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Complex Numbers Practice Questions

SAT · SAT Math · 136 free MCQs with instant results and detailed explanations.

136
Total
35
Easy
77
Medium
24
Hard

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Sample Questions from Complex Numbers

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Q1
Easy
Which of the following represents the complex number 5 - 2i in polar form?
A. √29 (cos θ + i sin θ)
B. 5 + 2i
C. √29 (cos 2 + i sin 2)
D. √29 (cos 1 + i sin 1)
Show Answer & Explanation
Correct Answer: A
To convert to polar form, calculate the modulus as √(5^2 + (-2)^2) = √29 and the angle θ = tan^(-1)(-2/5). Thus, it is expressed as √29 (cos θ + i sin θ).
Q2
Easy
What is the value of the complex number (3 + 4i) + (1 - 2i)?
A. 4 + 2i
B. 2 + 6i
C. 3 + 2i
D. 1 + 6i
Show Answer & Explanation
Correct Answer: A
To add two complex numbers, combine their real parts and their imaginary parts. Here, (3 + 1) for real parts gives 4, and (4 - 2) for imaginary parts gives 2, resulting in 4 + 2i.
Q3
Easy
What is the product of the complex numbers (2 + 3i) and (4 - i)?
A. 11 + 10i
B. 14 + 9i
C. 12 + 9i
D. 8 + 12i
Show Answer & Explanation
Correct Answer: A
To find the product of two complex numbers, use the distributive property (FOIL). Here, (2 * 4) + (2 * -i) + (3i * 4) + (3i * -i) = 8 - 2i + 12i + 3 = 11 + 10i.
Q4
Medium
Which of the following is equivalent to the expression (2 - 3i)(2 + 3i)?
A. 13
B. 7
C. 1
D. 6
Show Answer & Explanation
Correct Answer: A
The expression (2 - 3i)(2 + 3i) is a product of conjugates, which equals a² + b². Here, a = 2 and b = 3, so the result is 2² + 3² = 4 + 9 = 13.
Q5
Medium
What is the value of (3 + 4i) + (2 - 3i)?
A. 5 + i
B. 1 + i
C. 5 - i
D. 1 - i
Show Answer & Explanation
Correct Answer: A
When adding complex numbers, you add the real parts and the imaginary parts separately. Here, (3 + 2) + (4i - 3i) = 5 + i.
Q6
Medium
Which of the following represents the product of (1 + i)(1 - i)?
A. 2
B. 1
C. 0
D. 1 + 0i
Show Answer & Explanation
Correct Answer: A
The product (1 + i)(1 - i) uses the difference of squares formula, resulting in 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2.
Q7
Medium
What is the modulus of the complex number 7 - 24i?
A. 25
B. 17
C. 24
D. 31
Show Answer & Explanation
Correct Answer: A
The modulus of a complex number a + bi is calculated as √(a² + b²). Here, √(7² + (-24)²) = √(49 + 576) = √625 = 25.
Q8
Hard
If z = 5 + 12i, what is |z|^2?
A. 169
B. 25
C. 144
D. 157
Show Answer & Explanation
Correct Answer: A
The modulus squared of a complex number z = a + bi is given by |z|^2 = a^2 + b^2. Here, a = 5 and b = 12, so |z|^2 = 5^2 + 12^2 = 25 + 144 = 169.
Q9
Hard
What is the square of the complex number (3 + 4i)?
A. -7 + 24i
B. 9 + 24i - 16
C. 1 + 24i
D. -7 - 24i
Show Answer & Explanation
Correct Answer: A
To find the square of a complex number (a + bi), we use (a + bi)(a + bi) = a^2 + 2abi + (bi)^2. Here, a = 3 and b = 4. Thus, (3 + 4i)^2 = 3^2 + 2(3)(4i) + (4i)^2 = 9 + 24i - 16 = -7 + 24i.
Q10
Hard
If z = 5 - 12i, what is the modulus of z?
A. 13
B. 7
C. 17
D. 15
Show Answer & Explanation
Correct Answer: A
The modulus of a complex number z = a + bi is given by |z| = √(a^2 + b^2). Here, a = 5 and b = -12. So, |z| = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13.

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Complex Numbers — SAT SAT Math Practice Questions Online

This page contains 136 practice MCQs for the chapter Complex Numbers in SAT SAT Math. The questions are organized by difficulty — 35 easy, 77 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.