## 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 \(\sqrt { 4+3\sqrt { -20 } } +\sqrt { 4-3\sqrt { -20 } }\)

Solution:

From formula \(\sqrt { a+ib }\) =

Question 3.

Find the cube root of the following:

(i) -216

(ii) -512

Solution:

(i) -216 = (-6) (1)^{1/3} = -6, -6ω, -6ω^{2}

Hence, cube root of -216 = -6, -6ω, -6ω^{2}

(ii) -512 = (-8) (1)^{1/3} = -8, -8ω, -8ω^{2}

Hence, cube root of -512 = -8, -8ω, – 8ω^{2}

Question 4.

Prove that:

(i) 1 + ω^{n} + ω^{2n} = 0, whereas n = 2, 4

(ii) 1 + ω^{n} + ω^{2n} = 3, whereas n is multiple of 3.

Solution:

Question 5.

Prove that:

(ii) (1 + 5ω^{2} + ω) (1 + 5ω + ω^{2}) (5 + ω + ω^{2}) = 64

Solution:

(i) ω is cube root of unity.