L3 Complex Numbers
Question Bank
If $w = 2\text{cis}\frac{\pi}{6}$, write $w^3$ in polar form.
Write $\displaystyle \left(2\text{cis}\frac{\pi}{6}\right)^4$ as a complex number in the form $a + bi$.
Give exact values for $a$ and $b$.
Calculate the exact value of $\displaystyle \left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right)^6$.
Solve the equation $z^3 = 27\text{cis}\frac{3\pi}{4}$, where $z$ is a complex number.
Write your answers in polar form, $r\text{cis}\theta$.
Find the values of $r$ and $\theta$ that satisfy the equation $(r \text{cis} \theta)^3 = -27i$, where $r > 0$ and $0 \leq \theta < 2\pi$.
If $z = 2m\,\text{cis}\dfrac{5\pi}{12}$, where $m$ is a positive real constant, write $z^4$ in the form $a + bi$, giving $a$ and $b$ exactly in terms of $m$.
If $u = 2k - 2\sqrt{3}k\,i$, where $k$ is a positive real constant, write $u^4$ in the form $a + bi$, giving $a$ and $b$ in terms of $k$.
Solve the equation $z^5 + 32 = 0$.
Give your solutions in polar form, $r\,\text{cis}\,\theta$, with $-\pi < \theta \leq \pi$.
Solve the equation $z^4 = 16m^4 \text{cis} \frac{2\pi}{3}$, where $m > 0$.
Write your solutions in polar form.
Solve the equation $z^3 + 8m^6 = 0$, where $m$ is a positive real constant.
Write your solutions in polar form in terms of $m$.
Solve the equation $z^3 = 4\sqrt{2} + 4\sqrt{2}\,i$.
Give your solutions in polar form, $r\,\text{cis}\,\theta$, with $-\pi < \theta \leq \pi$.
Find all solutions of the equation $z^3 + 27ni = 0$, where $n$ is a positive real number.
Write your solutions in polar form.
Solve the equation $z^3 = k(1 - \sqrt{3}i)$, where $k$ is a positive real constant.
Write your solutions in polar form in terms of $k$.
Solve the equation $z^3 + 8i = 0$.
Give your solutions exactly, in the form $a + bi$, and show them on an Argand diagram.
Solve the equation $z^4 = k - ki$, where $k$ is a real, positive constant.
Write your solutions in polar form in terms of $k$.
If $w = 1 + \sqrt{3}i$, find the smallest positive integer $n$ such that $w^n$ is a purely real number.
Solve the equation $\displaystyle z^3 = \frac{4k}{1 + \sqrt{3}i} + \frac{4k}{1 - \sqrt{3}i}$ for $z$ in terms of $k$, where $k$ is a positive real constant.
Solve the equation $z^3 = 32k^6\left(-\sqrt{3} + i\right)$, where $k$ is a positive real constant.
Give your solutions in polar form, in terms of $k$, with arguments in the range $-\pi < \theta \leq \pi$.
One solution of the equation $z^4 = w$, where $w$ is a complex number, is $z = 2 + i$.
(a) Find $w$ in the form $a + bi$.
(b) Find the other three solutions of $z^4 = w$, in the form $a + bi$.
(c) Hence solve the equation $z^8 + 14z^4 + 625 = 0$, giving all eight solutions in the form $a + bi$.
(a) Find the three solutions of $u^3 = 8$. Give them exactly, in the form $a + bi$.
(b) Hence, or otherwise, solve the equation $(z + 1)^3 = 8(z - 1)^3$. Give your solutions exactly, in the form $a + bi$.