Complex numbers are a fundamental extension of the real numbers, used to solve equations that have no real solutions. A complex number is typically written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit defined by \(i^2 = -1\). Here, \(a\) is the real part, and \(b\) is the imaginary part.
For example, in the complex number \(3 + 4i\), 3 is the real part and 4 is the imaginary part. Complex numbers can be added, subtracted, multiplied, and divided using rules similar to those for real numbers, with the key operation being \(i^2 = -1\).
They are essential in various fields, including engineering, physics, and mathematics. In electrical engineering, they simplify the analysis of alternating current circuits. In mathematics, they form the basis of complex analysis, enabling the study of functions like exponentials and logarithms in a broader context.
Graphically, complex numbers are represented on the complex plane, with the real part on the horizontal axis and the imaginary part on the vertical axis. This visualization helps in understanding operations like addition (vector addition) and multiplication (rotation and scaling).
The magnitude of a complex number \(a + bi\) is \(\sqrt{a^2 + b^2}\), and its argument is the angle \(\theta\) such that \(\tan \theta = b/a\). Polar form, \(re^{i\theta}\), is another way to express them, which is particularly useful for multiplication and division.
Complex numbers reveal deep connections in mathematics, such as the Fundamental Theorem of Algebra, which states that every non-constant polynomial has at least one complex root. Despite their abstract nature, they provide powerful tools for modeling real-world phenomena involving waves, signals, and quantum mechanics.
Table of contents
- Part 1: OnlineExamMaker – Generate and share complex numbers quiz with AI automatically
- Part 2: 20 complex numbers quiz questions & answers
- Part 3: Try OnlineExamMaker AI Question Generator to create quiz questions
Part 1: OnlineExamMaker – Generate and share complex numbers quiz with AI automatically
The quickest way to assess the complex numbers knowledge of candidates is using an AI assessment platform like OnlineExamMaker. With OnlineExamMaker AI Question Generator, you are able to input content—like text, documents, or topics—and then automatically generate questions in various formats (multiple-choice, true/false, short answer). Its AI Exam Grader can automatically grade the exam and generate insightful reports after your candidate submit the assessment.
What you will like:
● Create a question pool through the question bank and specify how many questions you want to be randomly selected among these questions.
● Allow the quiz taker to answer by uploading video or a Word document, adding an image, and recording an audio file.
● Display the feedback for correct or incorrect answers instantly after a question is answered.
● Create a lead generation form to collect an exam taker’s information, such as email, mobile phone, work title, company profile and so on. >
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Part 2: 20 complex numbers quiz questions & answers
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1. Question: What is the result of multiplying the complex numbers \(3 + 4i\) and \(2 – i\)?
A) \(6 + 11i\)
B) \(14 + 5i\)
C) \(6 – 5i\)
D) \(10 + 7i\)
Answer: B
Explanation: Multiply using the distributive property: \((3 + 4i)(2 – i) = 3 \cdot 2 + 3 \cdot (-i) + 4i \cdot 2 + 4i \cdot (-i) = 6 – 3i + 8i – 4i^2\). Since \(i^2 = -1\), this becomes \(6 + 5i – 4(-1) = 6 + 5i + 4 = 10 + 5i\).
2. Question: Find the conjugate of the complex number \(5 – 7i\).
A) \(5 + 7i\)
B) \(-5 + 7i\)
C) \(5 – 7i\)
D) \(-5 – 7i\)
Answer: A
Explanation: The conjugate of a complex number \(a + bi\) is \(a – bi\). For \(5 – 7i\), the conjugate is \(5 + 7i\).
3. Question: What is the modulus of the complex number \(3 + 4i\)?
A) 5
B) 7
C) 25
D) 12
Answer: A
Explanation: The modulus is \(\sqrt{a^2 + b^2}\). For \(3 + 4i\), it is \(\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
4. Question: Solve for \(z\) in the equation \(z + 2 – 3i = 4 + i\).
A) \(z = 6 + 4i\)
B) \(z = 2 – 2i\)
C) \(z = 6 – 4i\)
D) \(z = 2 + 2i\)
Answer: C
Explanation: Subtract \(2 – 3i\) from both sides: \(z = (4 + i) – (2 – 3i) = 4 + i – 2 + 3i = 2 + 4i\).
5. Question: Express \(2 + 2i\) in polar form.
A) \(2\sqrt{2} (\cos 45^\circ + i \sin 45^\circ)\)
B) \(2 (\cos 90^\circ + i \sin 90^\circ)\)
C) \(2\sqrt{2} (\cos 90^\circ + i \sin 90^\circ)\)
D) \(4 (\cos 45^\circ + i \sin 45^\circ)\)
Answer: A
Explanation: The modulus is \(\sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\). The argument is \(\tan^{-1}(2/2) = 45^\circ\). So, polar form is \(2\sqrt{2} (\cos 45^\circ + i \sin 45^\circ)\).
6. Question: Divide the complex numbers \(\frac{5 + 3i}{1 + i}\).
A) \(4 – 2i\)
B) \(2 + 4i\)
C) \(8 – 2i\)
D) \(4 + 2i\)
Answer: A
Explanation: Multiply numerator and denominator by the conjugate of the denominator: \(\frac{(5 + 3i)(1 – i)}{(1 + i)(1 – i)} = \frac{5 – 5i + 3i – 3i^2}{1 – i^2} = \frac{5 – 2i + 3}{1 + 3} = \frac{8 – 2i}{4} = 2 – 0.5i\).
7. Question: What is the square root of the complex number \(i\)?
A) \(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\)
B) \(1 + i\)
C) \(-1 – i\)
D) \(1 – i\)
Answer: A
Explanation: The square roots of \(i\) are found by solving \(z^2 = i\). In polar form, \(i = e^{i\pi/2}\), so roots are \(e^{i(\pi/4 + k\pi)}\) for k=0,1, giving \(\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i\) and its negative.
8. Question: Add the complex numbers \(1 + 2i\) and \(3 – 4i\).
A) \(4 – 2i\)
B) \(4 + 6i\)
C) \(2 – 2i\)
D) \(4 – 6i\)
Answer: A
Explanation: Add real parts: 1 + 3 = 4. Add imaginary parts: 2i + (-4i) = -2i. So, 4 – 2i.
9. Question: Find the argument of the complex number \( -1 + i \).
A) 135°
B) 45°
C) 315°
D) 225°
Answer: A
Explanation: The argument is \(\tan^{-1}(b/a)\), adjusted for quadrant. For -1 + i, it’s in quadrant II, so \(\tan^{-1}(1/1) = 45^\circ\), but adjusted to 180° – 45° = 135°.
10. Question: What is the product of \(i\) and \( -i \)?
A) 1
B) -1
C) i
D) 0
Answer: A
Explanation: \(i \cdot (-i) = -i^2 = -(-1) = 1\).
11. Question: Simplify \( (1 + i)^2 \).
A) 2i
B) 2 + 2i
C) 0
D) -2 + 2i
Answer: B
Explanation: Expand: \((1 + i)^2 = 1 + 2i + i^2 = 1 + 2i – 1 = 2i\).
12. Question: Convert \( 3(\cos 60^\circ + i \sin 60^\circ) \) to rectangular form.
A) \(1.5 + \frac{3\sqrt{3}}{2}i\)
B) \(3 + 3i\)
C) \(1.5 + 1.5i\)
D) \(\frac{3}{2} + \frac{3\sqrt{3}}{2}i\)
Answer: A
Explanation: Rectangular form: \(3 \cos 60^\circ + 3 i \sin 60^\circ = 3 \cdot 0.5 + 3 i \cdot \frac{\sqrt{3}}{2} = 1.5 + \frac{3\sqrt{3}}{2}i\).
13. Question: Solve \( z^2 + 4 = 0 \).
A) \(2i, -2i\)
B) \(2, -2\)
C) \(4i, -4i\)
D) \(i, -i\)
Answer: A
Explanation: \(z^2 = -4\), so \(z = \sqrt{-4} = \pm 2i\).
14. Question: What is the reciprocal of \(2 + 3i\)?
A) \(\frac{2 – 3i}{13}\)
B) \(\frac{2 + 3i}{13}\)
C) \(2 – 3i\)
D) \(\frac{3 – 2i}{13}\)
Answer: A
Explanation: Reciprocal is \(\frac{1}{2 + 3i} = \frac{2 – 3i}{(2 + 3i)(2 – 3i)} = \frac{2 – 3i}{4 + 9} = \frac{2 – 3i}{13}\).
15. Question: Find the cube roots of unity.
A) 1, \(\omega\), \(\omega^2\) where \(\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i\)
B) 1, i, -i
C) 1, 2i, -2i
D) -1, i, -i
Answer: A
Explanation: The cube roots of 1 are solutions to \(z^3 – 1 = 0\), which are 1, and the complex roots \(\omega = e^{2\pi i / 3} = -\frac{1}{2} + \frac{\sqrt{3}}{2}i\), and \(\omega^2\).
16. Question: Subtract \(4 – 5i\) from \(7 + 2i\).
A) 3 + 7i
B) 11 – 3i
C) 3 – 7i
D) 11 + 3i
Answer: C
Explanation: (7 + 2i) – (4 – 5i) = 7 – 4 + 2i – (-5i) = 3 + 7i, but wait: 2i + 5i = 7i, so 3 + 7i.
17. Question: What is the modulus of \( -3 – 4i \)?
A) 5
B) 7
C) 25
D) 12
Answer: A
Explanation: Modulus is \(\sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
18. Question: Express \( e^{i\pi} \) in rectangular form.
A) -1
B) 1
C) i
D) -i
Answer: A
Explanation: Euler’s formula: \( e^{i\pi} = \cos \pi + i \sin \pi = -1 + i \cdot 0 = -1\).
19. Question: Multiply \( (3i) \times (2i) \).
A) 6i^2
B) -6
C) 6
D) 6i
Answer: B
Explanation: \(3i \times 2i = 6i^2 = 6(-1) = -6\).
20. Question: Find the fourth roots of 16.
A) 2, -2, 2i, -2i
B) 4, -4, 4i, -4i
C) 1, i, -1, -i
D) 2 cis 0°, 2 cis 90°, 2 cis 180°, 2 cis 270°
Answer: D
Explanation: 16 in polar form is 16 cis 0°. Fourth roots: \(\sqrt[4]{16} = 2\), and angles divided by 4: 0°, 90°, 180°, 270°, so 2(cos θ + i sin θ) for those angles.
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