Complex numbers: Cartesian form; complex plane; quadratic solutions practice questions.

Given z = 3 + 4i, calculate the modulus |z|.

• 7
• 5
• 25
• √7

Answer: 5

|z| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Determine the value of (2 + 3i) + (4 − 5i).

• 6 − 2i
• 6 + 8i
• −2 − 2i
• 8 − 15i

Answer: 6 − 2i

Add real and imaginary parts separately: (2 + 4) + (3 − 5)i = 6 − 2i.

Calculate the product (2 + i)(3 − 2i).

• 8 − i
• 6 − 2i
• 4 − i
• 8 + i

Answer: 8 − i

(2 + i)(3 − 2i) = 6 − 4i + 3i − 2i² = 6 − i + 2 = 8 − i, using i² = −1.

The quadratic equation x² − 4x + 13 = 0 has complex roots. Determine the roots.

• 2 ± 3i
• −2 ± 3i
• 4 ± 3i
• 2 ± 9i

Answer: 2 ± 3i

x = (4 ± √(16 − 52))/2 = (4 ± √(−36))/2 = (4 ± 6i)/2 = 2 ± 3i.

State the complex conjugate of z = −5 + 2i.

• −5 − 2i
• 5 − 2i
• 5 + 2i
• −5 + 2i

Answer: −5 − 2i

The conjugate negates only the imaginary part: z* = −5 − 2i.

A complex number z is plotted in the complex plane at the point (−3, 4), where the horizontal axis is real and the vertical axis is imaginary. State z in Cartesian form.

• −3 + 4i
• 4 − 3i
• 3 − 4i
• −3 − 4i

Answer: −3 + 4i

The real part is the horizontal coordinate (−3) and the imaginary part is the vertical coordinate (4), so z = −3 + 4i.