Further complex numbers

WFM02Topic 2.33 questions

Euler's relation, de Moivre's theorem, nth roots, loci and transformations of the complex plane.

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Original practice questions written in the style of the exam.

Question 15 marks
Solve the equation $z^3 = 8i$ , giving your answers in the form $a + bi$ where $a$ and $b$ are exact.
Mark scheme
- B1 $8i = 8\left(\cos\dfrac{\pi}{2} + i\sin\dfrac{\pi}{2}\right)$ (1)
Question 25 marks
The point $P$ represents $z$ on an Argand diagram, where $|z - 3| = 2|z + 3i|$ Show that the locus of is a circle, and find its centre and its radius.

2π

)

M1

z = 2\left(\cos\theta + i\sin\theta\right)

with

\theta = \dfrac{\pi}{6} + \dfrac{2k\pi}{3}

(1)

A1

\theta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, -\dfrac{\pi}{2}

(1)

A1

z = \sqrt{3} + i

and

z = -\sqrt{3} + i

(1)

A1

z = -2i

(1)

i

∣

P

Mark scheme

0

x^2 + y^2 + 2x + 8y + 9 = 0