L3 Complex Numbers
Question Bank
Write $\frac{14}{3+\sqrt{2}}$ in the form $a + b\sqrt{2}$, where $a$ and $b$ are integers.
Write $\frac{4}{\sqrt{5}-1}$ in the form $a + b\sqrt{5}$, where $a$ and $b$ are integers.
Write $(3 - \sqrt{7})^2$ in the form $a + b\sqrt{7}$, where $a$ and $b$ are integers.
Write $\dfrac{15k}{\sqrt{6k} - \sqrt{k}}$ in the form $a\sqrt{6k} + b\sqrt{k}$, where $k$ is a positive real constant and $a$ and $b$ are integers.
Write $(5 - \sqrt{3})(2 + 2\sqrt{3})$ in the form $a + b\sqrt{3}$, where $a$ and $b$ are integers.
Write $(2 + 3\sqrt{k})(4 - \sqrt{k})$ in the form $a + bk + c\sqrt{k}$, where $a, b$ and $c$ are integers.
Expand and simplify $\left(\sqrt{2m} - \sqrt{6m}\,i\right)^2$, where $m$ is a positive real constant.
Give your answer in the form $a + bi$, in terms of $m$.
Rewrite $\frac{3-\sqrt{2}}{1+\sqrt{2}}$ in the form $a + b\sqrt{2}$, where $a$ and $b$ are integers.
Write $\displaystyle \frac{10 + \sqrt{50}}{\sqrt{2}}$ in the form $a + b\sqrt{2}$, where $a$ and $b$ are integers.
By squaring both sides, find the integers $p$ and $q$ for which
$ \sqrt{14 + 6\sqrt{5}} = p + \sqrt{q} $
Write $\displaystyle \frac{5 + \sqrt{2}}{3 - 2\sqrt{2}}$ in the form $a + b\sqrt{2}$, where $a$ and $b$ are rational numbers.
Solve the equation
$ \sqrt{kx + 5k^2} = x - k $
for $x$ in terms of $k$, where $k$ is a positive real constant.
Solve the following equation for $x$ in terms of $m$.
$ 3\sqrt{x+2} - m\sqrt{x} = 0 $
Solve the equation $\sqrt{5x - 1} = x - 3$.
Solve the equation $2\sqrt{x+1} = x - 2$.
Solve the equation $\displaystyle \frac{4 + x}{x} = \sqrt{3}$, writing your solution in the form $x = a + b\sqrt{3}$.
Solve the equation $\displaystyle \frac{x}{x - 2} = \sqrt{3} + 1$.
Write your answer in the form $a + b\sqrt{3}$, where $a$ and $b$ are rational numbers.
Write $\displaystyle \frac{\sqrt{3k}}{\sqrt{3k} - \sqrt{k}}$ in the form $a + b\sqrt{3}$, where $a$ and $b$ are rational numbers.
Solve the following equation for $x$.
$ (2 + \sqrt{x})(3 + 2\sqrt{x}) = 5 + 10\sqrt{x} $
Solve the equation $k\sqrt{x+5} = 2\sqrt{x}$ for $x$ in terms of $k$.
Solve the equation
$ \sqrt{x + 1} - \sqrt{2x - 5} = 1 $
You must show algebraic working.
Solve the equation $\sqrt{x+k} - \sqrt{x} = 3$ for $x$ in terms of $k$.
Solve the equation
$ \sqrt{2x - 3} + 2\sqrt{3x - 2} = 5 $
You must show algebraic working.
Solve the equation
$ \sqrt{\frac{x}{x - 1}} + \sqrt{\frac{x - 1}{x}} = \frac{10}{3} $
Give your solutions exactly. You must show algebraic working.
Consider the equation
$ x - 2\sqrt{x + 1} = k $
where $k$ is a real constant.
Find all values of $k$ for which the equation has two distinct real solutions.
Solve the equation $\displaystyle{\frac{\sqrt{x+m} + \sqrt{x-m}}{\sqrt{x+m} - \sqrt{x-m}} = 5}$ for $x$ in terms of $m$.