Complex Numbers Practice Questions

A-Levels · A-Level Further Mathematics · 119 free MCQs with instant results and detailed explanations.

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Sample Questions from Complex Numbers

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Easy

Which of the following represents the complex number -2 + 3i in polar form?

A. √13 (cos 2.67 + i sin 2.67)

B. √13 (cos 2.36 + i sin 2.36)

C. √13 (cos 1.57 + i sin 1.57)

D. √13 (cos 3.14 + i sin 3.14)

Show Answer & Explanation

Correct Answer: B

To convert to polar form, we first find the modulus: |z| = √((-2)² + 3²) = √(4 + 9) = √13. The argument θ = tan⁻¹(3/-2) gives us approximately 2.36 radians in the correct quadrant.

Easy

If z = 1 + i, what is the value of z^2?

A. 2i

B. 1 + 2i - 1

C. 2 + 2i

D. 2 + 1i

Show Answer & Explanation

Correct Answer: C

To find z^2, we compute (1 + i)(1 + i) = 1² + 2(1)(i) + i² = 1 + 2i - 1 = 2i, so z^2 = 2 + 2i.

Easy

Which of the following represents the complex conjugate of the complex number z = 2 - 5i?

A. 2 + 5i

B. -2 + 5i

C. 2 - 5i

D. -2 - 5i

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Correct Answer: A

The complex conjugate of a complex number z = a + bi is given by a - bi. For z = 2 - 5i, the conjugate is 2 + 5i.

Medium

If z = 2 + 3i, what is the value of z^2?

A. -5 + 12i

B. 6 + 12i

C. -5 - 12i

D. -12 + 5i

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Correct Answer: A

To compute z^2 for z = 2 + 3i, we use the formula: (a + bi)² = a² + 2abi + (bi)². Thus, (2 + 3i)² = 2² + 2(2)(3i) + (3i)² = 4 + 12i - 9 = -5 + 12i.

Medium

If z = 4(cos(π/3) + isin(π/3)), what is the rectangular form of z?

A. 2 + 2√3 i

B. 4 + 4i

C. 2 - 2√3 i

D. 4 + 2√3 i

Show Answer & Explanation

Correct Answer: A

Using the formula z = r(cos(θ) + i sin(θ)), for r = 4 and θ = π/3, we have z = 4(cos(π/3) + i sin(π/3)) = 4(1/2) + 4(i√3/2) = 2 + 2√3 i.

Medium

What is the product of the complex numbers z1 = 1 + i and z2 = 2 - 3i?

A. 7 - i

B. -1 - i

C. -1 + i

D. 1 + 5i

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Correct Answer: A

To find the product of two complex numbers, use (a + bi)(c + di) = (ac - bd) + (ad + bc)i. Thus, (1 + i)(2 - 3i) = 1*2 + 1*(-3) + 2*i + (-3)*i = 2 + 3 + (-1)i = 7 - i.

Medium

Which of the following is a correct representation of the complex number -1 + i in polar form?

A. (√2, 135°)

B. (√2, -45°)

C. (1, 90°)

D. (2, 90°)

Show Answer & Explanation

Correct Answer: A

To convert to polar form, calculate the modulus r = √((-1)² + (1)²) = √2, and the argument θ = arctan(1/-1) = 135° (since it lies in the 2nd quadrant).

Hard

Given the complex number z = 3 + 4i, what is the modulus of z?

A. 5

B. 7

C. 25

D. 12

Show Answer & Explanation

Correct Answer: A

The modulus of a complex number z = a + bi is given by |z| = √(a² + b²). Here, a = 3 and b = 4, so |z| = √(3² + 4²) = √(9 + 16) = √25 = 5.

Hard

If z = e^(iπ/4), what is the value of z² in rectangular form?

A. 1 + i

B. 0 + 1

C. -1 + i

D. 1 - i

Show Answer & Explanation

Correct Answer: A

Using Euler's formula, z = cos(π/4) + i sin(π/4) = √2/2 + i√2/2. Squaring z gives z² = (√2/2 + i√2/2)² = (1 + 2i + (-1))/2 = 1 + i.

Q10

Hard

Given the complex number z = 3 + 4i, calculate the modulus of z and then determine which of the following represents |z|^2.

A. 25

B. 12

C. 7

D. 16

Show Answer & Explanation

Correct Answer: A

The modulus of a complex number z = a + bi is given by |z| = √(a^2 + b^2). Here, |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5. Therefore, |z|^2 = 25.

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Complex Numbers — A-Levels A-Level Further Mathematics Practice Questions Online

This page contains 119 practice MCQs for the chapter Complex Numbers in A-Levels A-Level Further Mathematics. The questions are organized by difficulty — 31 easy, 65 medium, 23 hard — so you can choose the right level for your preparation.

Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.