L3 Complex Numbers
Question Bank
$u = 4m^3\,\text{cis}\,\dfrac{\pi}{4}$ and $v = 2m\,\text{cis}\,\dfrac{\pi}{6}$, where $m$ is a positive real constant.
(a) Find $uv$ in polar form.
(b) Find $\dfrac{u}{v}$ in polar form.
If $u = 12 \text{cis} \frac{3\pi}{4}$ and $v = 4 \text{cis} \frac{\pi}{3}$, write $\displaystyle \frac{u}{v}$ in polar form.
$u$ and $v$ are complex numbers where $u = 12m \text{cis} \frac{3\pi}{4}$ and $v = 3m \text{cis} \frac{\pi}{4}$.
Find $u \div v$, expressing your answer in polar form $r\text{cis}\theta$.
Write $z = 4\,\text{cis}\left(-\dfrac{2\pi}{3}\right)$ in the form $a + bi$, giving the exact values of $a$ and $b$.
If $u = 3 - 2i$ and $w = 1 + 5i$, calculate the value of $2\bar{u} + w$.
Give your answer in the form $a + bi$.
Write $z = -\sqrt{3} + 3i$ in polar form, $r\,\text{cis}\,\theta$.
Give exact values of $r$ and $\theta$, where $-\pi < \theta \le \pi$.
If $u = k \text{cis} \frac{\pi}{5}$ and $v = k^2 \text{cis} \frac{\pi}{4}$, write the product $uv$ in polar form.
Let $u = 4 + i$ and $v = m - 2i$.
Find the value of the real number $m$ if the real part of the product $uv$ is 18.
Write $(k - 4i)^2$ in the form $a + bi$, giving $a$ and $b$ in terms of $k$.
Find the modulus and argument of the complex number $z = -3 - 3i$.
The complex number $z = x + 4i$ has a modulus of $|z| = \sqrt{41}$.
Find the possible values of the real number $x$.
Express $\displaystyle \frac{2k}{1 - i}$ in the form $a + bi$, giving $a$ and $b$ in terms of $k$.
Convert the complex number $z = 6\text{cis}\frac{2\pi}{3}$ into rectangular form $a + bi$.
Give exact values for $a$ and $b$.
If $w = 5 + ki$ and the argument of $w$ is $-\frac{\pi}{4}$, find the value of $k$.
$u = -6ki$ and $v = 2k^2\,\text{cis}\,\dfrac{\pi}{6}$, where $k$ is a positive real constant.
Find $\dfrac{u}{v}$ in polar form.
Write the complex number $u = -2\sqrt{3} - 2i$ in polar form $r\text{cis}\theta$.
Given that $z = 1 - 2i$, find the value of $z + \frac{10}{\bar{z}}$.
Give your answer in the form $a + bi$.
If $z = \sqrt{3} + i$ and $w = \frac{4}{z} + i$, find the exact value of $\text{arg}(w)$.
Given that $z = 2 - i$, find the value of $\bar{z}^2 - \frac{10}{z}$, giving your answer in the form $a + bi$.
Find the value of $p$ such that the complex number $\displaystyle \frac{2 + pi}{1 - i}$ is purely imaginary.
If $z = 2 + 3i$ and $w = 1 + mi$, find the value of $m$ if the real part of $\displaystyle \frac{z}{w}$ is zero.
$u = k + 2i$ and $v = k + 3i$, where $k$ is a real constant.
Given that $\arg(uv) = \dfrac{\pi}{4}$, find the possible value(s) of $k$.
The complex number $z = k + (k-1)i$ has a modulus of $\sqrt{13}$.
Find the possible values of $k$.
Write the complex number $\displaystyle \left(\frac{2i^6 + 4i}{1 + i}\right)^2$ in the form $a + bi$, where $a$ and $b$ are real numbers.
$z$ and $w$ are complex numbers such that $z = 1 - i$ and $zw = -4$.
Find the exact value of $\text{arg}(w)$.
Complex numbers $u$ and $v$ are $u = 3 + qi$ and $v = 1 - \sqrt{3}i$.
Given that $\displaystyle \left|\frac{u}{v}\right| = 2.5$, find all possible values of $q$.
Two complex numbers are defined by $u = 2 + i$ and $v = 1 + mi$.
Given that $\text{arg}\left(\frac{u}{v}\right) = \frac{\pi}{4}$, find the exact value of $m$.
The complex number $u = k + 2i$.
Given that $|u - 3| = 2\sqrt{5}$, find all possible values of $k$.
Find the value of $k$ such that $\displaystyle \frac{4 + ki}{1 - 2i}$ is a purely real number.
The complex numbers $u$ and $v$ are $u = 3 + i$ and $v = 1 - i$.
Determine the possible value(s) of the real constant $k$ if $\displaystyle \left| \frac{u}{v} + k \right| = \sqrt{13}$.
If $u = 3 + ki$ and $v = 1 - i$, find the value of $k$ if $\text{arg}\left(\frac{u}{v}\right) = \frac{\pi}{2}$.
Find all possible values of $m$ such that the complex number $z = \frac{m + 5i}{1 + mi}$ is purely real.
Two complex numbers are $u = 4 + ki$ and $v = 2 + ki$, where $k$ is a real constant.
Given that $\text{arg}(uv) = \frac{\pi}{4}$, find the exact value of $k$.