Complex Numbers in Polar Form: Precalculus Math Test – PDF Download

Question 1: Modulus and conjugate (2 points)

The complex number \(z\) is shown.

1. Find \(|z|\). (1 point)
2. Write the conjugate \(\overline{z}\). (1 point)

Question 2: Complex quotient (2 points)

Write \(\frac{10+5i}{2-i}\) in the form \(a+bi\).

Question 3: Polar form of a complex number (3 points)

Let \(z=2\sqrt{3}-2i\) (see figure).

1. Find the modulus of \(z\). (1 point)
2. Find the argument of \(z\) between \(0^{\circ}\) and \(360^{\circ}\). (1 point)
3. Write \(z\) in polar form. (1 point)

Question 4: Polar to rectangular (2 points)

Write each number in the form \(a+bi\).

1. \(8(\cos 150^{\circ}+i\sin 150^{\circ})\) (1 point)
2. \(6(\cos \frac{3\pi}{2}+i\sin \frac{3\pi}{2})\) (1 point)

Question 5: Products and quotients in polar form (2 points)

Let \(z_{1}=4(\cos 70^{\circ}+i\sin 70^{\circ})\) and \(z_{2}=2(\cos 20^{\circ}+i\sin 20^{\circ})\). Find \(z_{1}z_{2}\) and \(\frac{z_{1}}{z_{2}}\) in polar form. Then write \(z_{1}z_{2}\) in the form \(a+bi\).

Question 6: De Moivre's theorem (3 points)

Complete each part.

1. Evaluate \((1+i)^{10}\). (2 points)
2. Find the smallest positive integer \(n\) for which \((\cos 24^{\circ}+i\sin 24^{\circ})^{n}=1\). (1 point)

Question 7: Distance and midpoint (2 points)

The complex numbers \(z_{1}=-1-2i\) and \(z_{2}=7+4i\) are shown. Find the distance between them and their midpoint.

Question 8: Rotation and scaling (2 points)

The complex number \(z=2+2i\) is shown. It is multiplied by \(w=2(\cos 45^{\circ}+i\sin 45^{\circ})\). Write \(zw\) in polar form and in the form \(a+bi\).

Question 9: Circuit current (2 points)

An impedance \(Z=6+8i\) ohms has a voltage \(V=50\) volts. Find \(|Z|\) and the current \(I=\frac{V}{Z}\) in the form \(a+bi\).

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