Exponential form of a complex numberAQA A-Level Further Maths: Subtopic test

1. 1
   The complex number $z=2\mathrm{e}^{\mathrm{i}\pi/3}$.

   (a)

   Write $z$ in the form $a+b\mathrm{i}$.

   [1 mark]

   • A$\sqrt3+\mathrm{i}$
   • B$\frac12+\frac{\sqrt3}{2}\mathrm{i}$
   • C$1-\sqrt3\,\mathrm{i}$
   • D$1+\sqrt3\,\mathrm{i}$

   (b)

   Express $z^2$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$.

   [1 mark]

   • A$4\mathrm{e}^{2\mathrm{i}\pi/3}$
   • B$2\mathrm{e}^{2\mathrm{i}\pi/3}$
   • C$4\mathrm{e}^{\mathrm{i}\pi/3}$
   • D$4\mathrm{e}^{\mathrm{i}\pi/6}$

   (c)

   Find $z^3$ in the form $a+b\mathrm{i}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex numbers $w_1=3\mathrm{e}^{\mathrm{i}\pi/4}$ and $w_2=2\mathrm{e}^{-\mathrm{i}\pi/4}$.

   (a)

   Find $w_1w_2$.

   [1 mark]

   • A$5$
   • B$6$
   • C$6\mathrm{i}$
   • D$-6\mathrm{i}$

   (b)

   Find $\dfrac{w_1}{w_2}$.

   [1 mark]

   • A$\frac32$
   • B$-\frac32\mathrm{i}$
   • C$\frac32\mathrm{i}$
   • D$6\mathrm{i}$

   (c)

   Express $w_1^2w_2$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $z=1-\sqrt3\,\mathrm{i}$.

   (a)

   Express $z$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$, where $r>0$ and $-\pi<\theta\le\pi$.

   [3 marks]

   (b)

   Find the smallest positive integer $n$ for which $z^n$ is real and positive, and state the value of $z^n$ for this $n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $z=\mathrm{e}^{\mathrm{i}\theta}$, where $\theta$ is real.

   (a)

   (i) Show that $z+\dfrac1z=2\cos\theta$.
   (ii) Hence show that $z^2+\dfrac{1}{z^2}=4\cos^2\theta-2$.

   [6 marks]

   (b)

   Hence find all the values of $z$ that satisfy $z^2+\dfrac{1}{z^2}=-1$, giving each in the form $\mathrm{e}^{\mathrm{i}\theta}$ with $0\le\theta<2\pi$.

   [6 marks]

   Total for question 4: 12 marks

End of questions