Complex numbers (A2)AQA A-Level Further Maths: Topic test

1. 1
   The complex number $z=2\sqrt3-2\mathrm{i}$.

   (a)

   Which expression gives $z$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$ with $-\pi<\theta\le\pi$?

   [1 mark]

   • A$4\mathrm{e}^{\mathrm{i}\pi/6}$
   • B$16\mathrm{e}^{-\mathrm{i}\pi/6}$
   • C$4\mathrm{e}^{-\mathrm{i}\pi/3}$
   • D$4\mathrm{e}^{-\mathrm{i}\pi/6}$

   (b)

   Find $z^3$.

   [1 mark]

   • A$-64\mathrm{i}$
   • B$64\mathrm{i}$
   • C$-12\mathrm{i}$
   • D$64\mathrm{e}^{-\mathrm{i}\pi/6}$

   (c)

   Find the smallest positive integer $n$ for which $z^n$ is real and positive.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   For real $\theta$, consider the expansion of $(\cos\theta+\mathrm{i}\sin\theta)^4$.

   (a)

   Which expression is equal to $(\cos\theta+\mathrm{i}\sin\theta)^4$?

   [1 mark]

   • A$\cos^4\theta+\mathrm{i}\sin^4\theta$
   • B$\cos4\theta+\mathrm{i}\sin4\theta$
   • C$4\cos\theta+4\mathrm{i}\sin\theta$
   • D$\cos4\theta-\mathrm{i}\sin4\theta$

   (b)

   Which expression is the imaginary part of the binomial expansion of $(\cos\theta+\mathrm{i}\sin\theta)^4$?

   [1 mark]

   • A$4\cos^3\theta\sin\theta+4\cos\theta\sin^3\theta$
   • B$\cos^4\theta-6\cos^2\theta\sin^2\theta+\sin^4\theta$
   • C$4\cos^3\theta\sin\theta-4\cos\theta\sin^3\theta$
   • D$4\cos^3\theta\sin\theta-6\cos^2\theta\sin^2\theta$

   (c)

   Hence show that $\sin4\theta=4\sin\theta\cos\theta\left(1-2\sin^2\theta\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The equation $z^3=8\mathrm{i}$.

   (a)

   Express $8\mathrm{i}$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$, and hence write down the three roots of the equation in the form $\rho\mathrm{e}^{\mathrm{i}\phi}$ with $-\pi<\phi\le\pi$.

   [3 marks]

   (b)

   The three roots are the vertices of a triangle on an Argand diagram. Find the exact area of the triangle.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   For real $\theta$, the expression $\cos5\theta$ can be written as a polynomial in $\cos\theta$ by using de Moivre's theorem.

   (a)

   Show that $\cos5\theta=16\cos^5\theta-20\cos^3\theta+5\cos\theta$.

   [6 marks]

   (b)

   Use the result $\cos5\theta=16\cos^5\theta-20\cos^3\theta+5\cos\theta$ to show that $\cos^2\frac{\pi}{10}=\frac{5+\sqrt5}{8}$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The complex numbers $z_1=6\mathrm{e}^{2\mathrm{i}\pi/3}$ and $z_2=2\mathrm{e}^{\mathrm{i}\pi/6}$.

   (a)

   Find $\frac{z_1}{z_2}$.

   [1 mark]

   • A$3\mathrm{e}^{\mathrm{i}\pi/2}$
   • B$3\mathrm{e}^{5\mathrm{i}\pi/6}$
   • C$12\mathrm{e}^{\mathrm{i}\pi/2}$
   • D$3\mathrm{e}^{4\mathrm{i}}$

   (b)

   Find $z_1z_2$ in the form $x+\mathrm{i}y$.

   [1 mark]

   • A$6\sqrt3+6\mathrm{i}$
   • B$-6+6\sqrt3\mathrm{i}$
   • C$-3\sqrt3+3\mathrm{i}$
   • D$-6\sqrt3+6\mathrm{i}$

   (c)

   Find the value of $\left(\frac{z_1}{z_2}\right)^6$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   For real $\theta$ that is not a multiple of $2\pi$, let $z=\cos\theta+\mathrm{i}\sin\theta$, $C=\cos\theta+\cos2\theta+\cos3\theta+\cos4\theta$ and $S=\sin\theta+\sin2\theta+\sin3\theta+\sin4\theta$.

   (a)

   Which expression is equal to $z^3$?

   [1 mark]

   • A$\cos^3\theta+\mathrm{i}\sin^3\theta$
   • B$3\cos\theta+3\mathrm{i}\sin\theta$
   • C$\cos3\theta+\mathrm{i}\sin3\theta$
   • D$\cos3\theta-\mathrm{i}\sin3\theta$

   (b)

   Which expression is equal to $C+\mathrm{i}S$?

   [1 mark]

   • A$1+z+z^2+z^3$
   • B$z+z^2+z^3+z^4$
   • C$z^{10}$
   • D$z^{-1}+z^{-2}+z^{-3}+z^{-4}$

   (c)

   Find the exact values of $C$ and $S$ when $\theta=\frac{\pi}{3}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $\omega=\mathrm{e}^{2\mathrm{i}\pi/3}$, so that $1$, $\omega$ and $\omega^2$ are the cube roots of unity.

   (a)

   Show that $1+\omega+\omega^2=0$.

   [3 marks]

   (b)

   Hence show that $(2+\omega)\left(2+\omega^2\right)=3$, and deduce the distance between the points representing $-2$ and $\omega$ on an Argand diagram.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The equation $z^5=-32$.

   (a)

   (i) Solve the equation, giving the roots in the form $r\mathrm{e}^{\mathrm{i}\theta}$ with $-\pi<\theta\le\pi$. (ii) Hence show that $\cos\frac{\pi}{5}+\cos\frac{3\pi}{5}=\frac12$.

   [6 marks]

   (b)

   Show that $z^5+32=(z+2)\left(z^2-4z\cos\frac{\pi}{5}+4\right)\left(z^2-4z\cos\frac{3\pi}{5}+4\right)$.

   [6 marks]

   Total for question 8: 12 marks

End of questions