Complex numbers: Cartesian form, Argand diagram practice questions.

The principal argument (in radians) of z = 1 + i is

• 0
• π/4
• π/2
• π

Answer: π/4

z lies in the first quadrant with equal real and imaginary parts, so arg(z) = arctan(1/1) = π/4.

Calculate the product (2 + 3i)(1 − 4i), giving your answer in Cartesian form.

• −10 − 5i
• 14 − 5i
• 14 + 5i
• 2 − 5i

Answer: 14 − 5i

Expanding: 2 − 8i + 3i − 12i² = 2 − 5i + 12 = 14 − 5i; the distractor −10 − 5i comes from forgetting that i² = −1.

Express (3 + i)/(1 − 2i) in the form a + bi.

• 1/5 + 7i/5
• 1 − i
• 1/5 − 7i/5
• 1 + 7i

Answer: 1/5 + 7i/5

Multiply top and bottom by the conjugate 1 + 2i: numerator (3 + i)(1 + 2i) = 1 + 7i, denominator 1² + 2² = 5, giving 1/5 + 7i/5.

Solve the equation z² − 4z + 13 = 0.

• z = −2 ± 3i
• z = 2 ± 6i
• z = 4 ± 3i
• z = 2 ± 3i

Answer: z = 2 ± 3i

The quadratic formula gives z = (4 ± √(16 − 52))/2 = (4 ± 6i)/2 = 2 ± 3i; forgetting to halve √(−36) = 6i gives the distractor 2 ± 6i.

State the quadrant of the Argand diagram in which the point representing z = −2 + 2i lies.

• First quadrant
• Second quadrant
• Third quadrant
• Fourth quadrant

Answer: Second quadrant

The real part −2 is negative and the imaginary part 2 is positive, so the point (−2, 2) lies in the second quadrant.

For z = a + bi with a, b real and conjugate z*, the product z × z* is equal to which of the following?

• a² − b²
• (a + b)²
• a² + b²
• 2ab

Answer: a² + b²

(a + bi)(a − bi) = a² − (bi)² = a² + b², which is real and equals |z|²; a² − b² is the classic error of treating i² as +1.