = x3, xÎR
C) f(x) = sinx, 0
Which of the following function is non periodic ? A) f(x) = {x}, the fractional part of the number x B) f(x) = cot(x+7)
sin 2 x cos 2 x C) f(x) = 1 1 cot x 1 tan x D) f(x) = x+sinx
8.
The inverse of the function y =
10 x 10 x is 10 x 10 x
64
FUNCTION A) log10(2-x)
9.
If f(x) =
B)
1 log 10 (2 x 1) 2
C) (1, )
B) (-, -1)
B) [0, 1)
D) (0, )
C) (0, 1]
D) [-1, 1]
B) {728, 1474}
C) {0, 728}
D) none of these
12.
If f(x) = sin2x + x – [x], where [x] is the integral part of x, then f(x) is A) a periodic function with period B) a periodic function with period 2 C) a periodic function with period 1 D) not a periodic function
13.
Let f:RR defined by f(x) = x3 + x2 + 100x + 5sinx, then f is A) many-one onto
B) many-one into
B) f(1+x)=f(1-x)
1 1 1 4 log 2 x 2
(x y) y 4
B)
x2 y2 4
C) [-1, 1]
C)
x2 y2 4
D) R – [- 20 , 20 ]
D)
x2 y2 2
f(x) = sin-1x ; find the domain of f(logx) B) (1/e, e)
C) (1/e, 1)
D) (1/e, 2)
C) (0, 1]
D) (0, -1)
sin x f cos ecx then find the range of f(x) 1 sin x
A) (- -1) U [1 ) 65
1 1 1 4 log 2 x 2 D) none of these
A) [0, 1] B) [-1, 0] If f(x+2y, x-2y) = 4xy, then f(x, y) =
A) (1, e) 19.
D) none of these
x2 x 1 Let F:RR defined by f(x) = 2 , then the set of values of a for which f is onto is x ax 5
A) 18.
C) f(x+1) = f(x-1)
B)
C) 2 x ( x 1)
17.
D) one-one into
If the function f:[1, ) (1, ) is defined by f(x) = 2x(x-1), then f–1(x) is A)
16.
C) one-one onto
The graph of y = f(x) is symmetrical about the line x = 1, then A) f(-x) = f(x)
15.
1 2x log 4 2x
Range of f(x) = 16 x C 2 x 1 203 x C 4 x 5 is A) [728, 1474]
14.
D)
If [x] denotes the integral part of x, then the domain of f(x) = cos–1(x+|x|) is A) (0, 1)
11.
C)
1 x , the domain of f-1(x) contains 1 x
A) (-,) 10.
1 1 x log 10 2 1 x
B) (0, 1)
IIT MATHEMATICS 20.
f (sin x ) 1
[f(x)] = 3 then range of f (sin x ) 1 is A) [2, 5/3)
21.
B) (2, 5/3)
C) (2, 5/3]
D) [2, 5/3]
f(x) = 3+2sinx Find the interval in which h(x) attain maximum value such that h(x) = 1+[f(x)] if
A) 6 2 22.
B) 6 2
C) 6 2
x 2 2
D) [0, p/6]
The domain of the function 2x 1 3 tan x e is [a, b]. Find the value of a+b. 3
1/ 2 1 f(x) = (1 3x ) 3 sin
A) 1 23.
y=
sin 2 x sin x 1 sin 2 x sin x 1
A) (1/3, 3) 24.
C) 9
B) 17/15
D) 10 n 2 1 n 2 1
C) 1/15
B) 4
D) 2/15 x R then period of f(x) is
C) 8
D) 12
B) odd
C) constant
D) neither even nor odd
B) even
C) constant
B) even
C) constant
x(x-4)
f: [4, ) [4, ) be a function defined by f(x) = (5) A) 3/2
31.
B) 8
D) neither even nor odd
A function passes through origin and lie in II and IV quadrant is always A) odd
30.
D) (1/3, 1/2)
A function passes through origin and lie in I and II quadrant is always A) odd
29.
B) (1, 3)
The function f(x) = Sec 3 [log( x 1 x 2 )] is an A) even
28.
C) (1/3, 1)
find the range of y
If f:RR is a function satisfying the property f(2x+3) + f(2x+7) = 2 A) 2
27.
D) –2
If the period of f(x) = |sinnx| + |cosnx| is p/8 then find the value of A) 15/17
26.
C) –1
Period of f(x) = |sinx| + sinx is n find the value of g(n) if g(x) = |cosx| + [x] A) 7
25.
B) –2/3
B) 2/3
D) neither even nor odd
then find the value of
C) 1/3
f 1 (5 5 ) 1 f 1 (5 5 ) 1
D) 3
f(x+y, x-y) = xy then the Arithmetic mean of f(x, y) and f(y, x) will exist when 66
FUNCTION A) x, yR 32.
B) xI, yI
3f(x) + 5f(1/x) = A) 8
33.
36.
3|sinx| = x +
B) 1 1 x
C) 1/3
D) 1/4
C) 1
D) zero
The function f: (-, -1] ® (0, e5] defined by f(x) = is A) many one and onto B) many one and into C) one-one and onto D) one-one and into 1 x 3x x3 If f(x) = log then f(g(x)) is equal to and g(x) = 1 x 1 3x 2
B) [f(x)]3
If f(x) = 64x3 + A) f(a) = 12
38.
D) 4
No. of solutions B) 3
A) f(3x) 37.
C) 1/8
f(x) = (1+b2) x2 + 2bx + 1 and m(b) the minimum value of f(x) for a given value of b as b raises then the maximum value of m(b) is
A) 2 35.
D) xN, yN
R then f(3) =
B) 7
A) 1/2 34.
1 – 3x x(0) x
C) xI, yR
|x|+|y|=4; y=
C) 3f(x)
D) –f(x)
1 1 = 3 then 3 and a, b are the roots of 4x + x x
B) f(b) = 11
C) f(a) = f(b)
D) none of these
1 1 x then no.of values of x are four which satisfies both equation then x lie in the 2 x
interval A) [-4, 4] 39.
The range of the function f(x) = A) {1, 2, 3}
40.
B) (-4, 4) 7 x
Px 3
7 x
B) {1, 2, 3, 4}
C) (1, 4)
D) [-1, 4]
C) {1, 2, 3, 4, 5}
D) {2, 3, 4}
C x 3
No. of values of x for which below equation is satisfied Sin 1 x sin x A) 4
B) 2
C) 3
D) none
41.
Let f(x) = 1+x2+x3 and x = y+1. Let g be a function such that g(y) = f(x) and f be a function f(q) = g(cos2q). Then g(y) and f(q) are respectively A) y3 + 4y2 + 5y + 3 8cos6q + 4cos4q + 1 B) y3 – 4y2 + 5y + 1 8cos2q + 2cosq + 1 3 2 C) y – 4y + 5y + 1 8cos2q - 2cosq + 1 D) y3 – 4y2 – 5y –1 8cos2q - 2cosq - 1
42.
If g{f(x)} = |sinx| and f{g(x)} (sin x )2 then
67
IIT MATHEMATICS 2
A) f(x) = sin x B) f(x) = sinx C) f(x) = x2 D) f and g cannot be determined 43.
The domain of y = 1/ | x | x A) [0, )
44.
B) (-, 0)
C) (-, 0]
D) [1, )
If D is the domain of the function f(x) = then D contains A) [-1/3, ½]
45.
g(x) = g(x) = |x| g(x) = |x|
B) [-1/3, 0]
C) [-1/3, 1]
D) [1/2, 1]
1 If f(x) is a polynomial satisfying f(x).f(1/x) = f(x) + f(1/x), and f(3) = 1 2 x 3sin
3x 1 28, then 2
f(4) is given by A) 63
B) 65
C) 67
D) 68
46.
Which of the following sets of ordered pairs define a one to one function? A) R = {(x, y); x2 + y2 = 2} on R B) A = {1, 2, 3}, B = {1, 2, 3, 4, 5} and R = {(x, y); 5x+2y is a prime number on A C) A = {1,2,3,4}, B = {1,2,3,4,5,6,7} and R = {(x,y): y = x2 – 3x + 3} on A D) none of these
47.
is the period of the function
A) 48.
(1 sin x ) cos x (1 cos ecx)
The range of the function f(x) = A) (0, ½)
49.
B) [0, ½]
C) [0, )
D) [0, 2]
B) [1, 4)
C) (-, 3)
D) none of these
C) 62
D) none of these
2t 3 is 6
B) 6
The set G onto which the set F is mapped if y = log3x and F = (3, 27) is A) G = (0, 3)
52.
D) cos(sinx) + cos(cosx)
x2 is x4 1
The period of the function y = sin A) 2
51.
C) sin2x + cos3x
The domain of y = sin-1 - log10(4 – x) is A) (-,4)
50.
B) |sinx|+|cosx|
B) G = (1, 3)
C) G = (1, 4)
D) G = (0, 2)
C) 0, 3 3
D) none of these
2 x 2 is Range of f(x) = 3tan 9 A) 3 3, 3 3
B) 0, 3
68
FUNCTION 53.
Let f: R 0, defined by f(x) = tan–1 (x2 + x + a), then the set of values of a for which f is onto is 2
A) [0, )
54.
1 C) , 4
B) [2, 1]
If f(x) = sinx + tan
D) none of these
x x x x x + sin 2 + tan 3 + … + sin n 1 tan n is a periodic function with period k, 2 2 2 2 2
then k = A) 1 55.
56.
1 If af (x) + bf = x –1, x¹0 and ab, then f(2) = x a a b2
B)
2
59.
C) f(y)
C) (0, )
B) R – {0}
D) f(y)
3
B) – 2 x 2
3 x –1 +
3
D) –
Domain of
3 x
3
4x x 2 is B) R\(0, 4)
The domain of the function f(x) = log10 3 A) 0, 2
D) (-, 0)
2 2x x 2 is
3– x–2+
A) R\[0, 4]
69
D) none of these
B) one-one but not onto D) neither one-one nor onto
The domain of the function f(x) =
C) – 1 –
62.
a 2b a2 b2
If f:RR and f(x) = ax + sinx is one-one and onto, then the set of values of ‘a’ is
A) –2
61.
C)
B) f(x)
f:RR, f(x) = x|x| is A) one-one and onto C) not one-one but onto
A) (-,) 60.
a 2b a2 b2
Let f: {x,y,z} {1,2,3} be a one-one mapping such that only one of the following three statements is true and remaining two are false f(x) 2, f(y) = 2, f(z) 1, then A) f(x) > f(y) > f(z)
58.
D) 1/2n
If f(x+y) = f(x) + f(y) for all x, y R, then A) f(x) is an odd function B) f(x) is an even function C) f(x) is neither odd nor even function D) f(0) = 0
A) 57.
C) 2n
B) 2
B) (0, 3)
C) (0, 4)
D) [0, 14]
3 C) , 2
3 D) 0, 2
3 x is x
IIT MATHEMATICS 63.
64.
3 The domain of the function f (x) = 0, . 2
1 1 ,1 A) 1, 2 2
B) [–1, 1]
1 1 , C) , 2 2
1 ,1 D) 2
A) [–1, 2) [3, ) 65.
The domain of f (x)= A) R – {–1, –2}
67.
69.
D) none of these
x 1 5 x is
B) (–, 5)
C) (1, 5)
D) [1, 5]
C) R – {–1, –2, –3}
D) (–3, +) – {–1, –2}
log 2 ( x 3) is x 2 3x 2
B) (–2, +)
2 | x | The domain of the function f (x) = cos 4 +[log(3 – x)]–1 is
B) [–6, 2) (2, 3]
C) [–6, 3]
D) [–6, 3)
3 The domain of the function f (x) = cos–1is 4 2sin x
A) 2n , 2n 6 6
B) 2n , 2n 6 6
C) 2n , 2n 6 6
D) 2n , 2n 6 6
The domain of the function f (x) = log10 [1 – log10 (x2 – 5x + 16)] is A) (2, 3)
70.
C) [–1, 2] [3, )
–1
A) [–6, 3\{2}
68.
B) (–1, 2) [3, )
The domain of the function f (x) = A) [1, )
66.
( x 1)( x 3) is given by ( x 2)
The domain of the function f (x) =
B( [2, 3]
The domain of the function f (x) = 3 A) , 2 (2, 3) (3, ) 2 1 C) , 2
C) (2, 3] log
1 x 2
x2 5x 6
D) [2, 3)
is
3 B) , 2
D) None of these
70
