Find the real values of x and y , if ( 1 + i ) x - 2 i 3 + i + ( 2 - 3 i ) y + i 3 - i = i .

Given, ( 1 + i ) x - 2 i 3 + i + ( 2 - 3 i ) y + i 3 - i = i ⇒ ( 1 + i ) ( 3 - i ) x - 2 i ( 3 - i ) + ( 2 - 3 i ) ( 3 + i ) y + i ( 3 + i ) ( 3 + i ) ( 3 - i ) = i ⇒ 3 x - i x + 3 i x - i 2 x - 6 i + 2 i 2 + 6 y + 2 i y - 9 i y - 3 i 2 y + 3 i + i 2 9 - i 2 = i ⇒ 4 x + 2 i x - 3 i + 9 y - 7 i y - 3 10 = i ⇒ ( 4 x + 9 y - 3 ) + i ( 2 x - 3 - 7 y ) = 10 i Comparing both the sides: 4 x + 9 y - 3 = 0 ⇒ 4 x + 9 y = 3 ...(i) 2 x - 3 - 7 y = 10 ⇒ 2 x - 7 y = 13 ...(ii) Multiplying equation (ii) by 2 : 4 x - 14 y = 26 ...(iii) Subtracting equation (iii) from (i), we get, ∴ y = - 1 Substituting the value of y in equation (i): 4 x - 9 = 3 ⇒ 4 x = 12 ⇒ x = 3 ∴ x = 3 and y = - 1

Find the conjugate of the complex number 1 3 + 5 i .

Let z = 1 3 + 5 i = 1 3 + 5 i × 3 - 5 i 3 - 5 i = 3 - 5 i 9 - 25 i 2 = 3 - 5 i 9 + 25 = 3 - 5 i 34 ∴ z ¯ = 3 + 5 i 34

Find the conjugate of the complex number ( 3 - i ) 2 2 + i .

Let z = ( 3 - i ) 2 2 + i = ( 9 - 6 i - 1 ) 2 + i = 8 - 6 i 2 + i × 2 - i 2 - i = 16 - 8 i - 12 i + 6 i 2 4 - i 2 = 10 - 20 i 5 = 2 - 4 i ∴ z ¯ = 2 + 4 i

Find the conjugate of the complex number ( 1 + i ) ( 2 + i ) 3 + i .

Let z = ( 1 + i ) ( 2 + i ) 3 + i = 2 + i + 2 i + i 2 3 + i = 1 + 3 i 3 + i = 1 + 3 i 3 + i × 3 - i 3 - i = 3 - i + 9 i - 3 i 2 9 - i 2 = 6 + 8 i 10 = 3 + 4 i 5 ⇒ z ¯ = 3 - 4 i 5

Find the conjugate of the complex number ( 3 - 2 i ) ( 2 + 3 i ) ( 1 + 2 i ) ( 2 - i ) .

Let z = ( 3 - 2 i ) ( 2 + 3 i ) ( 1 + 2 i ) ( 2 - i ) = 6 + 9 i - 4 i - 6 i 2 2 - i + 4 i - 2 i 2 = 6 + 6 + 5 i 2 + 2 + 3 i = 12 + 5 i 4 + 3 i × 4 - 3 i 4 - 3 i = 63 - 16 i 25 ∴ z ¯ = 63 + 16 i 25