CAIE A-Level Mathematics A2 3.9 Complex numbers

[Maximum number: 10]

The complex numbers z and $\omega$ are defined by z=1-i and $\omega=-3+3 \sqrt{3} \mathrm{i}$ .

(a)

Express $z \omega$ in the form a+b i, where a and b are real and in exact surd form.

[ 1 ]

State $z \omega=(-3+3 \sqrt{3})+(3+3 \sqrt{3}) \mathrm{i}$
Or exact equivalent with real and imaginary parts collected. Need brackets around the coefficient of i.

Marking guidance:

Allow for a=, b= stated correctly.

(b)

Express z and $\omega$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$ , where r>0 and $-\pi<\theta \leqslant \pi$ . Give the exact values of r and $\theta$ in each case.

[ 4 ]

Obtain $|z|=\sqrt{2}$
Obtain $\arg z=-\frac{\pi}{4}$ final answer
Obtain $|\omega|=6$
Obtain $\arg \omega=\frac{2 \pi}{3}$ final answer

(c)

On an Argand diagram, the points representing $\omega$ and $z \omega$ are A and B respectively.

Prove that O A B is an isosceles right-angled triangle, where O is the origin.

[ 2 ]

Figure for Answer to Question (c) — CAIE A-Level Mathematics A2

Note: The question does not require the diagram.
If they use $\frac{5 \pi}{12}$ they need to demonstrate where it comes from.
Complex number equivalent to A B is $3 \sqrt{3}+3 \mathrm{i}$ .

Show |O A|=|A B|=6, hence isosceles
One mark for 'isosceles' and one mark for 'right angle'.
There will be alternatives e.g. use of Pythagoras (ratio of lengths is

$\angle A O B=\arg \omega-\arg z \omega=-\arg z=\frac{\pi}{4}$ hence third angle is a right
angle
$1: 1: \sqrt{2}$ ), expressing each number in "vector" form and using
scalar product or explaining the effect of multiplying by 1-i.

(d)

Using your answers to part (b), prove that $\tan \frac{5}{12} \pi=\frac{\sqrt{3}+1}{\sqrt{3}-1}$ .

[ 3 ]

$\arg z \omega=\arg z+\arg \omega\left(=\frac{2 \pi}{3}-\frac{\pi}{4}=\frac{5 \pi}{12}\right)$
For showing correct use of their angles from part (b).
Must demonstrate where $\frac{5 \pi}{12}$ comes from.

$\arg z \omega=\tan ^{-1} \frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$
Correct method for their z $\omega$ from part (a). Must link to point B on diagram or to $\arg z \omega$ .

Need to see $\tan ^{-1} \frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$ or $\tan \theta=\frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$ and not just $\tan ^{-1} \frac{1+\sqrt{3}}{-1+\sqrt{3}}$ .

$\Rightarrow \tan \left(\frac{5}{12} \pi\right)=\frac{\sqrt{3}+1}{\sqrt{3}-1}$
Obtain given answer from full and correct working.

[Maximum number: 11]

The complex number $-1+\sqrt{7} \mathrm{i}$ is denoted by u. It is given that u is a root of the equation

2 x^{3}+3 x^{2}+14 x+k=0,

where k is a real constant.

(a)

Find the value of k.

[ 3 ]

Substitute $x=-1+\sqrt{7} \mathrm{i}$ in the equation and attempt expansions of $x^{2}$ and
$x^{3}$

*M1

Use $\mathrm{i}^{2}=-1$ correctly at least once and solve for k
$2(20-4 \sqrt{7} \mathrm{i})+3(-6-2 \sqrt{7} \mathrm{i})+14(-1+\sqrt{7} \mathrm{i})+k=0$

Obtain answer k=-8
SC B1 only for those who show no working for the cube
and square and obtain answer k=-8.

Alternative method for question 10(a)

Attempt division by $(x+1-\sqrt{7} \mathrm{i})$ as far as $2 x^{2}+z_{1} x+\ldots$

*M1

See division on next page.

Use $\mathrm{i}^{2}=-1$ correctly at least once and obtain $2 x^{2}+z_{1} x+z_{2}+$ remainder
Obtain answer k=-8

(b)

Find the other two roots of the equation.

[ 4 ]

State answer $-1-\sqrt{7} \mathrm{i}$
Can be seen simply stated on its own, or in a list of roots.

Marking guidance:

Allow if stated clearly in part 10(a).

Carry out a method for finding a quadratic factor with zeros $-1+\sqrt{7} \mathrm{i}$ and
$-1-\sqrt{7} \mathrm{i}$
Or state $(x-(-1+\sqrt{7} \mathrm{i}))(x-(-1-\sqrt{7} \mathrm{i}))(2 x-p)$

Obtain $x^{2}+2 x+8$
Or obtain $(-1+\sqrt{7} \mathrm{i})(-1-\sqrt{7} \mathrm{i}) p=-8$
Or obtain $(-1+\sqrt{7} \mathrm{i})+(-1-\sqrt{7} \mathrm{i})+\frac{p}{2}=-\frac{3}{2}$

Obtain root $x=\frac{1}{2}$ , or equivalent, via division or inspection
Needs to follow from the working.

(c)

On an Argand diagram, sketch the locus of points representing complex numbers z satisfying the equation |z-u|=2.

[ 2 ]

Official markscheme Argand diagram for Q10(c).

|z-u|=2 is the circle of radius 2 centred at u. Since
$u=-1+\sqrt7\,i,$
the locus is a circle centred at $(-1,\sqrt7)$ on the Argand diagram with radius 2.

B1 for centre $-1+\sqrt7 i$ . B1 for a circle of radius 2, with scale or radius indicated;
if more than one circle is shown, maximum B1.

(d)

Determine the greatest value of $\arg z$ for points on this locus, giving your answer in radians.

[ 2 ]

Carry out a complete method for calculating the maximum value of $\arg z$ for
correct circle
e.g. $\frac{\pi}{2}+\tan ^{-1} \frac{1}{\sqrt{7}}+\frac{\pi}{4}$ Can be implied by $155.7^{\circ}$ .

Obtain answer 2.72 radians
CAO. The question requires radians.

[Maximum number: 8]

(a)

Solve the equation $z^{2}-6 \mathrm{i} z-12=0$ , giving the answers in the form x+i y, where x and y are real and exact.

[ 3 ]

Use quadratic formula, or completing the square $\left((z-3 i)^{2}-3=0\right)$
and use $\mathrm{i}^{2}=-1$ to find a root
Obtain a root, e.g. $\sqrt{3}+3 \mathrm{i}$
Or exact 2 term equivalent e.g. $\frac{6 \mathrm{i}}{2}+\frac{\sqrt{12}}{2}$ ISW.

Obtain the other root, e.g. $-\sqrt{3}+3 \mathrm{i}$
Or exact 2 term equivalent ISW.

(b)

On a sketch of an Argand diagram with origin O, show points A and B representing the roots of the equation in part (a).

[ 1 ]

Show points representing the roots correctly

B1 FT

2 roots consistent with their (a) and with no errors seen on the diagram.

B0 if they only have one root or more than 2 roots Must match their scale and $1<\sqrt{3}<2$

Linear scales seen or implied.

Need some indication of scale (numbers or dashes). Scales along an axis must be approximately consistent but scales may be different on the 2 axes.

(c)

Find the exact modulus and argument of each root.

[ 3 ]

State modulus of either root is $2 \sqrt{3}$ , or simplified exact equivalent

B1 FT

ISW if converted to decimal .

Marking guidance:

Ignore modulus of second root if seen.
Follow their root(s) not on either axis (from (a) or (b)).

Find the argument of one of their roots - get as far as $\tan ^{-1}(\ldots)$
SOI but must be correct for their root.

Obtain correct arguments $\frac{1}{3} \pi$ and $\frac{2}{3} \pi$ , or simplified exact equivalents
Must obtain values. Allow degrees.

(d)

Hence show that the triangle O A B is equilateral.

[ 1 ]

Give a complete justification that the correct triangle is equilateral
Check their diagram in (b).
Possible justifications: 3 equal sides, or all angles equal
to $\frac{\pi}{3}$ , or isosceles and an angle of $\frac{\pi}{3}$ .

[Maximum number: 4]

The complex numbers z and $\omega$ are defined by z=1-i and $\omega=-3+3 \sqrt{3} \mathrm{i}$ .

Express z and $\omega$ in the form $r \mathrm{e}^{\mathrm{i} \theta}$ , where r>0 and $-\pi<\theta \leqslant \pi$ . Give the exact values of r and $\theta$ in each case.

Obtain $|z|=\sqrt{2}$
Obtain $\arg z=-\frac{\pi}{4}$ final answer
Obtain $|\omega|=6$
Obtain $\arg \omega=\frac{2 \pi}{3}$ final answer

[Maximum number: 1]

The complex numbers z and $\omega$ are defined by z=1-i and $\omega=-3+3 \sqrt{3} \mathrm{i}$ .

Express $z \omega$ in the form a+b i, where a and b are real and in exact surd form.

State $z \omega=(-3+3 \sqrt{3})+(3+3 \sqrt{3}) \mathrm{i}$
Or exact equivalent with real and imaginary parts collected. Need brackets around the coefficient of i.

Marking guidance:

Allow for a=, b= stated correctly.

[Maximum number: 4]

The complex number $-1+\sqrt{7} \mathrm{i}$ is denoted by u. It is given that u is a root of the equation

2 x^{3}+3 x^{2}+14 x+k=0,

where k is a real constant.

Find the other two roots of the equation.

State answer $-1-\sqrt{7} \mathrm{i}$
Can be seen simply stated on its own, or in a list of roots.

Marking guidance:

Allow if stated clearly in part 10(a).

Carry out a method for finding a quadratic factor with zeros $-1+\sqrt{7} \mathrm{i}$ and
$-1-\sqrt{7} \mathrm{i}$
Or state $(x-(-1+\sqrt{7} \mathrm{i}))(x-(-1-\sqrt{7} \mathrm{i}))(2 x-p)$

Obtain $x^{2}+2 x+8$
Or obtain $(-1+\sqrt{7} \mathrm{i})(-1-\sqrt{7} \mathrm{i}) p=-8$
Or obtain $(-1+\sqrt{7} \mathrm{i})+(-1-\sqrt{7} \mathrm{i})+\frac{p}{2}=-\frac{3}{2}$

Obtain root $x=\frac{1}{2}$ , or equivalent, via division or inspection
Needs to follow from the working.

[Maximum number: 3]

The complex numbers z and $\omega$ are defined by z=1-i and $\omega=-3+3 \sqrt{3} \mathrm{i}$ .

Using your answers to part (b), prove that $\tan \frac{5}{12} \pi=\frac{\sqrt{3}+1}{\sqrt{3}-1}$ .

$\arg z \omega=\arg z+\arg \omega\left(=\frac{2 \pi}{3}-\frac{\pi}{4}=\frac{5 \pi}{12}\right)$
For showing correct use of their angles from part (b).
Must demonstrate where $\frac{5 \pi}{12}$ comes from.

$\arg z \omega=\tan ^{-1} \frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$
Correct method for their z $\omega$ from part (a). Must link to point B on diagram or to $\arg z \omega$ .

Need to see $\tan ^{-1} \frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$ or $\tan \theta=\frac{3+3 \sqrt{3}}{-3+3 \sqrt{3}}$ and not just $\tan ^{-1} \frac{1+\sqrt{3}}{-1+\sqrt{3}}$ .

$\Rightarrow \tan \left(\frac{5}{12} \pi\right)=\frac{\sqrt{3}+1}{\sqrt{3}-1}$
Obtain given answer from full and correct working.

[Maximum number: 7]

The polynomial $x^{3}+5 x^{2}+31 x+75$ is denoted by p(x).

Hence find the complex numbers z which are roots of $\mathrm{p}\left(z^{2}\right)=0$ .

State $z_{1}=\sqrt{3} \mathrm{i}$ and $z_{2}=-\sqrt{3} \mathrm{i}$
Expand $(x+\mathrm{i} y)^{2}=-1+2 \sqrt{6} \mathrm{i}$ and compare real and imaginary parts
Marking guidance:

Allow for use of $z^{2}=-1-2 \sqrt{6}$ .

Obtain $x^{2}-y^{2}=-1$ and $x y=\sqrt{6}$
Solve to obtain x and y
Obtain $z_{3}=\sqrt{2}+\sqrt{3} \mathrm{i}$ and $z_{4}=-\sqrt{2}-\sqrt{3} \mathrm{i}$
Use $z^{2}=-1-2 \sqrt{6}$ i to obtain $z_{5}$ and $z_{6}$
Allow for use of $z^{2}=-1+2 \sqrt{6}$ .

Obtain $z_{5}=\sqrt{2}-\sqrt{3} \mathrm{i}$ and $z_{6}=-\sqrt{2}+\sqrt{3} \mathrm{i}$

[Maximum number: 2]

The complex numbers z and $\omega$ are defined by z=1-i and $\omega=-3+3 \sqrt{3} \mathrm{i}$ .

On an Argand diagram, the points representing $\omega$ and $z \omega$ are A and B respectively.

Prove that O A B is an isosceles right-angled triangle, where O is the origin.

Figure for Answer to Question 9 — CAIE A-Level Mathematics A2

Note: The question does not require the diagram.
If they use $\frac{5 \pi}{12}$ they need to demonstrate where it comes from.
Complex number equivalent to A B is $3 \sqrt{3}+3 \mathrm{i}$ .

Show |O A|=|A B|=6, hence isosceles
One mark for 'isosceles' and one mark for 'right angle'.
There will be alternatives e.g. use of Pythagoras (ratio of lengths is

$\angle A O B=\arg \omega-\arg z \omega=-\arg z=\frac{\pi}{4}$ hence third angle is a right
angle
$1: 1: \sqrt{2}$ ), expressing each number in "vector" form and using
scalar product or explaining the effect of multiplying by 1-i.

[Maximum number: 6]

(a)

On a single Argand diagram sketch the loci given by the equations |z-3+2 i|=2 and $|w-3+2 \mathrm{i}|=|w+3-4 \mathrm{i}|$ where z and w are complex numbers.

[ 4 ]

(b)

Hence find the least value of |z-w| for points on these loci. Give your answer in an exact form.

[ 2 ]

Carry out a correct method for finding the least value of |z-w|
(Distance (3,-2) to (0,1))-2.

Obtain answer $\sqrt{18}-2$ or $3 \sqrt{2}-2$

CAIE A-Level Mathematics A2 3.9 Complex Numbers Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)

Question (d)

Question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question 3

Question (a)

Question (b)

Question (c)

Question (d)

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question (a)

Question (b)