Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Complex Numbers, Exercise 9: END-OF-CHAPTER REVIEW EXERCISE 11

The complex number $z$ is defined by $z=\frac{k-4i}{2k-i}$ where $k$ is an integer.

The imaginary part of $z$ is $Imz=\frac{7}{5}.$Find the value of $k.$

The complex number $z$ is defined by $z=\frac{k-4i}{2k-i}$ where $k$ is an integer.

Find the argument of $z.$

Without using a calculator, solve the equation
$3w+2i{w}^{*}=17+8i$
where $w^{*}$ denotes the complex conjugate of $w.$ Give your answer in the form $a+bi.$

In an Argand diagram, the loci $\arg (z-2i)=\frac{1}{6}\pi \text{ and }\left|z-3\right|=\left|z-3i\right|$
intersect at the point $P.$ Express the complex number represented by $P$ in the form $e^{\mathrm{i}\theta },$ giving the exact value of $\theta$ and the value of $r$ correct to $3$ significant figures.

The complex numbers $u$ and $v$ satisfy the equations $u+2v=2\mathrm{i}$ and $\mathrm{i}u+v=3$.
Solve the equations for $v$ and $v,$ giving both answers in the form $x+\mathrm{i}y,$ where $x$ and $y$ are real.

On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $\left|z+i\right|=1$ and the locus representing complex numbers $w$ satisfying $\arg (w-2)=\frac{3}{4}\pi$. Find the least value of $\left|z-w\right|$ for points on these loci.

Throughout this question the use of a calculator is not permitted.

The complex numbers $w$ and $z$ satisfy the relation $w=\frac{z+i}{iz+2}.$

Given that $z=1+\mathrm{i},$ find $w,$ giving your answer in the form $x+\mathrm{i}y,$ where $x$ and $y$ are real.

Throughout this question the use of a calculator is not permitted.

The complex numbers $w$ and $z$ satisfy the relation $w=\frac{z+i}{iz+2}.$

Given instead that $w=z$ and the real part of $z$ is negative, find $z,$ giving your answer in the form $x+\mathrm{iy},$ where $x$ and $y$ are real.