Operations with Complex Numbers (Edexcel IAL Further Maths): Exam Questions

$2z+z^{\ast}=\frac{3+4i}{7+i}$

Find $z$, giving your answer in the form $a+bi$, where $a$ and $b$ are real constants. You must show all your working.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The complex number $z$ is defined by

$z=-3+4i$

Determine $\left| z^{2} - 3 \right|$

Express $\frac{50}{z^{\ast}}$ in the form $kz$ , where $k$ is a positive integer.

Hence find the value of $\text{arg} ( \frac{50}{z^{asterisk}} \times )$

Give your answer in radians to 3 significant figures.

Given that

$\frac{3z-1}{2}=\frac{λ+5i}{λ-4i}$

where $λ$ is a real constant,

determine $z$, giving your answer in the form $x+yi$, where $x$ and equation$y$ are real and in terms of $λ$.

Given also that $\mathrm{arg}z=\frac{π}{4}$

find the possible values of equation$λ$.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

$z_{1}=3+2iz_{2}=2+3iz_{3}=a+bia,b\inℝ$

Determine the exact value of $\vert z_{1} + z_{2} \vert$

Given that $w=\frac{z_{2}z_{3}}{z_{1}}$

Determine $w$in terms of $a$ and $b$, giving your answer in the form $x+iy$ , where $x,y\inℝ$

Given also that $w=\frac{4}{13}+\frac{58}{13}i$

Determine the value of $a$ and the value of $b$

Determine $\mathrm{arg}w$, giving your answer in radians to 4 significant figures.

$z_{1}=3+3iz_{2}=p+qip,q\inℝ$

Given that $\vert z_{1} z_{2} \vert = 15 \sqrt{2}$

determine $\vert z_{2} \vert$

Given also that $p=-4$

determine the possible values of $q$

Show equation$z_{1}$ and the possible positions for $z_{2}$ on the same Argand diagram.

The complex numbers $z_{1}$ and $z_{2}$ are given by

$z_{1}=3+5i\text{and}z_{2}=-2+6i$

Show $z_{1}$ and $z_{2}$ on a single Argand diagram.

Without using your calculator and showing all stages of your working,

(i) determine the value of $\vert z_{1} \vert$

(ii) Express $\frac{z_{1}}{z_{2}}$ in the form $a+bi$

Hence determine the value of $\mathrm{arg}\frac{z_{1}}{z_{2}}$

Give your answer in radians to 2 decimal places.

The complex numbers $z_{1}$, $z_{2}$ and $z_{3}$ are given by

$z_{1}=2-iz_{2}=p-iz_{3}=p+i$

where $p$ is a real number.

Find $\frac{z_{2}z_{3}}{z_{1}}$ in the form $a+bi$ where $a$ and $b$ are real. Give your answer in its simplest form in terms of $p$.

Given that $\left| \frac{z_{2} z_{3} }{z_{1} } \right| = 2 \sqrt{5}$

find the possible values of p.

The complex number $z$ is defined by

$z=-λ+3i$ where $λ$is a positive real constant.

Given that the modulus of $z$ is 5,

write down the value of $λ$

determine the argument of $z$, giving your answer in radians to one decimal place.

In part (c) you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.

Express in the form $a+ib$ where $a$ and $b$ are real,

(i) $\frac{z+3i}{2-4i}$

(ii) $z^{2}$

Show on a single Argand diagram the points $A$, $B$, $C$ and $D$ that represent the complex numbers

$z,z^{\ast},\frac{z+3i}{2-4i}\text{and}z^{2}$