Core Pure: Complex numbersEdexcel A-Level Further Maths: Topic test

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

   (a)

   What is the modulus of $z$?

   [1 mark]

   • A$3\sqrt2$
   • B$36$
   • C$6$
   • D$3\sqrt3+3$

   (b)

   What is the principal argument of $z$?

   [1 mark]

   • A$\frac{\pi}{6}$
   • B$\frac{5\pi}{6}$
   • C$-\frac{\pi}{6}$
   • D$\frac{2\pi}{3}$

   (c)

   Find the modulus and the principal argument of $z^3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The quadratic equation $z^2-6z+13=0$ has roots $\alpha$ and $\beta$, where $\mathrm{Im}(\alpha)>0$.

   (a)

   What are the roots of the equation?

   [1 mark]

   • A$3\pm2\mathrm{i}$
   • B$-3\pm2\mathrm{i}$
   • C$3\pm4\mathrm{i}$
   • D$6\pm4\mathrm{i}$

   (b)

   What is the modulus of $\alpha$?

   [1 mark]

   • A$13$
   • B$5$
   • C$\sqrt5$
   • D$\sqrt{13}$

   (c)

   Find $\alpha^2$ in the form $a+b\mathrm{i}$ and verify that $\left|\alpha^2\right|=13$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The cubic equation $z^3-5z^2+17z-13=0$ has $2+3\mathrm{i}$ as one root. All of its coefficients are real.

   (a)

   Find the other two roots of the equation.

   [3 marks]

   (b)

   The three roots are represented by the points $P$, $Q$ and $T$ in an Argand diagram. Find the area of triangle $PQT$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The locus $C$ of a point $P$ representing the complex number $z$ is given by $|z-3-4\mathrm{i}|=5$. The locus $\ell$ is given by $\arg(z-3)=\frac{\pi}{2}$.

   (a)

   Show that $C$ passes through the origin, and find its Cartesian equation. Find also the greatest value of $|z|$ for points on $C$.

   [6 marks]

   (b)

   Find the complex number represented by the point where $\ell$ meets $C$. Give it in the form $r(\cos\theta+\mathrm{i}\sin\theta)$, with $\theta$ in radians to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The complex number $z=\sqrt2\,\mathrm{e}^{\mathrm{i}\pi/8}$.

   (a)

   Which expression is equal to $z^4$?

   [1 mark]

   • A$2\mathrm{i}$
   • B$4\mathrm{e}^{\mathrm{i}\pi/32}$
   • C$-4$
   • D$4\mathrm{i}$

   (b)

   Which expression is equal to $z^{-1}$?

   [1 mark]

   • A$\sqrt2\,\mathrm{e}^{-\mathrm{i}\pi/8}$
   • B$\frac{1}{\sqrt2}\mathrm{e}^{-\mathrm{i}\pi/8}$
   • C$\frac{1}{\sqrt2}\mathrm{e}^{\mathrm{i}\pi/8}$
   • D$-\sqrt2\,\mathrm{e}^{\mathrm{i}\pi/8}$

   (c)

   Use de Moivre's theorem to find $z^6$ in the form $a+b\mathrm{i}$, giving $a$ and $b$ in exact form.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The equation $z^4=-16$ has four distinct complex roots.

   (a)

   What is the modulus of each root?

   [1 mark]

   • A$4$
   • B$16$
   • C$2$
   • D$\sqrt2$

   (b)

   Which of the following is one of the roots?

   [1 mark]

   • A$\sqrt2+\sqrt2\,\mathrm{i}$
   • B$2+2\mathrm{i}$
   • C$2\mathrm{i}$
   • D$-2$

   (c)

   The roots are represented by the vertices of a square in an Argand diagram. Find the area of the square.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Let $\theta$ be a real angle and let $x=\sin\theta$.

   (a)

   Use de Moivre's theorem to show that $\sin3\theta=3x-4x^3$.

   [3 marks]

   (b)

   Hence show that $\sin\frac{\pi}{18}$ is a root of the equation $8x^3-6x+1=0$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The complex number $w=2\sqrt2\,(1-\mathrm{i})$.

   (a)

   Write $w$ in the form $r\mathrm{e}^{\mathrm{i}\theta}$, where $-\pi<\theta\leqslant\pi$, and find the three cube roots of $w$ in the same form.

   [6 marks]

   (b)

   The three cube roots of $w$ are represented by the vertices of a triangle in an Argand diagram. Show that the triangle is equilateral and find its area, giving the answer to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions