Roots of complex numbers and roots of unityEdexcel A-Level Further Maths: Subtopic test

1. 1
   The complex number $w=8\mathrm{i}$ and the equation $z^3=8\mathrm{i}$.

   (a)

   What is the modulus of each root of $z^3=8\mathrm{i}$?

   [1 mark]

   • A$8$
   • B$2$
   • C$\frac{8}{3}$
   • D$2\sqrt2$

   (b)

   Which of the following is the smallest positive argument of a root of $z^3=8\mathrm{i}$?

   [1 mark]

   • A$\frac{\pi}{6}$
   • B$\frac{\pi}{2}$
   • C$\frac{\pi}{3}$
   • D$\frac{2\pi}{3}$

   (c)

   Find the three roots of $z^3=8\mathrm{i}$, giving each in the form $x+\mathrm{i}y$ where $x$ and $y$ are exact.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\omega=\mathrm{e}^{2\pi\mathrm{i}/5}$, so that $1,\omega,\omega^2,\omega^3,\omega^4$ are the fifth roots of unity.

   (a)

   What is the argument of $\omega^3$, in the interval $-\pi<\theta\le\pi$?

   [1 mark]

   • A$\frac{6\pi}{5}$
   • B$\frac{3\pi}{5}$
   • C$-\frac{4\pi}{5}$
   • D$-\frac{6\pi}{5}$

   (b)

   What is the value of $1+\omega+\omega^2+\omega^3+\omega^4$?

   [1 mark]

   • A$1$
   • B$5$
   • C$-1$
   • D$0$

   (c)

   Show that $\omega^4=\overline{\omega}$, and hence show that $\omega+\omega^4=2\cos\frac{2\pi}{5}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $w=-4+4\sqrt3\,\mathrm{i}$.

   (a)

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

   [3 marks]

   (b)

   Find the cube roots of $w$, giving each in the form $r\mathrm{e}^{\mathrm{i}\theta}$ where $r>0$ and $-\pi<\theta\le\pi$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The roots of the equation $z^6=64$ are represented in an Argand diagram by the points $A_0,A_1,\dots,A_5$, taken in order of increasing argument from $A_0$, which represents $z=2$.

   (a)

   Find the six roots of $z^6=64$ in the form $x+\mathrm{i}y$, with $x$ and $y$ exact. Hence show that $A_0A_1A_2A_3A_4A_5$ is a regular hexagon, and state the length of its sides.

   [6 marks]

   (b)

   Show that $A_0A_2A_4$ is an equilateral triangle and find its exact area.

   [6 marks]

   Total for question 4: 12 marks

End of questions