Roots of complex numbersEdexcel International A Level Further Maths: Subtopic test

1. 1
   Consider the equation $z^3=8$.

   (a)

   What is the modulus of each root of the equation?

   [1 mark]

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

   (b)

   Which of the following is the complete set of roots of the equation?

   [1 mark]

   • A$2,\;2e^{2\pi i/3},\;2e^{-2\pi i/3}$
   • B$2,\;2e^{i\pi/3},\;2e^{2\pi i/3}$
   • C$2e^{i\pi/3},\;-2,\;2e^{-i\pi/3}$
   • D$8,\;8e^{2\pi i/3},\;8e^{4\pi i/3}$

   (c)

   Find the non-real roots of the equation in the form $a+ib$, giving $a$ and $b$ as exact values.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex number $w=-8+8\sqrt3\,i$ and the equation $z^4=w$.

   (a)

   Which gives the modulus and the principal argument of $w$?

   [1 mark]

   • A$16$ and $-\dfrac\pi3$
   • B$16$ and $\dfrac\pi3$
   • C$16$ and $\dfrac{2\pi}{3}$
   • D$4$ and $\dfrac{2\pi}{3}$

   (b)

   What is the angle between the arguments of adjacent roots of $z^4=w$?

   [1 mark]

   • A$\dfrac\pi4$
   • B$\dfrac\pi2$
   • C$\dfrac{2\pi}{3}$
   • D$\dfrac\pi6$

   (c)

   Find the four roots of $z^4=w$ in the form $re^{i\theta}$, where $-\pi<\theta\le\pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $z^3=-27i$.

   (a)

   Solve the equation, giving the roots in the form $re^{i\theta}$, where $-\pi<\theta\le\pi$.

   [3 marks]

   (b)

   Hence solve $(w+2)^3=-27i$, giving each root in the form $a+ib$ with exact values of $a$ and $b$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Consider the equation $z^5=1$.

   (a)

   Find the roots of the equation in the form $e^{i\theta}$, where $-\pi<\theta\le\pi$, and hence show that $\cos\frac{2\pi}{5}+\cos\frac{4\pi}{5}=-\frac12$.

   [6 marks]

   (b)

   Use the roots of $z^5=1$ to show that the roots of $(w+1)^5=w^5$ are $w=-\frac12-\frac i2\cot\frac{k\pi}{5}$ for $k=1,2,3,4$.

   [6 marks]

   Total for question 4: 12 marks

End of questions