Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions Chapter-9. Step by step Solutions of OP Malhotra S.Chand ISC Class-11 Mathematics with Exe-9 (a), 9 (b), 9 (c), 9 (d), 9 (e), 9 (f) With Chapter Test. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Class: 11th
Subject: Mathematics
Chapter  : Ch-9 Complex Number of Section -A
Board ISC
Writer  OP Malhotra
Publications S.Chand Publications 2020-21

-: Select Topics :-

Exe-9 (a)

Exe-9 (b)

Exe-9 (c)

Exe-9 (d)

Exe-9 (e)

Exe-9 (f)

Chapter Test


Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Imaginary Numbers
The square root of a negative real number is called an imaginary number, e.g. √-2, √-5 etc.
The quantity √-1 is an imaginary unit and it is denoted by ‘i’ called Iota.

Integral Power of IOTA (i)
i = √-1, i2 = -1, i3 = -i, i4 = 1
So, i4n+1 = i, i4n+2 = -1, i4n+3 = -i, i4n = 1

  • For any two real numbers a and b, the result √a × √b : √ab is true only, when atleast one of the given numbers i.e. either zero or positive.
    √-a × √-b ≠ √ab
    So, i2 = √-1 × √-1 ≠ 1
  • ‘i’ is neither positive, zero nor negative.
  • in + in+1 + in+2 + in+3 = 0

Complex Number

A number of the form x + iy, where x and y are real numbers, is called a complex number, x is called real part and y is called imaginary part of the complex number i.e. Re(Z) = x and Im(Z) = y.

Purely Real and Purely Imaginary Complex Number
A complex number Z = x + iy is a purely real if its imaginary part is 0, i.e. Im(z) = 0 and purely imaginary if its real part is 0 i.e. Re (z) = 0.

Equality of Complex Number
Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are equal, iff x1 = x2 and y1 = y2 i.e. Re(z1) = Re(z2) and Im(z1) = Im(z2)
Note: Order relation “greater than’’ and “less than” are not defined for complex number.

Algebra of Complex Numbers

Addition of complex numbers
Let z1 = x1 + iy1 and z2 = x2 + iy2 be any two complex numbers, then their sum defined as
z1 + z= (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i (y1 + y2)

Properties of Addition

  • Commutative: z1 + z2 = z2 + z1
  • Associative: z1 + (z2 + z3) = (z1 + z2) + z3
  • Additive identity z + 0 = z = 0 + z
    Here, 0 is additive identity.

Subtraction of complex numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their difference is defined as
z1 – z2 = (x1 + iy1) – (x2 + iy2) = (x1 – x2) + i(y1 – y2)

Multiplication of complex numbers
Let z1 = (x1 + iy1) and z2 = (x2 + iy2) be any two complex numbers, then their multiplication is defined as
z1z2 = (x1 + iy1) (x2 + iy2) = (x1x2 – y1y2) + i (x1y2 + x2y1)

Properties of Multiplication

  • Commutative: z1z2 = z2z1
  • Associative: z1(z2z3) = (z1z2)z3
  • Multiplicative identity: z . 1 = z = 1 . z
    Here, 1 is multiplicative identity of an element z.
  • Multiplicative inverse: For every non-zero complex number z, there exists a complex number z1 such that z . z1 = 1 = z1 . z
  • Distributive law: z1(z2 + z3) = z1z2 + z1z3

Exe-9 (a)

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Page 9-3 to 9-4

Express each of the following in the form b or bi. where b is a real number :

Question 1:

3i . 2

Question 2:

i(-1)

Question 3:

…………………..

……………………

……………………

Question 27:

If i = √-1, prove that  the following :


Exe-9 (b)

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Page 9-16 to 9-18

Question 1:

In each of the following find r + s, r – s, rs, r/s if r denote the first complex number and s denote the …………….

……………….

Question 2:

Solve each of the following equations for real x and y :

………….

………..

Question 3:

……………….

………………

………………

Question 25:

If z = -3 + √-2, then prove that ………………


Exe-9 (c)

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Page 9-35 to 9-36

Question 1:

If (-2 + √-3)…………………………….. also find the module of a + bi.

Question 2:

Find the modulus of …………….

Question 3:

………………….

………………….

………………….

Question 9:

Solve : |z| + z = 2+ i, where z is complex number.


Exe-9 (d)

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Page 9-42 

Find the modulus and amplitude of the following complex number and   hence express them into polar form.

Question 1:

√3 + i

Question 2:

-√3 + i

Question 3:

……………….

………………..

……………….

Question 14:

Given the complex number ……………….

(i) ……………….

(ii)……………..

(iii) …………………..

(iv) …………….


Exe-9 (e)

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Page 9-48

Question 1:

Illustrate in the complex plane, the set  of points satisfying the following conditions. Explain your answer :

(i)………….

(ii)…

(iii)…..

Question 2:

Illustrate and ……………………….. |z + 2|.

Question 3:

…………………..

…………………..

…………………..

Question 10:

W hat is the region represented ………………………….. Argand plane.



Exe-9 (f)

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Page 9-54 to 9-55

Question 1:

Find the square root of the following complex number.

……………….

………………..

Question 2:

If ω is a cube root of  unity, then

…………….

………………

Question 3:

……………….

………………..

………………….

Question 15:

If 1, ω, ω² are cube roots of unity, prove that, ……………….. triangle.


Chapter Test

Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions

Page 9-58 to 9-59

Question 1:

Find the square root of 5 – 12i

Question 2:

Find the locus of a complex……………… Argand plane.

Question 3:

………………..

…………………

………………….

Question 25:

The locus of the point z is the Argand plane for which ………………………..

…………….

……………..

 

-: End of Complex Number Solution :-

Return to :-  OP Malhotra S. Chand ISC Class-11 Maths Solutions


Thanks

Please share with your friends

2 thoughts on “Complex Number OP Malhotra S.Chand ISC Class-11 Maths Solutions”

Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.