CAIE A-Level Mathematics 3.9 Complex numbers Question Bank

Multiply numerator and denominator by 3-i
M1
OE
Obtain numerator - 10+10 i or denominator 10
A1
Obtain final answer -1+i
A1

State or imply $r=\sqrt{2}$
B1 FT
State or imply that $\theta=\frac{3}{4} \pi$
B1 FT

State that O A and B C are parallel
B1
State that B C=2 O A
B1

Use angle $A O B=\arg u-\arg v=\arg \frac{u}{v}$
M1
Obtain the given answer
A1
Alternative method for question 8(d)
Obtain $\tan A O B$ from gradients of O A and O B and the $\tan (A \pm B)$ formula
M1
Obtain the given answer
A1
Alternative method for question 8(d)
Obtain $\cos A O B$ by using the cosine rule or a scalar product
M1
Obtain the given answer
A1

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

DM1

$2(20-4 \sqrt{7} \mathrm{i})+3(-6-2 \sqrt{7} \mathrm{i})+14(-1+\sqrt{7} \mathrm{i})+k=0$

Obtain answer k=-8

A1

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

DM1

Obtain answer k=-8

A1

State answer $-1-\sqrt{7} \mathrm{i}$

B1

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
$-1-\sqrt{7} \mathrm{i}$

M1

Or state $(x-(-1+\sqrt{7} \mathrm{i}))(x-(-1-\sqrt{7} \mathrm{i}))(2 x-p)$

Obtain $x^{2}+2 x+8$

A1

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

A1

Needs to follow from the working.

|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.

Carry out a complete method for calculating the maximum value of $\arg z$ for
correct circle

M1

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

A1

CAO. The question requires radians.