Class 11 Maths Chapter 5 Complex Numbers and Quadratic Equations

Ex 5.1

Exercise 5.1

Express each of the complex number given in the Exercises 1 to 10 in the form a + ib.
 Question 1.

Solution.

= -3i2 = -3(-1)                    [∵ i2 = -1]
= 3 = 3 + 0i

 Question 2.
i9+ i19
Solution.

 Question 3.
i-39
Solution.

 Question 4.
3(7 + i7) + i(7 + i7)
Solution.
3(7 + i7) + i(7 + i7) = 21 + 21i + 7i + 7i2
= 21 + (21 + 7)i + (-1)7 = 21 – 7 + 28i
= 14 + 28i.

 Question 5.
(1 – i) – (- 1 +i6)
Solution.
(1 – i) – (-1 + i6) = 1 – i + 1 – 6i
= (1 +1) – i(1 + 6)
= 2 – 7i

 Question 6.

Solution.

 Question 7.

Solution.

 Question 8.
(1 -i)4
Solution.
(1 -i)4 = [(1 – i)2]2 = [1 – 2i + i2]2
= [1 – 2i + (-1)]2
= (-2i)2 = 4i2 = 4(-1) = – 4
= – 4 + 0i

 Question 9.

Solution.

 Question 10.

Solution.

Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13.
 Question 11.
4 – 3i
Solution.

 Question 12.

Solution.

 Question 13.
-i
Solution.

 Question 14.
Express the following expression in the form of a + ib:

Solution.
We have, 

Exercise 5.2

Find the modulus and the arguments of each of the complex numbers in Exercises 1 to 2.
Question 1.

Solution.
We have, 

 Question 2.

Solution.
We have, 

Convert each of the complex numbers given in Exercises 3 to 8 in the polar form:
 Question 3.
1 – i
Solution.
We have, z = 1 – i

 Question 4.
-1 + i
Solution.
We have, z = -1 + i

 Question 5.
-1 – i
Solution.
We have, z = -1 – i

 Question 6.
-3
Solution.
We have, z = -3, i.e., z = -3 + 0i

 Question 7.

Solution.
We have, 

 Question 8.
i
Solution.
We have, z = i, i.e., z = 0 + 1.i

Exercise 5.3

Solve each of the following equations:
 Question 1.
x2 + 3 = 0
Solution.
We have, x2 + 3 = 0 ⇒ x2 = -3
⇒  ⇒ x = 

 Question 2.
2x2 + x + 1 = 0
Solution.
We have, 2x2 + x + 1 = 0
Comparing the given equation with the general form ax2 + bx + c = 0, we get

 Question 3.
x2 + 3x + 9 = 0
Solution.
We have, x2 + 3x + 9 = 0
Comparing the given equation with the general form ax2 + bx + c = 0,we get a = 1, b = 3, c = 9

 Question 4.
-x2 + x – 2 = 0
Solution.
We have, -x2 + x – 2 = 0
Comparing the given equation with the general form ax2 + bx + c = 0,we get
a = 1, b = 1, c = -2

 Question 5.
x2 + 3x + 5 = 0
Solution.
We have, x2 + 3x + 5 = 0
Comparing the given equation with the general form ax2 + bx + c = 0, we get a = 1, b = 3, c = 5.

 Question 6.
x2 – x + 2 = 0
Solution.
We have, x2 – x + 2 = 0
Comparing the given equation with the general form ax2 + bx + c = 0, we get a = 1, b = -1, c = 2.

 Question 7.

Solution.
We have, 
Comparing the given equation with the general form ax2 + bx + c = 0, we get

 Question 8.

Solution.
We have, 
Comparing the given equation with the general form ax2 + bx + c = 0, we get

 Question 9.

Solution.
We have, 
Comparing the given equation with the general form ax2 + bx + c = 0, we get

 Question 10.

Solution.
We have, 
Comparing the given equation with the general form ax2 + bx + c = 0, we get