Euler's relation and polar formEdexcel International A Level Further Maths: Subtopic test

1. 1
   Let $z=2e^{i\pi/3}$ and $w=3e^{i\pi/6}$.

   (a)

   Find $zw$ in the form $re^{i\theta}$.

   [1 mark]

   • A$6e^{i\pi/2}$
   • B$5e^{i\pi/2}$
   • C$6e^{i\pi/6}$
   • D$6e^{i\pi^2/18}$

   (b)

   Find $\dfrac{z}{w}$ in the form $re^{i\theta}$.

   [1 mark]

   • A$\frac23e^{i\pi/2}$
   • B$\frac23e^{i\pi/6}$
   • C$\frac32e^{i\pi/6}$
   • D$\frac23e^{-i\pi/6}$

   (c)

   Find $z+w$ in the form $a+ib$, giving $a$ and $b$ as exact values.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\theta$ be a real number and let $z=e^{i\theta}$.

   (a)

   Find $z+\dfrac1z$.

   [1 mark]

   • A$2i\sin\theta$
   • B$2e^{i\theta}$
   • C$2$
   • D$2\cos\theta$

   (b)

   Find $z-\dfrac1z$.

   [1 mark]

   • A$2\sin\theta$
   • B$-2i\sin\theta$
   • C$2i\sin\theta$
   • D$2i\cos\theta$

   (c)

   By expanding $\left(z+\dfrac1z\right)^2$ and using $z^2=e^{2i\theta}$, show that $\cos^2\theta=\frac12(1+\cos2\theta)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $z=-1+i\sqrt3$.

   (a)

   Write $z$ in the form $re^{i\theta}$, where $-\pi<\theta\le\pi$.

   [3 marks]

   (b)

   Given that $w=\sqrt2\,e^{-i\pi/4}$, find $zw$ and $\dfrac zw$ in the form $re^{i\theta}$, where $-\pi<\theta\le\pi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Euler's relation states that $e^{i\theta}=\cos\theta+i\sin\theta$ for real $\theta$.

   (a)

   Use the exponential forms $\cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}$ to show that $\cos A+\cos B=2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)$ for real $A$ and $B$.

   [6 marks]

   (b)

   Use the result of part (a) to solve $\cos\theta+\cos3\theta=0$ for $0\le\theta<2\pi$.

   [6 marks]

   Total for question 4: 12 marks

End of questions