De Moivre's theorem and exponential formEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $z=\cos\frac{\pi}{9}+i\sin\frac{\pi}{9}$.

   (a)

   Find $z^6$.

   [1 mark]

   • A$\frac12+\frac{\sqrt3}{2}i$
   • B$-\frac12-\frac{\sqrt3}{2}i$
   • C$-\frac12+\frac{\sqrt3}{2}i$
   • D$\cos\frac{\pi}{54}+i\sin\frac{\pi}{54}$

   (b)

   Which expression is equal to $z+\frac1z$?

   [1 mark]

   • A$2\cos\frac{\pi}{9}$
   • B$2i\sin\frac{\pi}{9}$
   • C$\cos\frac{\pi}{9}$
   • D$2\cos\frac{2\pi}{9}$

   (c)

   Find the exact value of $z^3+\frac{1}{z^3}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $z=2e^{i\pi/6}$ is given.

   (a)

   Write $z$ in the form $x+iy$.

   [1 mark]

   • A$1+\sqrt3\,i$
   • B$\frac{\sqrt3}{2}+\frac12i$
   • C$2+\frac{\pi}{6}i$
   • D$\sqrt3+i$

   (b)

   Which is $z^4$ in exponential form?

   [1 mark]

   • A$8e^{2\pi i/3}$
   • B$16e^{2\pi i/3}$
   • C$2e^{2\pi i/3}$
   • D$16e^{i\pi/24}$

   (c)

   Find $z^4$ in the form $x+iy$, giving exact values.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $z=\cos\theta+i\sin\theta$, where $\theta$ is real.

   (a)

   Use de Moivre's theorem to show that $\sin3\theta=3\sin\theta-4\sin^3\theta$.

   [3 marks]

   (b)

   Given also that $\cos3\theta=4\cos^3\theta-3\cos\theta$, show that $\tan3\theta=\frac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question, $\theta$ is a real number and $n$ is a positive integer.

   (a)

   Given that $z=\cos\frac{\pi}{n}+i\sin\frac{\pi}{n}$, show that $1+z+z^2+\dots+z^{n-1}=1+i\cot\frac{\pi}{2n}$.

   [6 marks]

   (b)

   (i) Use $z+\frac1z=2\cos\theta$, where $z=\cos\theta+i\sin\theta$, to show that $\cos^4\theta=\frac18(\cos4\theta+4\cos2\theta+3)$.
   (ii) Hence find $\int_0^{\pi/2}\cos^4\theta\,\mathrm{d}\theta$.

   [6 marks]

   Total for question 4: 12 marks

End of questions