Level 3 Complex Numbers

(a)

Write $\dfrac{6k}{\sqrt{5k}+\sqrt{2k}}$ in the form $a\sqrt{5k}+b\sqrt{2k}$, where $a$ and $b$ are integers and $k$ is a positive real constant.

(b)

If $u=k-k\sqrt3\,\mathrm{i}$, where $k$ is a positive real constant, write $u^5$ in the form $a+b\mathrm{i}$, where $a$ and $b$ are real.

Give your answer in terms of $k$.

(c)

The complex numbers $u$ and $v$ are $u=k-\mathrm{i}$ and $v=3+k\mathrm{i}$, where $k$ is a real constant.

Find the exact value(s) of $k$ for which $\arg(uv)=\dfrac{3\pi}{4}$.

(d)

One solution of the equation $z^3+az^2+20z+b=0$ is $z=2\sqrt2\operatorname{cis}\left(-\dfrac{\pi}{4}\right)$.

If $a$ and $b$ are real, find the values of $a$ and $b$, and the other two solutions of the equation.

(e)

Find all the complex numbers $z$ that satisfy the equation $z^2=2k\mathrm{i}\,\bar z$, where $k$ is a positive real constant.

Give your solutions in the form $a+b\mathrm{i}$, in terms of $k$.

(a)

The complex numbers $u$ and $v$ are shown on the Argand diagram below.

If $w=\dfrac{u}{v}$, find $w$, and clearly show it on the Argand diagram above.

(b)

Solve the equation $x^2-2\sqrt5\,kx+k^2=0$ for $x$.

Simplify your answer as far as possible, giving your answer in terms of $k$.

(c)

Solve the equation $\sqrt{2x-3k}-\sqrt{x-k}=2\sqrt k$ for $x$, in terms of $k$, where $k$ is a positive real constant.

(d)

$z=x+y\mathrm{i}$ is a non-zero complex number, where $x$ and $y$ are real.

Prove that if $z+\dfrac1z$ is a real number, then either $z$ is real or $|z|=1$.

(e)

The locus of a complex number $z$ is described by $|z-4+2\mathrm{i}|=2|z-4-\mathrm{i}|$.

Find the Cartesian equation of the locus of $z$, giving your answer in the form $(x-a)^2+(y-b)^2=k$.

Then find the greatest value of $\arg z$ for $z$ on this locus.

(a)

Given that $(2x-1)$ is a factor of $4x^3+kx^2-11x+6$, find the value of $k$.

(b)

Find the values of $k$ for which the solutions of the equation $x^2+(k-3)x+k=0$ are not real.

(c)

Find the possible values of the real number $m$, given that $\left|\dfrac{m+2\mathrm{i}}{1-\mathrm{i}}\right|=|m-\mathrm{i}|$.

(d)

Solve the equation $z^4+8k^4+8\sqrt3\,k^4\mathrm{i}=0$, where $k$ is a positive real constant.

Write your solution(s) in polar form, in terms of $k$.

(e)

Solve the equation $z^2-(4-\mathrm{i})z+5-5\mathrm{i}=0$.

Give your solutions in the form $a+b\mathrm{i}$, and show your algebraic working clearly.