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Placement Test Practice — Complex Numbers
Placement Test Practice — Complex Numbers
Practice Test — 20 Questions
1. Compute $(3+2i)+(1-5i)$.
2. Compute $(4-i)(2+3i)$.
3. Simplify $i^{17}$.
4. Find $\bar{z}$ for $z = 7-3i$.
5. Find $|z|$ for $z = 5+12i$.
6. Compute $\frac{3+i}{1-2i}$.
7. Convert $z = -1 + \sqrt{3}i$ to polar form.
8. Compute $(1+i)^6$ using De Moivre.
9. Find the cube roots of $1$.
10. Solve $x^2 + 9 = 0$.
11. $z \cdot \bar{z}$ for $z = 4+3i$?
12. Simplify $\frac{1}{i}$.
13. Convert $4\operatorname{cis}(\pi/2)$ to rectangular.
14. Solve $x^2+2x+2 = 0$.
15. What are the 4th roots of $16$?
16. Compute $(2i)^3$.
17. Find $\arg(-2)$.
18. Are the roots of $x^2+1=0$ real?
19. If $3-2i$ is a root of a real-coefficient polynomial, what other root exists?
20. Find $|z|$ for $z = -6+8i$.
Show Answer Key
1. $4-3i$
2. $8+12i-2i-3i^2 = 11+10i$
3. $17 = 4(4)+1$; $i^{17} = i$
4. $7+3i$
5. $\sqrt{25+144} = 13$
6. $\frac{(3+i)(1+2i)}{5} = \frac{1+7i}{5}$
7. $r = 2$, $\theta = 2\pi/3$; $2\operatorname{cis}(2\pi/3)$
8. $(\sqrt{2})^6\operatorname{cis}(6\pi/4) = 8\operatorname{cis}(3\pi/2) = -8i$
9. $1$, $-\frac{1}{2}+\frac{\sqrt{3}}{2}i$, $-\frac{1}{2}-\frac{\sqrt{3}}{2}i$
10. $x = \pm 3i$
11. $16+9 = 25$
12. $-i$
13. $4i$
14. $x = -1 \pm i$
15. $2, 2i, -2, -2i$
16. $8i^3 = -8i$
17. $\pi$
18. No, $x = \pm i$
19. $3+2i$
20. $\sqrt{36+64} = 10$