$$|z| = 2,\ \text{Arg}(z) = \frac{5\pi}{6}$$

$$\frac{r}{s} e$$^{i(\omega - \theta)}$$

Question 1

Which of the following statements best describes why the set of real numbers $$\mathbb{R}$$ does not solve the equation $$x^2 = -2$?

Accepted Answer

Because the square of any real number is always positive or zero.

Question 2

Given $$z = -\sqrt{3} + i$$, what is the modulus $$|z|$$ and the principal argument $$	ext{Arg}(z)$$?

Accepted Answer

$$|z| = 2,\ 	ext{Arg}(z) = \frac{5\pi}{6}$$

Question 3

If $$z = 3 - 5i$$, what are the real and imaginary parts of $$z$$?

Accepted Answer

Re$$(z) = 3$$, Im$$(z) = -5$$

Question 4

Which of the following is the correct rectangular form for $$z if$$ $$|z| = 5$$ and $$\arg(z) = 	an$$^{-1}$$(3/4)$$?

Accepted Answer

$$z = 4 + 3i$$

Question 5

Simplify $$\frac{i}{1 + i\sqrt{3}}$$ into standard form $$a + bi.$$

Accepted Answer

$$\frac{\sqrt{3}}{4} + \frac{1}{4}i$$

Question 6

Which geometric locus is described by $$|z - (3 - 4i)| \leq 5 in $$the complex plane?

Accepted Answer

A closed disk centered at $$(3, -4)$$ with radius $$5$$

Question 7

Given $$z = 1 + i$$, use De Moivre’s Theorem to compute $$z^5.$$

Accepted Answer

$$-4 - 4i$$

Question 8

Which of the following is the correct expression for the $$n-th $$roots of unity in the complex plane?

Accepted Answer

$$w_k = \cos\left(\frac{2k\pi}{n}\right) + i\sin\left(\frac{2k\pi}{n}\right)$$ for $$k = 0, 1, ..., n-1$$

Question 9

If $$z = a + bi$$, what is the geometric interpretation of its complex conjugate $$\overline{z} in $$the Argand diagram?

Accepted Answer

It is the reflection of $$z$$ across the real axis.

Question 10

Which of the following is true about the modulus and argument of the reciprocal $$w^{-1} of a $$nonzero complex number $$w$$?

Accepted Answer

The modulus is the reciprocal of the modulus of $$w$$, and the argument is the negative of the argument of $$w.$$

Question 11

Given $$z = 2 + 3i$$ and $$w = 1 - i$$, compute $$z \cdot w.$$

Accepted Answer

$$5 + i$$

Question 12

If $$z = r e$$^{i\omega}$$ and w = s$$ $$e^{i	heta}$$, what is $$\frac{z}{w} in $$polar form?

Accepted Answer

$$\frac{r}{s} e$$^{i(\omega - 	heta)}$$

Question 13

Which of the following is the rectangular form of $$z if$$ $$|z| = \frac{1}{2}$$ and $$\arg(z) = -\frac{\pi}{3}$$?

Accepted Answer

$$\frac{1}{4} - i\frac{\sqrt{3}}{4}$$

Question 14

If $$z = a + bi$$, which of the following is always true about $$|z|$$?

Accepted Answer

$$|z| = \sqrt{a^2$ + b^2$}$$

Question 15

Which of the following best describes the geometric arrangement of the cube roots of unity in the complex plane?

Accepted Answer

They form the vertices of an equilateral triangle inscribed in the unit circle.

Question 16

Given $$z = 2 + 3i$$ and $$w = 2 - i$$, simplify $$\frac{z}{w}$$ into standard form $$a + bi.$$

Accepted Answer

$$\frac{1}{5} + i$$

Question 17

If $$z = x + yi$$ and $$\overline{z} is $$its conjugate, which of the following is always true?

Accepted Answer

$$z \cdot \overline{z} = x^2$ + y^2$

Complex Numbers: Foundations, Properties, and Calculus Applications

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