[Maximum number: 3]

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities $|z+2-3 \mathrm{i}| \leqslant 2$ and $\arg z \leqslant \frac{3}{4} \pi$ .

[Maximum number: 4]

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities $|z| \geqslant 2$ and $|z-1+\mathrm{i}| \leqslant 1$ .

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities $|z+1-i| \leqslant 1$ and $\arg (z-1) \leqslant \frac{3}{4} \pi$ .

(a)

Given the complex numbers u=a+i b and w=c+i d, where a, b, c and d are real, prove that $(u+w)^{*}=u^{*}+w^{*}$ .

[ 2 ]

(b)

Solve the equation $(z+2+\mathrm{i})^{*}+(2+\mathrm{i}) z=0$ , giving your answer in the form x+i y where x and y are real.

[ 4 ]

The complex number w is defined by w=2+i.

(a)

Showing your working, express $w^{2}$ in the form x+i y, where x and y are real. Find the modulus of $w^{2}$ .

(b)

Shade on an Argand diagram the region whose points represent the complex numbers z which satisfy

\left|z-w^{2}\right| \leqslant\left|w^{2}\right|

[Maximum number: 7]

Throughout this question the use of a calculator is not permitted.
It is given that the complex number $-1+(\sqrt{ } 3) \mathrm{i}$ is a root of the equation

k x^{3}+5 x^{2}+10 x+4=0

where k is a real constant.

(a)

Write down another root of the equation.

[ 1 ]

(b)

Find the value of k and the third root of the equation.

[ 6 ]

[Maximum number: 6]

Find the complex numbers w which satisfy the equation $w^{2}+2 \mathrm{i} w^{*}=1$ and are such that $\operatorname{Re} w \leqslant 0$ . Give your answers in the form x+i y, where x and y are real.

(a)

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities $|z-3-2 \mathbf{i}| \leqslant 1$ and $\operatorname{Im} z \geqslant 2$ .

[ 4 ]

(b)

Find the greatest value of $\arg z$ for points in the shaded region, giving your answer in degrees.

[ 3 ]

[Maximum number: 5]

The complex number u is given by $u=10-4 \sqrt{6} \mathrm{i}$ .
Find the two square roots of u, giving your answers in the form a+i b, where a and b are real and exact.

(a)

Solve the equation $z^{2}-2 p \mathrm{i} z-q=0$ , where p and q are real constants.

[ 2 ]

(b)

In an Argand diagram with origin O, the roots of this equation are represented by the distinct points A and B.

Given that A and B lie on the imaginary axis, find a relation between p and q.

[ 2 ]

(c)

Given instead that triangle O A B is equilateral, express q in terms of p.

[ 3 ]