De Moivre's theoremAQA A-Level Further Maths: Subtopic test

1. 1
   The complex number $z=\cos\frac{\pi}{12}+\mathrm{i}\sin\frac{\pi}{12}$.

   (a)

   Find $z^4$.

   [1 mark]

   • A$\frac{\sqrt3}{2}+\frac12\mathrm{i}$
   • B$\cos\frac{\pi}{48}+\mathrm{i}\sin\frac{\pi}{48}$
   • C$\frac12+\frac{\sqrt3}{2}\mathrm{i}$
   • D$4\cos\frac{\pi}{12}+4\mathrm{i}\sin\frac{\pi}{12}$

   (b)

   Find $z^{12}$.

   [1 mark]

   • A$1$
   • B$-1$
   • C$\mathrm{i}$
   • D$-\mathrm{i}$

   (c)

   Express $z^{-5}$ in the form $\cos\alpha-\mathrm{i}\sin\alpha$, stating the exact value of $\alpha$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $w=1+\mathrm{i}$.

   (a)

   Write $w$ in modulus-argument form.

   [1 mark]

   • A$\sqrt2\left(\cos\frac{\pi}{4}+\mathrm{i}\sin\frac{\pi}{4}\right)$
   • B$2\left(\cos\frac{\pi}{4}+\mathrm{i}\sin\frac{\pi}{4}\right)$
   • C$\sqrt2\left(\cos\frac{\pi}{2}+\mathrm{i}\sin\frac{\pi}{2}\right)$
   • D$\sqrt2\left(\cos\frac{\pi}{4}-\mathrm{i}\sin\frac{\pi}{4}\right)$

   (b)

   Find $w^8$.

   [1 mark]

   • A$256$
   • B$16\mathrm{i}$
   • C$-16$
   • D$16$

   (c)

   Find $w^{10}$ in the form $a+b\mathrm{i}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The equation $8x^3-6x-1=0$ is to be solved.

   (a)

   Use de Moivre's theorem to show that $\cos3\theta=4\cos^3\theta-3\cos\theta$.

   [3 marks]

   (b)

   By substituting $x=\cos\theta$, find the three roots of the equation in the form $\cos(q\pi)$, where $q$ is a rational number.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For real $\theta$, let $C=\displaystyle\sum_{k=0}^{\infty}\frac{\cos k\theta}{2^k}$ and $S=\displaystyle\sum_{k=0}^{\infty}\frac{\sin k\theta}{2^k}$.

   (a)

   (i) By considering $C+\mathrm{i}S$ as a geometric series, show that $C+\mathrm{i}S=\dfrac{2}{2-\cos\theta-\mathrm{i}\sin\theta}$, explaining why the series converges.
   (ii) Hence show that $C=\dfrac{4-2\cos\theta}{5-4\cos\theta}$.

   [6 marks]

   (b)

   (i) Find an expression for $S$ in terms of $\theta$.
   (ii) Show that $C^2+S^2=\dfrac{4}{5-4\cos\theta}$, and hence find the greatest value of $C^2+S^2$ as $\theta$ varies.

   [6 marks]

   Total for question 4: 12 marks

End of questions