Rajasthan Board RBSE Class 11 Maths Chapter 5 Complex Numbers Ex 5.3

Question 1.
Find the square root of the following complex numbers:
(i) -5 + 12i
(ii) 8 – 6i
(iii) -i
Solution:
Formula

Question 2.
Find the value of
Solution:
From formula =

Question 3.
Find the cube root of the following:
(i) -216
(ii) -512
Solution:
(i) -216 = (-6) (1)^1/3 = -6, -6ω, -6ω²
Hence, cube root of -216 = -6, -6ω, -6ω²
(ii) -512 = (-8) (1)^1/3 = -8, -8ω, -8ω²
Hence, cube root of -512 = -8, -8ω, – 8ω²

Question 4.
Prove that:
(i) 1 + ωⁿ + ω²ⁿ = 0, whereas n = 2, 4
(ii) 1 + ωⁿ + ω²ⁿ = 3, whereas n is multiple of 3.
Solution:

Question 5.
Prove that:

(ii) (1 + 5ω² + ω) (1 + 5ω + ω²) (5 + ω + ω²) = 64
Solution:
(i) ω is cube root of unity.