L3 Complex Numbers
Question Bank
$u = a + 3i$ and $v = 2 - i$, where $a$ is a real number.
Given that $uv = 11 + 2i$, find the value of $a$. You must show algebraic working.
Find the real numbers $p$ and $q$ such that $(p + 2i)(3 - i) = 17 + qi$.
You must show algebraic working.
Solve the equation $z + 3\bar{z} = 12 + 4i$.
Give your answer in the form $a + bi$. You must show algebraic working.
$x$ and $y$ are real numbers such that
$ (x + iy)(3 - i) = 10i $
Find the values of $x$ and $y$. You must show algebraic working.
Solve the equation $2z + 3\bar{z} = 10 + i$ for the complex number $z$.
Give your answer in the form $x + iy$.
Solve the equation $\displaystyle \frac{z + i}{z - 2} = i$ for the complex number $z$.
Solve the following simultaneous equations for the complex numbers $u$ and $v$:
$ u + 2v = 5 + 3i $
$ 2u - v = 5 - 4i $
Solve the following system of equations for the complex numbers $z$ and $w$:
$ z + iw = 0 $
$ iz + w = 4 + 4i $
Find the values of real numbers $x$ and $y$ such that $(2 + i)x + (3 - i)y = 7 - 4i$.
Solve the equation $3z - i\bar{z} = 7 - 5i$.
Give your answer in the form $x + iy$.
$u$ is the complex number $3 + i$.
$v$ is the complex number $2 - 3i$.
Find the real numbers $k$ and $m$ such that $ku + mv = 13 - 3i$.
Given that
$ \frac{z + k}{z - i} = 2 + i $
where $k$ is a real constant and $z \ne i$, find $z$ in terms of $k$.
Give your answer in the form $a + bi$.
$v$ is the complex number $2 + 3i$.
$w$ is the complex number $4 - i$.
Find the real numbers $p$ and $q$ such that $pv + qw = -2 + 11i$.
Find $\text{arg}(z)$ if $\displaystyle \frac{5z}{z + 2} = 3 + i$.
Find the square roots of $5 - 12i$.
Hint: Let $\sqrt{5 - 12i} = x + iy$.
Solve $z^2 = 8 - 6i$ where $z = x + iy$.
Find the value(s) of the real constant $m$, given that $z = 2 + mi$ and $\bar{z} = 5mz^{-1}$.
Find the values of $x$ and $y$ in terms of $k$, where $k$ is a non-zero real constant, given that $x$ and $y$ are real and:
$ \frac{1}{x + iy} = 1 - \frac{1}{1 - ki} $
Find the values of real numbers $x$ and $y$ given that:
$ \frac{13}{x - iy} - \frac{5}{2 + i} = 1 - i $
Find the complex numbers $z$ and $w$ that satisfy the following system of equations:
$ iz + w = 1 $
$ z + (1 + i)w = 4 + 2i $
Solve the equation $z^2 + |z|^2 = 8 + 6i$.
Give your answers in the form $x + iy$.
Find the complex numbers $u$ and $v$ that satisfy:
$ (1 + i)u + v = 5 + i $
$ 2u - iv = 2 $
If $z + 3i = iz + m$, where $m$ is a real number, and $\displaystyle \frac{w}{z} = 1 + i$, where $\text{Im}(w) = 10$, find the value of $m$.
If $z = a + bi$ is a non-zero complex number, and $\displaystyle \frac{2}{\bar{z}} + \frac{i}{z} = 1 + i$, find the values of $a$ and $b$.
Find the exact value of $\text{arg}(z)$ if $z = x + iy$ is a non-zero complex number such that:
$ \frac{3}{z} + \frac{1}{\bar{z}} = 2 - \sqrt{3}i $
Find all the complex numbers $z$ that satisfy $z^2 = -2k\bar{z}$, where $k$ is a positive real constant.
Give your solutions in the form $a + bi$, in terms of $k$.