Complex Numbers: Mixed Practice

Before you begin#

Twelve ques­tions on com­plex num­bers and qua­drat­ics. One rule saves a lot of grief: sim­plify to the form $a + ib$ before you try to find a mod­u­lus or an argu­ment. For the argu­ment, sketch the point so you know its quad­rant.

Ques­tions#

1. Express $(1 - 2i)^{-3}$ in the form $a + ib$.
2. Find $x, y$ if $(x + iy)(2 - 3i) = 4 + i$.
3. Eval­u­ate $i^{17} + i^{22} + i^{35} + i^{100}$.
4. Find the mul­ti­plica­tive inverse of $\sqrt{5} + 3i$.
5. If $\displaystyle x + iy = \frac{a + ib}{a - ib}$, prove $x^2 + y^2 = 1$.
6. Find the mod­u­lus of $\displaystyle \frac{(3 + 4i)(5 - 12i)}{6 + 8i}$.
7. Con­vert to polar form: $-2 + 2\sqrt{3}i$; $\displaystyle \frac{1 - i}{\cos\frac{\pi}{3} + i\sin\frac{\pi}{3}}$.
8. Solve: $5x^2 - 6x + 2 = 0$; $27x^2 - 10x + 1 = 0$; $x^2 - x + 2 = 0$.
9. If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, find $\alpha^2 + \beta^2$ and show $\alpha^3 = 1$.
10. For which real $k$ does $x^2 + kx + 9 = 0$ have non-real roots?
11. If $\displaystyle \left(\tfrac{1 + i}{1 - i}\right)^3 - \left(\tfrac{1 - i}{1 + i}\right)^3 = x + iy$, find $(x, y)$.
12. Let $z_1, z_2$ be com­plex num­bers with $\lvert z_1 \rvert = \lvert z_2 \rvert = 1$. Show that $\displaystyle \left\lvert \tfrac{z_1 - z_2}{1 - \bar z_1 z_2} \right\rvert = 1$ when­ever the denom­i­na­tor is non-zero. (Hint: $\lvert w \rvert^2 = w\bar w$.)

Answers#