Find the square root of complex number - 1 + 2 2 i .

Let - 1 + 2 2 i = ± x + i y , x , y ∈ R Squaring on both sides, we get ⇒ - 1 + 2 2 i = ( x + i y ) 2 ⇒ - 1 + 2 2 i = x 2 + ( i y ) 2 + 2 x i y ⇒ - 1 + 2 2 i = x 2 - y 2 + 2 x i y ∵ i 2 = - 1 Now equate real and imaginary parts on each side, we get ⇒ x 2 - y 2 = - 1 → 1 , 2 x y = 2 2 → 2 Now, x 2 + y 2 2 = x 2 - y 2 2 + 4 x 2 y 2 ⇒ x 2 + y 2 2 = x 2 - y 2 2 + 2 x y 2 ⇒ x 2 + y 2 2 = 1 + 8 ⇒ x 2 + y 2 = 9 ⇒ x 2 + y 2 = ± 3 As x and y are real numbers, x 2 + y 2 cannot be negative. ∴ x 2 + y 2 = 3 → 3 Now, add 1 and 3 , we get ⇒ 2 x 2 = 2 ⇒ x 2 = 1 ⇒ x = ± 1 From 2 , we have y = 2 ± 1 = ± 2 ∴ - 1 + 2 2 i = ± ( 1 + 2 i ) .

Find the square root of complex number 4 + 4 3 i .

Let 4 + 4 3 i = ± x + i y , x , y ∈ R Squaring on both sides, we get ⇒ 4 + 4 3 i = ( x + i y ) 2 ⇒ 4 + 4 3 i = x 2 + ( i y ) 2 + 2 x i y ⇒ 4 + 4 3 i = x 2 - y 2 + 2 x i y ∵ i 2 = - 1 Now equate real and imaginary parts on each side, we get ⇒ x 2 - y 2 = 4 → 1 , 2 x y = 4 3 → 2 Now, x 2 + y 2 2 = x 2 - y 2 2 + 4 x 2 y 2 ⇒ x 2 + y 2 2 = x 2 - y 2 2 + 2 x y 2 ⇒ x 2 + y 2 2 = 16 + 48 ⇒ x 2 + y 2 2 = 64 ⇒ x 2 + y 2 = ± 8 As x and y are real numbers, x 2 + y 2 cannot be negative. ∴ x 2 + y 2 = 8 → 3 Now, add 1 and 3 , we get ⇒ 2 x 2 = 12 ⇒ x 2 = 6 ⇒ x = ± 6 From 2 , we have y = 2 3 ± 6 = 2 3 2 3 = ± 2 ∴ 4 + 4 3 i = ± ( 6 + 2 i ) .

Find the square root of complex number 7 - 30 - 2 .

Let 7 - 30 - 2 = 7 - 30 i 2 = ± x + i y , x , y ∈ R Squaring on both sides, we get ⇒ 7 - 30 i 2 = ( x + i y ) 2 ⇒ 7 - 30 i 2 = x 2 + ( i y ) 2 + 2 x i y ⇒ 7 - 30 i 2 = x 2 - y 2 + 2 x i y ∵ i 2 = - 1 Now equate real and imaginary parts on each side, we get ⇒ x 2 - y 2 = 7 → 1 , 2 x y = - 30 2 → 2 Now, x 2 + y 2 2 = x 2 - y 2 2 + 4 x 2 y 2 ⇒ x 2 + y 2 2 = x 2 - y 2 2 + 2 x y 2 ⇒ x 2 + y 2 2 = 49 + 900 × 2 ⇒ x 2 + y 2 2 = 49 + 1800 ⇒ x 2 + y 2 2 = 1849 ⇒ x 2 + y 2 = ± 43 As x and y are real numbers, x 2 + y 2 cannot be negative. ∴ x 2 + y 2 = 43 → 3 Now, add 1 and 3 , we get ⇒ 2 x 2 = 50 ⇒ x 2 = 25 ⇒ x = ± 5 From 2 , we have y = - 15 2 x = - 15 2 ± 5 = ∓ 3 2 ∴ 7 - 30 - 2 = ± ( 5 - 3 2 i ) .

E M B I B E

Understanding ISC Mathematics Class 11 Volume 1>Chapter 5 - Complex Numbers>EXERCISE 5.5

M L Aggarwal Solutions for Chapter: Complex Numbers, Exercise 5: EXERCISE 5.5

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Complex Numbers, Exercise 5: EXERCISE 5.5

Attempt the practice questions on Chapter 5: Complex Numbers, Exercise 5: EXERCISE 5.5 with hints and solutions to strengthen your understanding. Understanding ISC Mathematics Class 11 Volume 1 solutions are prepared by Experienced Embibe Experts.