Operations with Complex Numbers (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

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

$z=-3+4\mathrm{i}$

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

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

Hence find the value of $\arg \left(\frac{50}{z^{*}}\right)$

Give your answer in radians to 3 significant figures.

Given that

$\frac{3z-1}{2}=\frac{\lambda +5\mathrm{i}}{\lambda -4\mathrm{i}}$

where $\lambda$ is a real constant,

determine $z$, giving your answer in the form $x+y\mathrm{i}$, where $x$ and $y$ are real and in terms of $\lambda$.

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

find the possible values of $\lambda$.

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

Solutions relying entirely on calculator technology are not acceptable.

$z_1=3+2\mathrm{i}{z}_2=2+3\mathrm{i}{z}_3=a+b\mathrm{i}a,b\in \mathrm{ℝ}$

Determine the exact value of $|z_1+z_2|$

Given that $w=\frac{z_2{z}_3}{z_1}$

Determine $w$in terms of $a$ and $b$, giving your answer in the form $x+\mathrm{i}y$ , where $x,y\in \mathrm{ℝ}$

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

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

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

$z_1=3+3\mathrm{i}{z}_2=p+q\mathrm{i}p, q\in \mathrm{ℝ}$

Given that $\left|z_1{z}_2\right|=15\sqrt{2}$

determine $\left|z_2\right|$

Given also that $p=-4$

determine the possible values of $q$

Show $z_1$ 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 $\left|z_1\right|$

[1]

(ii) Express $\frac{z_1}{z_2}$ in the form $a+bi$

[3]

Hence determine the value of $\arg \frac{z_1}{z_2}$

Give your answer in radians to 2 decimal places.

The complex numbers $z_1$, $z_2$ and $z_3$ are given by

$z_1=2-i{z}_2=p-i{z}_3=p+i$

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

$z=-\lambda +3\mathrm{i}$ where $\lambda$is a positive real constant.

Given that the modulus of $z$ is 5,

write down the value of $\lambda$

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

(ii) $z^2$

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

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