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Complex numbers: board exam questions with solutions

On the ECE exam under Mathematics and the REE exam under Mathematics. Try each question before you open its solution, then practise with fresh numbers below: 2 computed question types on this topic.

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Questions

1. Simplify j²⁷, where j = √−1.

  1. −j
  2. j
  3. −1
  4. 1
Show the answer and solution

Answer: A. −j

27 = 4(6) + 3, so j²⁷ = j³ = −j.

2. What is the magnitude of 3 − j4?

  1. 7
  2. 1
  3. 5
  4. 25
Show the answer and solution

Answer: C. 5

|3 − j4| = √(3² + 4²) = 5.

3. Evaluate (2 + j3)(1 − j2).

  1. −4 − j
  2. 8 − j
  3. 2 − j6
  4. 8 + j7
Show the answer and solution

Answer: B. 8 − j

2 − j4 + j3 − j²6 = 2 − j + 6 = 8 − j.

4. Express 10∠30° in rectangular form.

  1. 8.66 + j5.00
  2. 5.00 + j8.66
  3. 8.66 − j5.00
  4. 10.00 + j30.00
Show the answer and solution

Answer: A. 8.66 + j5.00

10cos 30° = 8.66 and 10sin 30° = 5.00.

5. Evaluate (6 + j8)/(3 − j4).

  1. 2.00 + j0.00
  2. −0.56 + j1.92
  3. −1.20 + j1.60
  4. 0.56 − j1.92
Show the answer and solution

Answer: B. −0.56 + j1.92

In polar form, 10∠53.13° ÷ 5∠−53.13° = 2∠106.26° = −0.56 + j1.92.

6. A 30 Ω resistor is in series with an inductive reactance of 40 Ω. What is the impedance in polar form?

  1. 70∠53.13° Ω
  2. 50∠36.87° Ω
  3. 50∠−53.13° Ω
  4. 50∠53.13° Ω
Show the answer and solution

Answer: D. 50∠53.13° Ω

Z = 30 + j40, so |Z| = 50 Ω and θ = arctan(40/30) = 53.13° (positive because the circuit is inductive).

7. Find the three cube roots of 8∠90°.

  1. 2∠30°, 2∠120°, 2∠210°
  2. 2∠90°, 2∠210°, 2∠330°
  3. 8∠30°, 8∠150°, 8∠270°
  4. 2∠30°, 2∠150°, 2∠270°
Show the answer and solution

Answer: D. 2∠30°, 2∠150°, 2∠270°

By De Moivre's theorem, the roots are 8^(1/3)∠(90° + 360°k)/3 = 2∠(30° + 120°k) for k = 0, 1, 2: 30°, 150° and 270°.

8. Impedances Z₁ = 4 + j3 Ω and Z₂ = 4 − j3 Ω are connected in parallel. What is the equivalent impedance?

  1. 8 Ω
  2. 4 Ω
  3. 3.125 Ω (purely resistive)
  4. 2.5 Ω
Show the answer and solution

Answer: C. 3.125 Ω (purely resistive)

Z = Z₁Z₂/(Z₁ + Z₂) = (16 + 9)/8 = 25/8 = 3.125 + j0 Ω.

Worked examples with computed numbers

These numbers are fixed on this page; in practice they change every time.

9. Express z = -7 + j4 in polar form.

  1. 8.062∠-29.74°
  2. 65∠150.3°
  3. 8.062∠150.3°
  4. 8.062∠-60.26°
Show the answer and solution

Answer: C. 8.062∠150.3°

|z| = √((-7)² + 4²) = 8.062; the angle is atan2(4, -7) = 150.3° (mind the quadrant).

10. Multiply (-2 − j2)(3 + j4).

  1. 2 − j14
  2. 2 − j2
  3. -6 − j8
  4. -14 − j14
Show the answer and solution

Answer: A. 2 − j14

(a + jb)(c + jd) = (ac − bd) + j(ad + bc) = 2 − j14, since j² = −1.

Other topics in the same subjects

Algebra · Trigonometry · Analytic geometry · Differential calculus · Integral calculus · Differential equations · Probability and statistics · Matrices and determinants · Laplace transforms and Fourier series

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