Complex numbers: polar and Euler form; applications (e.g. AC circuits) practice questions.

State the modulus and argument of the complex number z = 1 + i.

• |z| = √2, arg(z) = π/4
• |z| = 2, arg(z) = π/4
• |z| = √2, arg(z) = π/2
• |z| = 1, arg(z) = π/4

Answer: |z| = √2, arg(z) = π/4

|z| = √(1² + 1²) = √2 and arg(z) = arctan(1/1) = π/4 since z lies in the first quadrant.

Determine the modulus of z = 3 − 4i.

• 7
• 1
• 5
• 25

Answer: 5

|z| = √(3² + (−4)²) = √(9 + 16) = √25 = 5.

A complex number is given in polar form as z = 2(cos(π/3) + i sin(π/3)). Determine z in Cartesian form.

• 1 + i√3
• √3 + i
• 2 + 2i
• 1 − i√3

Answer: 1 + i√3

z = 2cos(π/3) + 2i sin(π/3) = 2(1/2) + 2i(√3/2) = 1 + i√3.

Express z = 4e^(iπ/2) in Cartesian form.

• 4
• −4
• 4i
• −4i

Answer: 4i

e^(iπ/2) = cos(π/2) + i sin(π/2) = 0 + i(1) = i, so z = 4i.

Let z₁ = 3e^(iπ/6) and z₂ = 2e^(iπ/3). Determine z₁z₂ in Euler form.

• 6e^(iπ/2)
• 5e^(iπ/2)
• 6e^(iπ/18)
• 6e^(iπ/9)

Answer: 6e^(iπ/2)

When multiplying in Euler form, moduli multiply (3×2 = 6) and arguments add (π/6 + π/3 = π/2), giving 6e^(iπ/2).

Given z₁ = 12e^(iπ/2) and z₂ = 3e^(iπ/6), calculate z₁/z₂ in Euler form.

• 4e^(iπ/3)
• 9e^(iπ/3)
• 4e^(i2π/3)
• 15e^(iπ/3)

Answer: 4e^(iπ/3)

When dividing in Euler form, moduli divide (12/3 = 4) and arguments subtract (π/2 − π/6 = π/3), giving 4e^(iπ/3).