Complex Numbers Practice Questions

HSC/SSC (Bangladesh) · HSC Mathematics · 113 free MCQs with instant results and detailed explanations.

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Sample Questions from Complex Numbers

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Easy

If z = 1 - 2i, what is the conjugate of z?

A. 1 + 2i

B. -1 - 2i

C. 1 - 2i

D. 2 + 1i

Show Answer & Explanation

Correct Answer: A

The conjugate of a complex number z = a + bi is given by a - bi. For z = 1 - 2i, its conjugate is 1 + 2i.

Easy

What is the value of (2 + 3i) + (4 - 5i)?

A. 6 - 2i

B. 6 + 2i

C. 2 + 8i

D. 8 - 2i

Show Answer & Explanation

Correct Answer: A

To add complex numbers, add the real parts and the imaginary parts separately. Thus, (2 + 3i) + (4 - 5i) = (2 + 4) + (3 - 5)i = 6 - 2i.

Easy

Which of the following is the conjugate of the complex number 5 + 2i?

A. 5 - 2i

B. -5 + 2i

C. -5 - 2i

D. 2 + 5i

Show Answer & Explanation

Correct Answer: A

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

Medium

Which of the following represents the complex number z = -5 + 12i in polar form?

A. 13(cos(2.2) + i sin(2.2))

B. 13(cos(0.85) + i sin(0.85))

C. 13(cos(-2.2) + i sin(-2.2))

D. 13(cos(1.67) + i sin(1.67))

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Correct Answer: A

To convert to polar form, find r = √((-5)² + (12)²) = 13, and θ = tan⁻¹(12/-5) which is in the second quadrant, giving θ ≈ 2.2 radians. Hence, the polar form is 13(cos(2.2) + i sin(2.2)).

Medium

Which of the following represents the principal argument of the complex number z = -3 + 4i?

A. arctan(-4/3)

B. arctan(4/3) + π

C. arctan(3/4)

D. arctan(-3/4) + π

Show Answer & Explanation

Correct Answer: B

The principal argument of a complex number in the second quadrant is found by adding π to the arctangent of the imaginary part divided by the real part, hence arctan(4/3) + π.

Medium

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

A. √13

B. √17

C. 5

D. √10

Show Answer & Explanation

Correct Answer: B

The magnitude of a complex number z = a + bi is given by |z| = √(a² + b²). Here, |z| = √(2² + 3²) = √(4 + 9) = √13.

Medium

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

A. -1 + 5i

B. 7 - i

C. 8 - i

D. -7 + i

Show Answer & Explanation

Correct Answer: A

Multiplying (1 + i)(2 - 3i) gives 2 - 3i + 2i - 3(i²), simplifying to 2 + 3 + (-3 + 2)i = -1 + 5i.

Hard

If z = 3 + 4i, find the modulus of z and the argument of z in radians.

A. 5, 0.927

B. 7, 0.643

C. 5, 0.927 + 2πk

D. 5, 1.107

Show Answer & Explanation

Correct Answer: C

The modulus of z = √(3^2 + 4^2) = √25 = 5. The argument is tan^(-1)(4/3) which is approximately 0.927 radians. Since arguments can differ by multiples of 2π, the complete representation is 0.927 + 2πk.

Hard

Solve the equation z^2 + (3 + 2i)z + (1 - i) = 0 for z, where z is a complex number.

A. -1 + i

B. -1 - i

C. 2 - i

D. -2 + 2i

Show Answer & Explanation

Correct Answer: A

Using the quadratic formula z = [-b ± sqrt(b² - 4ac)] / 2a, where a=1, b=3+2i, c=1-i, we calculate the discriminant and find two roots. One of the roots simplifies to -1 + i.

Q10

Hard

If z1 = 2 + 3i and z2 = 4 - i, what is the product z1 * z2?

A. 11 + 10i

B. 14 + 10i

C. 10 + 1i

D. 5 + 15i

Show Answer & Explanation

Correct Answer: A

To find the product of two complex numbers, multiply them using the distributive property: z1 * z2 = (2 + 3i)(4 - i) = 8 - 2i + 12i - 3 = 11 + 10i.

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Complex Numbers — HSC/SSC (Bangladesh) HSC Mathematics Practice Questions Online

This page contains 113 practice MCQs for the chapter Complex Numbers in HSC/SSC (Bangladesh) HSC Mathematics. The questions are organized by difficulty — 33 easy, 61 medium, 19 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.