Syllabus JEE MAINS

Complex numbers as ordered pairs of real’s, Representation of complex numbers in the form a+ib and

their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument

(or amplitude) of a complex number, square root of a complex number, triangle inequality.

Quick Revision

1. Iota

1.1 Definition

1 is a purely imaginary number which is denoted by i and read as ‘iota’.

i.e., iota i 1

1.2 Integral powers of iota :

1.2.1 Since 1i hence we have 12i , ii3 and 14i .

1.2.2 To find the value of ),4(nin first divide n by 4. Let q be the quotient and r be the

remainder. i.e., rqn 4 where 30 r

rrqrqrqn iiiiii )(.)1()(.)( 44

In general we have the following results iiiiii nnnn 3424144 ,1,,1 , where n is

any integer.

Brain Demur …

The sum of four consecutive powers of i is always zero i.e.,

.,0321 Iniiii nnnn

,,1,,1 iiin where n is any integer.

iiii 2)1(,2)1( 22

ii

ii

i

ii

i

i1

1

2,

1

1,

1

1

2. Complex Number.

2.1 Definition

if x and y be two real numbers

Then the number z = x + iy is called a complex number.

x is called real part and y is called imaginary part of z

i.e, x = Re(z) and y = Im (z)

2.2 Ordered Pair Form

In the form of ordered pair the complex number z = x + iy is denoted as (x, y)

i.e., x + iy = (x, y)

2.3 Algebraic operations with complex numbers

2.3.1 Addition : (a + ib) + (c + id) = (a + c) + i(b + d)

2.3.2 Subtraction : (a + ib) - (c + id) = (a – c) + i(b – d)

2.3.3 Multiplication: (a + ib).(c + id) = (ac – bd) + i(ad + bc)

2.3.4 Division : (when at least one of c and d is non-zero)

Division (when at least one of c and d is non-zero)

3. Modulus of a complex Number

3.1 Definition: Modulus of a complex number z = x + iy is denoted as mod (z) or | z |, is

defined as

|z| = , where x = Re(z), y = Im(z). Sometimes, |z| is called absolute value of z.

3.2 Properties of modulus

(i) |z| 0 and |z| = 0 if and only if z = 0, i.e., x = 0, y = 0

(ii) |z| = | | = |– z| = |– |.

idc

iba

2222dc

adbci

dc

bdac

22 yx

z z

(iii) z = |z|2

(iv) – |z| ≤ Re (z) ≤ | z| and – |z| ≤ Im(z) ≤ |z|

(v) |z1z2| = |z1| |z2|

(vi) |z1 ± z2| ≤ |z1| + |z2|

(vii) |z1 – z2| | z1| - |z2| |

(viii) |z2| = |z|2 or |zn| = |z|n also |z1z2 .... zn| = |z1| |z2| ...... |zn|

4. The conjugate of the complex number

4.1 Definition : The conjugate of the complex number z = a+ib is defined to be a-ib and is

denoted by z .

4.2 Properties of the Conjugate of a Complex

Number:

is the mirror image of z in the real axis.

(i) (ii)

(iii) (iv)

(v)

(vi) (vii)

(viii)

(ix)

(x) (xi) If is a function of

complex number z.

Brain Demur …

Complex conjugate is obtained by just changing the sign of i.

Conjugate of ii

Conjugate of ziiz

)( 21 zz and ).( 21 zz real 21 zz or 12 zz

z

z

zz zR2zz e

zli2zz m2

zzz

2121 zzzzn21n21 z...zzz...zz

2121 zzzz2121 z.zzz

2

1

2

1

z

z

z

znn

zz

zz ωωω where,zfthen,zf

2121 zzzz

5. Argument of a Complex Number:

5.1 Definition:

The amplitude or argument of a complex number z is the inclination of the directed line

segment representing z, with real axis.

x < 0, y > 0 y > 0, y > 0

= 8 – tan-1y

x = tan 1

y

x

x < 0, y < 0 x > 0, y < 0

= + tan-2 y

x = tan-1 y

x

5.2 Principal value of arg (z) : Principal value of argument of any complex number lies

between

5.3 Properties of arguments

(i) 0k(,k2)z()z()zz( 2121 argargarg or 1 or – 1)

In general

)1or10k(,k2)z(..........)z()z()z()z.........zzz( n321n321 orargargargargarg

(ii) )()()(2121

zargzargzzarg

(iii) 0(,221

2

1 kkzargzargz

zarg or 1 or – 1)

(iv) 0(,22 kkzargz

zarg or 1 or – 1) (v) 0(,2)( kkzargnzarg n or 1 or –

1)

x

ytan)z(amp 1

(vi) If ,1

2

z

zarg then k2

z

z

2

1arg , where Ik

(vii) z

argzargzarg1

(viii) 2/)zz(Arg

(ix) )z()z( argarg (x) 0)z()z( argarg or

)z()z( argarg

(xi) )z()z( argarg

(xii) ),(cos|z||z|2zzzz 21212121 where )z( 11 arg and )z( 22 arg

Brain Demur …

If 2121 zzzz and arg 1z = arg z2.

2121 || zzzz arg )( 1z arg )( 2z i.e., z1 and z2 are parallel.

2121 || zzzz arg )( 1z arg ,2)( 2 nz where n is some integer.

nzargzargzzzz 2)()(|||||||| 212121 , where n is some integer.

2121 zzzz arg )( 1z – arg 2/)( 2z .

If 1||,1|| 21 zz then (i) 2

21

2

21 zzzz (arg )( 1z arg 22))(z

(ii) 2

21

2

21 zzzz 221 )()( zrgazrga

).cos(||||2 2121

2

2

2

1

2

21 zzzzzz

).cos(||||2 2121

2

2

2

1

2

21 zzzzzz

If |||| 21 zz and ,0)()( 21 zampzamp then 21 zz are conjugate complex numbers of

each other.

0)(,0 zzampz or .2/)(;0)(; zzampzzamp

arg ,0)1( arg ;)1( arg ,2/)(i arg .2/)( i

arg ).Im()(Re4

)( zzz

Amplitude of complex number in I and II quadrant is always positive and in IIIrd and IVth

quadrant is always negative.

If a complex number multiplied by i (Iota) its amplitude will be increased by 2/ and

will be decreased by 2/ , if multiplied by –i, i.e. )(2

)( zrgaizrga and .2

)()( zrgaizrga

6. Triangle Inequalities

In any triangle, sum of any two sides is greater than the third side and difference of any

two side is less than the third side. By applying this basic concept to the set of complex

numbers we are having the following results.

(1) |||||| 2121 zzzz (2) |||||| 2121 zzzz

(3) |||||||| 2121 zzzz (4) |||||||| 2121 zzzz

Note : In a complex plane || 21 zz is the distance between the points 1z and 2z .

The equality |||||| 2121 zzzz holds only when arg )( 1z = arg )( 2z i.e., 1z and 2z

are parallel.

The equality |||||||| 2121 zzzz holds only when arg )( 1z – arg )( 2z = i.e., 1z and

2z are antiparallel.

In any parallelogram sum of the squares of its sides is equal to the sum of the squares of

its diagonals i.e. )|||(|2|||| 22

21

221

221 zzzzzz

Law of polygon i.e., ||....|||||....| 2121 nn zzzzzz

7. Square Root of a complex Number :

The square root of a complex number is given by

Where

,

Hence the required square roots as follows :

z x iy

z = x + iy = a + ib

z xa

2

z xb

2

and

8. DE' MOIVRE'S THEOREM

8.1 If n is any rational number, then n(cos isin ) cosn isinn .

8.2 If 1 1 2 2 3 3 n nz (cos isin )(cos isin )(cos isin ).....(cos isin )

then 1 2 3 n 1 2 3 nz cos( ..... ) isin( ..... ) , where 1 2 3 n, , ..... R .

8.3 If z r(cos isin ) and n is a positive integer, then

1/n 1/n 2k 2kz r cos isin

n n, where k 0,1,2,3,.....(n 1) .

8.4 If p,q z and q 0, then p / q 2k p 2k p(cos isin ) cos isin

q q, where

k 0,1,2,3.....(q 1) .

8.5 Deductions: If n Q, then

(i) n(cos isin ) cosn isinn

(ii) n(cos isin ) cosn isinn

(iii) n(cos isin ) cosn isinn

(iv) n(sin icos ) cosn isinn2 2

Note : This theorem is not valid when n is not a rational number or the complex number

is not in the form of cos isin .

Powers of complex numbers : Let z x iy r(cos isin )

n n n nz r (cos isin ) r (cosn isinn )

9. ROOTS OF UNITY

9.1 The n th Root of Unity

Let x be the n th root of unity. Then

(where k is an integer)

z x z xi if y 0

2 2

z x z xi if y 0

2 2

ππ k2sinik2cos1xn

. Then the n n th roots of unity are t

(t = 0,1,2,....,n-1), i.e. the nth roots of unity are .

9.3 Cube roots of unity: Cube roots of unity are the solution set of the equation 3x 1 0

1/ 3x (1) 1/ 3x (cos0 isin0)

2k 2k

x cos isin3 3

, where k 0,1,2

Therefore roots are 2 2 4 4

1,cos isin ,cos isin3 3 3 3

or 2 i / 3 4 i / 31,e ,e .

Alternative : 1/ 3x (1) 3x 1 0 2(x 1)(x x 1) 0

1 i 3 1 i 3

x 1, ,2 2

If one of the complex roots is , then other root will be 2 or vice-versa.

9.4 Properties of cube roots of unity

(i) 1 + + 2 = 0 (ii) 3 = 1

(iii) r 2r 0,if r not a multiple of 31

3,if r is a multiple of 3 (iv) 2 and 2( ) and 3. .

(v) Cube roots of unity from a G.P.

(vi) Imaginary cube roots of unity are square of each other i.e., 2 2( ) and

2 2 3( ) . .

(vii) Imaginary cube roots of unity are reciprocal to each other i.e., 21 and

2

1.

(viii) The cube roots of unity by, when represented on complex plane, lie on vertices of

an equilateral triangle inscribed in a unit circle having centre at origin, one vertex being

on positive real axis.

n

k2sini

n

k2cosx

ππ

1n2,....,,,1 ααα

Ox

y

1

2

(ix) A complex number a ib, for which | a : b | 1: 3 or 3 :1, can always be expressed in

terms of 2i, , .

Note : If 2 i / 31 i 3e

2, then 2 4 i / 3 2 i / 31 i 3

e e2

or vice-versa 3. .

2a b c 0 a b c,if a, b, c are real.

9.5 Fourth roots of unity: The four, fourth roots of unity are given by the solution

set of the equation 4x 1 0. 2 2(x 1)(x 1) 0 x 1, i

Note : Sum of roots = 0 and product of roots = –1.

10. Representation of a Complex Number :

10.1 Cartesian Representation : In Cartesian Representation a complex number z = x + iy

can be represented by a point P whose cartesian coordinates are (x, y)

Note : | z | = , Thus modulus of complex number denotes the distance of point

P(z) from origin.

The angle which OP makes with the positive direction of real axis is known as the

amplitude or argument of the complex number z.

= arg(z) = tan–1

10.2 Polar Representation : If z = x + iy is a complex number then z = r(cos + i sin ) is a

polar form of complex number z where x = r cos , y = r sin ; r = = | z | and =

tan-1

(where is the principle argument of z, i.e. – < ≤ )

10.3 Vector representation : If P is the point (a, b) on the arg and plane corresponding to the

complex number ibaz .

22 yx

x

y

22 yx

x

y

Then jbiaOP , |z|ba|OP| 22 and arg z = direction of the vector

a

btanOP 1

Therefore, complex number z can also be represented by OP .

10.4 Eulerian representation (Exponential form): Since we have ie = sincos i and

thus z can be expressed as irez , where r|z| and arg (z)

Note : )sini(cose i

sini2ee,cos2ee iiii

11. Application of Complex Numbers in Co-ordinate Geometry.

11.1 Distance formula: The distance between two points )( 1zP and )( 2zQ is given by

|| 12 zzPQ = |affix of Q – affix of P|

Note: Three points )z(B),z(A 21 and )z(C 3 are collinear then ACBCAB

i.e., |zz||zz||zz| 313221 .

11.2 Section formula: If R(z) divides the joining of )z(P 1 and )z(Q 2 in the ratio

)0m,m(m:m 2121

(i) If R(z) divides the segment PQ internally in the ratio of 21 m:m then 21

1221

mm

zmzmz

(ii) If R(z) divides the segment PQ externally in the ratio of 21 m:m then 21

1221

mm

zmzmz

Note : 1. mid point =2

21 zz

2. If 321 ,, zzz are affixes of the vertices of a triangle, then affix of its centroid is

.3

321 zzz

11.3 Equation of a straight line

11.4 Equation of a circle : The equation of a circle whose centre is at point having affix oz

and radius r is r|zz| o

11.4.1 General equation of a circle :

0bzazazz

where a is complex number and Rb .

Centre and radius are – a and b|a| 2 respectively.

11.4.3 Equation of circle in diametric form: If end points of diameter represented by )z(A 1

and )z(B 2 and )z(P be any point on circle then, 0)zz()zz()zz( 121 which is required

equation of circle in diametric form.

11.5 Equation of parabola : Now for parabola PMSP

2

|a2zz||az| or })z(z{

2

1)zz(a4zz 22

where Ra (focus)

Directrix is 0a2zz

11.6 Equation of ellipse : For ellipse aPSSP 2'

azzzz 2|||| 21

where ||2 21 zza (since eccentricity <1)

Then point z describes an ellipse having foci at 1z and 2z and Ra .

11.7 Equation of hyperbola : For hyperbola a2P'SSP

azzzz 2|||| 21

M P(z)

N A S(a+i.0)

z+z+

2a

=0

S'(z2) S(z1)

P(z)

S'(z2) S(z1)

P(z)

O

where ||2 21 zza (since eccentricity >1)

Then point z describes a hyperbola having foci at 1z and 2z and Ra

11.8 Standard Loci in the Argand Plane.

(1) If z is a variable point in the arg and plane such that arg )(z , then locus of z is

a straight line (excluding origin) through the origin inclined at an angle with x–

axis.

(2) If z is a variable point and z1 is a fixed point in the argand plane such that arg

)( 1zz , then locus of z is a straight line passing through the point

representing z1 and inclined at an angle with x-axis. Note that the point z1 is

excluded from the locus.

(3) If z is a variable point and 21 z,z are two fixed points in the argand plane, then

(i) |zz||zz| 21

Locus of z is the perpendicular bisector of the line segment joining 1z and 2z

(ii) |zz||zz| 21 = constant |zz| 21

Locus of z is an ellipse

(iii) |zz||zz||zz| 2121

Locus of z is the line segment joining 1z and 2z

(iv) |zz||zz||zz| 2121

Locus of z is a straight line joining 1z and 2z but z does not lie between 1z and 2z .

(v) |zz|constant|zz||zz| 2121 Locus of z is a hyperbola.

(vi) |zz||zz||zz| 212

22

1

Locus of z is a circle with 1z and 2z as the extremities of diameter.

(vii) 1k|zz|k|zz| 21 Locus of z is a circle.

(viii) arg 2

1

zz

zz)fixed( Locus of z is a segment of circle.

(ix) arg2

1

zz

zz = 2/

Locus of z is a circle with 1z and 2z as the vertices of diameter.

(x) arg 2

1

zz

zz= 0 or

Locus z is a straight line passing through 1z and 2z .