Q: Which of the following is the correct polar form of the complex number $$z = 3 + 4i$$?

$$5\left(\cos\theta + i\sin\theta\right)$$, where $$\theta = \arctan\left(\frac{4}{3}\right)$$

Question 1

Given the point A plotted on the complex plane at coordinates (3, 4), what is the modulus (absolute value) of the complex number represented by A?

Accepted Answer

$$\sqrt{3^2$ + 4^2$} = \sqrt{25} = 5$$

Question 2

If the point A on the complex plane represents the complex number $$z = 3 + 4i$$, what is the argument (angle with the positive real axis) of $$z in $$radians?

Accepted Answer

$$\arctan\left(\frac{4}{3}\right)$$

Question 3

Which of the following is the correct polar form of the complex number $$z = 3 + 4i$$?

Accepted Answer

$$5\left(\cos	heta + i\sin	hetaight)$$, where $$	heta = \arctan\left(\frac{4}{3}ight)$$

Question 4

If a complex number $$z is $$represented by the point (x, y) on the complex plane, which formula gives its modulus?

Accepted Answer

$$|z| = \sqrt{x^2$ + y^2$}$$

Question 5

A point is plotted at (-5, 12) on the complex plane. What is the modulus of the corresponding complex number?

Accepted Answer

$$\sqrt{(-5)^2$ + 12^2$} = \sqrt{25 + 144} = \sqrt{169} = 13$$

Question 6

Which quadrant does the complex number $$z = -3 + 4i$$ lie in on the complex plane?

Accepted Answer

Second quadrant

Question 7

If $$z = 3 + 4i$$, what is the value of $$z^* ($$the complex conjugate of $$z$$)?

Accepted Answer

$$3 - 4i$$

Question 8

A complex number $$z is $$plotted at (0, -7) on the complex plane. What is the argument of $$z$$?

Accepted Answer

$$\frac{3\pi}{2}$$

Question 9

If $$z = 6 - 8i$$, what is $$|z|$$?

Accepted Answer

$$\sqrt{6^2$ + (-8)^2$} = \sqrt{36 + 64} = \sqrt{100} = 10$$

Question 10

Which of the following is the correct rectangular form of a complex number with modulus $$r = 10$$ and argument $$	heta = \frac{\pi}{6}$$?

Accepted Answer

$$10\cos\left(\frac{\pi}{6}\right) + 10i\sin\left(\frac{\pi}{6}\right)$$

Complex Numbers and the Complex Plane

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