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 :-
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 + z2 = (x1 + iy1) + (x2 + iy2) = (x1 + x2) + i (y1 + y2)
- 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
When is this going to be uploaded
up to 7 ch uploaded and rest will upload soon