Edexcel A-Level Mathematics AS Fp1 1 Complex Numbers Questions

Practise complex-number algebra, roots and Argand diagrams, including conjugates, modulus, argument and geometric interpretations.

Exam points

convert complex roots into a + ib form and use conjugate pairs in polynomial factors
interpret roots on an Argand diagram to find positions, moduli or areas
use real and imaginary parts to solve for parameters under an argument condition

Covered syllabus points

FP1.1.1 - Definition of complex numbers FP1.1.2 - Sum, product and quotient of complex numbers FP1.1.3 - Argand diagrams and complex operations FP1.1.4 - Complex solutions of quadratic equations FP1.1.5 - Finding conjugate complex roots FP1.1.6 - Finding conjugate complex roots

[Maximum number: 8]

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

\mathrm{f}(z)=4 z^{3}+p z^{2}-24 z+108

where p is a constant.
Given that -3 is a root of the equation f(z)=0

(a)

using algebra, solve f(z)=0 completely, giving the roots in simplest form,

[ 4 ]

$4z^3-8z^2-24z+108=(z+3)\left(4z^2+\ldots z+36\right)$ $=(z+3)\left(4z^2-20z+36\right)$ $4z^2-20z+36=0 \Rightarrow z=\frac{20\pm\sqrt{400-4\times4\times36}}{8}=\ldots$
Roots are $-3,\;\frac{5\pm \mathrm{i}\sqrt{11}}{2}$
(4)

(b)

determine the modulus of the complex roots of f(z)=0

[ 2 ]

Product of complex roots is $\frac{36}{4}=9$ , so modulus is $\sqrt{9}$ .
or
$\text{Modulus}=\sqrt{\left(\frac{5}{2}\right)^2+\left(\frac{\sqrt{11}}{2}\right)^2}$
Hence modulus is 3

(c)

show the roots of f(z)=0 on a single Argand diagram.

[ 2 ]

[Maximum number: 8]

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

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

(a)

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

[ 2 ]

(b)

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

[ 4 ]

(i)

determine the value of $\left|z_{1}\right|$

[ 1 ]

$\left|z_{1}\right|=\sqrt{3^{2}+5^{2}}=\sqrt{34}$
(1)

(ii)

express $\frac{z_{1}}{z_{2}}$ in the form a+b i, where a and b are fully simplified fractions.

[ 3 ]

$\frac{z_{1}}{z_{2}}=\frac{3+5 \mathrm{i}}{-2+6 \mathrm{i}} \times \frac{-2-6 \mathrm{i}}{-2-6 \mathrm{i}}=\ldots$ $=\frac{-6-18 \mathrm{i}-10 \mathrm{i}+30}{40}$ $=\frac{3}{5}-\frac{7}{10} \mathrm{i}$
(3)

(c)

Hence determine the value of $\arg\left(\frac{z_1}{z_2}\right)$

Give your answer in radians to 2 decimal places.

[ 2 ]

$\arg\left(\frac{z_1}{z_2}\right)=\tan^{-1}\left(\frac{-7/10}{3/5}\right) =\tan^{-1}\left(-\frac76\right)$ $=-0.86\ \text{radians}\quad\text{or}\quad 5.42\ \text{radians}$

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question 2

Question (a)

Question (b)

Question (i)

Question (ii)

Question (c)