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 $\arg z \leqslant \frac{3}{4} \pi$ .

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)

On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities $-\frac{1}{3} \pi \leqslant \arg (z-1-2 \mathrm{i}) \leqslant \frac{1}{3} \pi$ and $\operatorname{Re} z \leqslant 3$ .

[ 3 ]

(b)

Calculate the least value of $\arg z$ for points in the region from (a). Give your answer in radians correct to 3 decimal places.

[ 2 ]

[Maximum number: 4]

On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities $|z-2 \mathrm{i}| \leqslant|z+2-\mathrm{i}|$ and $0 \leqslant \arg (z+1) \leqslant \frac{1}{4} \pi$ .

[Maximum number: 5]

On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities $|z-1+2 i| \leqslant|z|$ and $|z-2| \leqslant 1$ .

[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 $|z-1+\mathrm{i}| \leqslant 1$ .

[Maximum number: 4]

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

(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 ]

[Maximum number: 5]

The square roots of 24-7i can be expressed in the Cartesian form x+i y, where x and y are real and exact.

By first forming a quartic equation in x or y, find the square roots of 24-7 i in exact Cartesian form.

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|

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Question 2

Question 2

Question 2

Question 2(a)

Question 2(b)

Question 2

Question 2

Question 2

Question 3

Question 3

Question 3(a)

Question 3(b)

Question 3

Question 3

Question 3(i)

Question 3(ii)