A Level Maths: Pure 3 chapter 9: Complex numbers MCQs with answers

By Team Megalecture · Updated 4 Nov 2026

29 exam-style questions on Complex numbers, each with the answer and a short explanation. Ten of them are below to try. The chapter test draws ten fresh questions each time and is marked the moment you finish, with your score by topic.

29 questions · 1 topics · 10 per test

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What chapter 9 covers

Ten Complex numbers questions with answers

Try each one, then open the answer. These ten stay the same; the test draws new ones every time.

1. A cubic equation with real coefficients has roots 2 and 1 − 3i. What is its third root?

  1. A
    −1 + 3i
  2. B
    1 + 3i
  3. C
    −1 − 3i
  4. D
    −2
Show answer

Answer: B. Non-real roots of a polynomial equation with real coefficients occur in conjugate pairs, so the conjugate 1 + 3i must also be a root. The conjugate changes the sign of the imaginary part only.

2. Solve the equation z2 − 4z + 13 = 0.

  1. A
    z = −2 ± 3i
  2. B
    z = 2 ± 3i
  3. C
    z = 4 ± 6i
  4. D
    z = 2 ± 6i
Show answer

Answer: B. z = [4 ± √(16 − 52)]/2 = (4 ± √(−36))/2 = (4 ± 6i)/2 = 2 ± 3i. Forgetting to divide the 6i by 2 gives 2 ± 6i; −2 ± 3i comes from a sign slip with −b.

3. Find the two square roots of −15 + 8i.

  1. A
    4 + i and −4 − i
  2. B
    1 − 4i and −1 + 4i
  3. C
    1 + 4i and −1 + 4i
  4. D
    1 + 4i and −1 − 4i
Show answer

Answer: D. Let (a + ib)2 = −15 + 8i, so a2 − b2 = −15 and ab = 4. Substituting b = 4/a gives a4 + 15a2 − 16 = 0, so a2 = 1: a = 1, b = 4 or a = −1, b = −4. Check: (1 + 4i)2 = 1 + 8i − 16 = −15 + 8i. The two square roots are always negatives of each other.

4. Find the complex number z such that (1 + i)z + 2z* = 5 + 3i.

  1. A
    1 + 2i
  2. B
    1 − 2i
  3. C
    2 − i
  4. D
    1.8 + 0.4i
Show answer

Answer: B. Let z = x + iy. Then (1 + i)(x + iy) = (x − y) + i(x + y) and 2z* = 2x − 2iy, so the left side is (3x − y) + i(x − y). Equating parts: 3x − y = 5 and x − y = 3, so x = 1 and y = −2. Using 2z in place of 2z* gives 1.8 + 0.4i.

5. In an Argand diagram the point P represents the complex number z. Which describes the point that represents iz?

  1. A
    P rotated through 90° clockwise about the origin
  2. B
    the reflection of P in the real axis
  3. C
    P translated 1 unit parallel to the imaginary axis
  4. D
    P rotated through 90° anticlockwise about the origin
Show answer

Answer: D. i has modulus 1 and argument ½π, so multiplying by i keeps |z| the same and adds ½π to the argument: a rotation of 90° anticlockwise about O. Reflection in the real axis gives z*, and adding i (not multiplying) would be the translation.

6. A quartic equation with real coefficients and leading coefficient 1 has roots 3i and 1 + i. Which is the equation?

  1. A
    (z2 − 9)(z2 − 2z + 2) = 0
  2. B
    (z2 + 9)(z2 + 2z + 2) = 0
  3. C
    (z2 + 9)(z2 − 2z + 2) = 0
  4. D
    (z2 + 9)(z2 − 2z) = 0
Show answer

Answer: C. The conjugates −3i and 1 − i must also be roots. (z − 3i)(z + 3i) = z2 + 9 and (z − 1 − i)(z − 1 + i) = (z − 1)2 + 1 = z2 − 2z + 2. z2 − 9 comes from taking (3i)2 = 9.

7. Which describes the locus of points z = x + iy satisfying arg(z − 1 − i) = ¼π?

  1. A
    the whole of the line y = x
  2. B
    the part of the line y = x for which x > 1
  3. C
    the part of the line y = x for which x < 1
  4. D
    the part of the line y = x for which x > −1
Show answer

Answer: B. arg(z − (1 + i)) = ¼π is a half-line that starts at the point (1, 1), which is not included, and goes at 45° to the positive real direction, up and to the right. It lies along y = x, but only where x > 1; points of y = x with x < 1 have argument −¾π from (1, 1).

8. z = 3 − 4i. Find |z| and Im z.

  1. A
    |z| = 5 and Im z = −4
  2. B
    |z| = 5 and Im z = −4i
  3. C
    |z| = 25 and Im z = −4
  4. D
    |z| = 7 and Im z = −4
Show answer

Answer: A. |z| = √(32 + 42) = 5. The imaginary part is the real number −4 (the coefficient of i), not −4i; 25 is |z|2.

9. Find arg(−1 + √3 i), where −π < arg z ≤ π.

  1. A
    −π/3
  2. B
    2π/3
  3. C
    π/3
  4. D
    5π/6
Show answer

Answer: B. The point (−1, √3) is in the second quadrant. The angle it makes with the negative real axis is tan−1(√3/1) = π/3, so arg = π − π/3 = 2π/3. A calculator gives tan−1(√3/(−1)) = −π/3, which is in the wrong quadrant.

10. The complex number z has |z| = 2 and arg z = π/5. Find |z3| and arg(z3).

  1. A
    6 and 3π/5
  2. B
    8 and π/5
  3. C
    8 and 3π/5
  4. D
    6 and π/5
Show answer

Answer: C. Multiplying complex numbers multiplies the moduli and adds the arguments, so z3 has modulus 23 = 8 and argument 3 × π/5 = 3π/5, which is already between −π and π.

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How many Complex numbers MCQs are there?

29, each with the answer and a short explanation, across 1 topics: Complex numbers. Every chapter test draws ten of them, so you can take it many times.

Is the A Level Maths: Pure 3 chapter 9 test free?

Yes. The test, the answers and the explanations are free and need no sign-up. Sign in only if you want your scores kept across devices.

How is the chapter test marked?

The moment you press Finish: your score, the right answer and an explanation for every question, and the ones you got wrong saved in My mistakes to retry.

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