Practise Questions

Question 1

i⁵ = ?

Answer 1

i

Question 2

i⁷ = ?

Answer 2

-i

Question 3

Solve x² + 9 = 0

Answer 3

x² - 3²i² = 0

(x + 3i)(x - 3i) = 0

x = ±3i

Question 4

Solve -2x² + 2x - 13 = 0

Answer 4

2x² - 2x + 13 = 0

x = ^{2 - √<span>4 - 104</span>}⁄₄

x = ^{2 - √<span>100</span>}⁄₄

x = ^{2 - 10i}⁄₄

x = ^{1 - 5i}⁄2

x = ¹⁄2 - ⁵⁄2i

Question 5

Simplify (4 + 7i) - (-2 + 9i)

Answer 5

4 + 7i + 2 - 9i = 6 - 2i

Question 6

Simplify (5 + 2i)(3 - 4i)

Answer 6

= 15 - 20i + 6i - 8i²

= 15 - 14i + 8

= 23 - 14i

Question 7

If z = 5 - 2i

z = ?

Answer 7

5 + 2i

Question 8

If z₁ = 3 + i and z₂ = 2 - 3i, find (z₁ - z₂)²

Answer 8

= (z₁ - z₂)²

= (3 + i - 2 + 3i)²

= (1 + 4i)²

= 1 + 8i + 16i²

= 1 + 8i - 16

= -15 + 8i

Question 9

Find real numbers x and y if (1 + i)x + (2 - 3i)y = 10

Answer 9

(1 + i)x + (2 - 3i)y = 10

• x* + ix + 2y - 3iy = 10
  (1) x + 2y = 10
  (2) x - 3y = 0

5y = 10 Subtract (2) from (1)

• y* = 2
• x* = 6

Question 10

Solve 2z - 1 = (4 - i)²

Answer 10

2x + 2iy - 1 = 16 - 8i + i²

2x + 2iy - 1 = 16 - 8i - 1

2x + 2iy - 1 = 15 - 8i

2x + 2iy - 16 + 8i = 0

• x* - 8 + iy + 4i = 0
• x* = 8
• y* = -4

Question 11

Factorise z² + 16

Answer 11

z² - 16i² = (z + 4i)(z - 4i)

(z = ±4i)

Question 12

Solve z² + 2z + 26 = 0

Answer 12

• z* = ^{-2 ±√<span>4 - 104</span>}⁄₂
• z* = ^{-2 ±10i}⁄₂
• z* = -1 ± 10i

Question 13

Find the square root of 12 + 5i

CURRENTLY WRONG REVISE

Answer 13

(x + iy)² = 12 + 5i

• x² - y² + 2ixy* = 12 + 5i
  (1) x² - y² = 12
  (2) 2xy = 5
• x* = ⁵⁄_2<em>y</em>

²⁵⁄_{4<em>y</em>² - <em>y</em>²} = 12

25 - 4y⁴ = 48y²

4y⁴ + 48y² - 25 = 0

• y*² = ^{-48 ±√2304 + 400}⁄₂
• y*² = 2 OR -50
• y* = ±√2 (y is an element of the real)
• x* = ±⁵⁄_{2√<span>2</span>}
• z* = ±(⁵⁄_{2√<span>2</span>} + √2i)

Question 14

Solve z² - (3 - 2i)z + (1 - 3i) = 0

Answer 14

z² - (3 - 2i)z + (1 - 3i) = 0

(x² - y²- 3x - 2y + 1) + i(2xy - 3y + 2x - 3) = 0

(1) x² - y²- 3x - 2y + 1 = 0
(2) 2xy - 3y + 2x - 3 = 0