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Complex Numbers Practice Questions

Thanaweya Amma (Egypt) · Thanaweya Mathematics · 114 free MCQs with instant results and detailed explanations.

114
Total
27
Easy
63
Medium
24
Hard

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Sample Questions from Complex Numbers

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Q1
Easy
Which of the following is a purely imaginary number?
A. 5 + 0i
B. 0 + 7i
C. 3 - 2i
D. 1 + 1i
Show Answer & Explanation
Correct Answer: B
A purely imaginary number has no real part, meaning it is of the form 0 + bi. The option 0 + 7i fits this definition, while the others contain a non-zero real part.
Q2
Easy
Which of the following is the sum of the complex numbers 2 + 3i and 4 - 5i?
A. 6 - 2i
B. 6 + 2i
C. 2 + 8i
D. 2 - 2i
Show Answer & Explanation
Correct Answer: A
To find the sum of two complex numbers, add their real parts and imaginary parts separately. (2 + 4) + (3i - 5i) = 6 - 2i.
Q3
Easy
If z = 3 + 4i, what is the modulus of z?
A. 5
B. 7
C. 1
D. 8
Show Answer & Explanation
Correct Answer: A
The modulus of a complex number z = a + bi is given by |z| = √(a² + b²). Here, |z| = √(3² + 4²) = √(9 + 16) = √25 = 5.
Q4
Medium
Which of the following represents the complex conjugate of the number z = 2 - 5i?
A. 2 + 5i
B. -2 + 5i
C. 5 - 2i
D. 2 - 5i
Show Answer & Explanation
Correct Answer: A
The complex conjugate of a complex number z = a + bi is given by a - bi. Therefore, for z = 2 - 5i, the conjugate is 2 + 5i.
Q5
Medium
If z = 1 + i and w = 2 - 3i, what is the product zw?
A. -1 + 5i
B. 7 - i
C. 8 - i
D. 5 + 7i
Show Answer & Explanation
Correct Answer: A
To find the product of two complex numbers, we use the distributive property. (1 + i)(2 - 3i) = 1*2 + 1*(-3i) + i*2 + i*(-3i) = 2 - 3i + 2i + 3 = -1 + 5i.
Q6
Medium
If z = 4 + 3i, what is z^2?
A. 7 + 24i
B. 16 + 24i
C. 7 - 24i
D. 16 - 24i
Show Answer & Explanation
Correct Answer: A
To square a complex number, use (a + bi)² = a² + 2abi + (bi)². For z = 4 + 3i, z² = 4² + 2(4)(3i) + (3i)² = 16 + 24i - 9 = 7 + 24i.
Q7
Medium
Which of the following is the Cartesian form of the complex number z = 5(cos 45° + i sin 45°)?
A. 5 + 5i
B. 2.5 + 2.5i
C. 5√2 + 5√2 i
D. 0 + 5i
Show Answer & Explanation
Correct Answer: A
Using Euler's formula, we convert z = 5(cos 45° + i sin 45°) to Cartesian form by calculating cos 45° = sin 45° = √2/2. Therefore, z = 5(√2/2 + i(√2/2)) = 5(√2/2) + 5(√2/2)i = 5 + 5i.
Q8
Hard
If z1 = 2 + 3i and z2 = 4 - 5i, what is the product z1 * z2?
A. 23 - 7i
B. 23 + 7i
C. 2 - 15i
D. 7 - 22i
Show Answer & Explanation
Correct Answer: A
To find the product of two complex numbers z1 = a + bi and z2 = c + di, we use the formula (a + bi)(c + di) = ac + adi + bci + bdi². Here, a = 2, b = 3, c = 4, d = -5. Therefore, z1 * z2 = (2 * 4) + (2 * -5)i + (3 * 4)i + (3 * -5)(-1) = 8 - 10i + 12i + 15 = 23 - 7i.
Q9
Hard
Given two complex numbers z1 = 3 + 4i and z2 = 1 - 2i, what is the product z1 * z2?
A. -5 + 10i
B. 11 - 10i
C. 11 + 10i
D. 5 + 10i
Show Answer & Explanation
Correct Answer: B
The product of two complex numbers z1 = a + bi and z2 = c + di is given by (ac - bd) + (ad + bc)i. Here, a = 3, b = 4, c = 1, and d = -2. Thus, z1 * z2 = (3*1 - 4*(-2)) + (3*(-2) + 4*1)i = (3 + 8) + (-6 + 4)i = 11 - 2i.
Q10
Hard
If z = 4 + 3i, what is the modulus of z and the argument of z in radians?
A. 5, 0.6435
B. 5, 0.9273
C. 3.5, 0.6435
D. 5, 1.1071
Show Answer & Explanation
Correct Answer: B
The modulus of z is calculated as √(4² + 3²) = √(16 + 9) = √25 = 5. The argument is found using tan⁻¹(3/4) which is approximately 0.6435 radians. Thus, the correct answer combining both values is 5 and approximately 0.6435 radians.

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Complex Numbers — Thanaweya Amma (Egypt) Thanaweya Mathematics Practice Questions Online

This page contains 114 practice MCQs for the chapter Complex Numbers in Thanaweya Amma (Egypt) Thanaweya Mathematics. The questions are organized by difficulty — 27 easy, 63 medium, 24 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.