Welcome to SRM Gateway. We thank you very much for joining SRM Gateway. We are all committed to your satisfaction and fulfill your expectations in training, quality in our services, which is our primary goal. We are sure that you will become very skilled and will be positioning yourself well in the IT industry. We shall always ensure that you are guided well during the course. We wish you all the best for the complete learning of the technology Stay ahead in the technology. Way ahead of others.

Warm Regards, Management SRM Gateway

Document Number : SRMGW - IIT-JEE - MATH-02 Course Code : AIEEE

SRM Gateway This document is the property of SRM Gateway. This document has been prepared exclusively for the use of the students. No part of this document shall be copied or transferred in any form or by any means which would initiate legal proceedings, if found.

SRM Gateway No. 9, 3rd Avenue, Ashok Nagar, Chennai – 600 083

Preface About the book

This book will be an effective training supplement for students to master the subject. This material has in-depth coverage and review questions to give you a better understanding. A panel of experts have complied the material, which has been rigorously reviewed by industry experts and we hope that this material will be a value addition. We also would value your feedback, which will be useful for us to fine-tune it better.

Wishing you all success.

SRM Gateway

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

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close