9709. P3. Complex Numbers. Past Exam Questions

9709/31/M/J/25q3 – Mark Scheme

Find the complex numbers z for which $\frac{z + 5i}{z - 5}$ is real and $\vert x \vert = \sqrt{17}$ . Give your answers in the form z = x + iy, where x and y are real. [6 marks]

9709/31/M/J/25q6 – Mark Scheme

It is given that $z_1 = 3e^{ \frac{1}{4} \pi i }$ , $z_2 = \frac{3}{2} e^{ \frac{1}{6} \pi i}$ and $\omega = 2e^{ \frac{1}{2} \pi i }$ .

(a) State the value of $\omega z_1$ and $\omega z_2$ . Give your answers in the form $re^{ i \theta}$ , where r>0 and $- \pi < \theta \leq \pi$ [2 marks]

(b) On a sketch of an Argand diagram with origin O, show the points A, B, C and D representing the complex numbers $z_1$ , $z_2$ , $\omega z_1$ and $\omega z_2$ respectively. [2 marks]

9709/32/M/J/25q3 – Mark Scheme

On an Argand diagram shade the region whose points represent complex numbers z which satisfy both the inequalities $\vert z - 3i \vert \leq 2$ and $\frac{1}{4} \pi \leq arg(z-1-2i) \leq \frac{3}{4} \pi$ [5 marks]

9709/32/M/J/25q5 – Mark Scheme

The square roots of $-1-4 \sqrt{5} i$ can be expressed in the Cartesian form x + iy, where x and y are real and exact.

By first forming a quartic equation in x or y, find the square roots of $-1 - 4 \sqrt{5} i$ in exact Cartesian form. [5 marks]

9709/33/M/J/25q4 – Mark Scheme

It is given that $z_1 = r_1 e^{i \theta _1}$ and $z_2 = r_2 e^{i \theta _2}$ .

Show that $(z_1 z_2 )^{*} = z_1^{*}z_2^{*}$ . [3 marks]

(b) $z = 3e^{ \frac{1}{4} \pi i}$ is a root of the equation z²+bz+c = 0, where b and c are real.

State the other root and hence find the values of b and c. [3 marks]

9709/33/M/J/25q6 – Mark Scheme

Find the complex numbers z for which $\frac{z+4}{z+4i}$ is real and $\vert z \vert = \sqrt{10}$ . Give your answers in the form z = x + iy, where x and y are real. [6 marks]

9709. P3. Complex Numbers. Past Exam Questions

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