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

1. 1
   The equation $z^6=1$.

   (a)

   Which of these is a root of the equation?

   [1 mark]

   • A$\mathrm{e}^{\mathrm{i}\pi/3}$
   • B$\mathrm{e}^{\mathrm{i}\pi/6}$
   • C$\mathrm{e}^{\mathrm{i}\pi/4}$
   • D$\mathrm{e}^{2\mathrm{i}\pi/5}$

   (b)

   What is the sum of all six roots?

   [1 mark]

   • A$1$
   • B$6$
   • C$-1$
   • D$0$

   (c)

   Write down all six roots in the form $\mathrm{e}^{\mathrm{i}\theta}$ with $-\pi<\theta\le\pi$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $w=-8$.

   (a)

   Find the modulus of each of the cube roots of $w$.

   [1 mark]

   • A$8$
   • B$2\sqrt2$
   • C$2$
   • D$\frac83$

   (b)

   Which of these is not a cube root of $w$?

   [1 mark]

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

   (c)

   Find the three cube roots of $w$ in the form $a+b\mathrm{i}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The equation $z^4=-8+8\sqrt3\,\mathrm{i}$.

   (a)

   Find the modulus and the arguments, in the range $-\pi<\theta\le\pi$, of the four roots of the equation.

   [3 marks]

   (b)

   The four roots are plotted in the Argand diagram. Show that they are the vertices of a square, and find its area.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The points $P_k$ represent the complex numbers $2\omega^k$ for $k=0,1,2,3,4$, where $\omega=\mathrm{e}^{2\pi\mathrm{i}/5}$. They are the vertices of a regular pentagon.

   (a)

   (i) Show that $1+\omega+\omega^2+\omega^3+\omega^4=0$.
   (ii) Hence show that $\cos\frac{2\pi}{5}+\cos\frac{4\pi}{5}=-\frac12$.

   [6 marks]

   (b)

   Show that the product $P_0P_1\times P_0P_2\times P_0P_3\times P_0P_4$ of the four lengths from $P_0$ to the other vertices is $80$.

   [6 marks]

   Total for question 4: 12 marks

End of questions