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

1. 1
   The complex numbers $z$ and $w$ are given by $z=4+3\mathrm{i}$ and $w=1-2\mathrm{i}$.

   (a)

   Find $zw$.

   [1 mark]

   • A$-2-5\mathrm{i}$
   • B$10-5\mathrm{i}$
   • C$4-6\mathrm{i}$
   • D$5+\mathrm{i}$

   (b)

   Find $\frac{z}{w}$.

   [1 mark]

   • A$2-\mathrm{i}$
   • B$\frac{2}{3}-\frac{11}{3}\mathrm{i}$
   • C$4-\frac{3}{2}\mathrm{i}$
   • D$-\frac{2}{5}+\frac{11}{5}\mathrm{i}$

   (c)

   Find $z^2-2w$, giving your answer in the form $x+\mathrm{i}y$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The quadratic equation $z^2+4z+8=0$ has roots $\alpha$ and $\beta$, where $\alpha$ has positive imaginary part.

   (a)

   Which of these gives the two roots of the equation?

   [1 mark]

   • A$-2\pm2\mathrm{i}$
   • B$2\pm2\mathrm{i}$
   • C$-2\pm4\mathrm{i}$
   • D$-4\pm4\mathrm{i}$

   (b)

   Find $\alpha^2$.

   [1 mark]

   • A$8\mathrm{i}$
   • B$8-8\mathrm{i}$
   • C$-8\mathrm{i}$
   • D$4+4\mathrm{i}$

   (c)

   Show that $\frac{\alpha}{\beta}=-\mathrm{i}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The quartic $f(z)=z^4-4z^3+6z^2-4z+5$ has real coefficients, and $z=2+\mathrm{i}$ is a root of the equation $f(z)=0$.

   (a)

   Write down a second root of $f(z)=0$, and show that $z^2-4z+5$ is a factor of $f(z)$.

   [3 marks]

   (b)

   Hence find all four roots of $f(z)=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The cubic equation $z^3+az^2+bz+20=0$, where $a$ and $b$ are real constants, has a root $z=1+3\mathrm{i}$. The points $A$, $B$ and $C$ on an Argand diagram represent the three roots, where $A$ represents $1+3\mathrm{i}$ and $B$ represents the other non-real root.

   (a)

   Find the values of $a$ and $b$, and find the third root.

   [6 marks]

   (b)

   (i) Find the area of triangle $ABC$. (ii) Show that the locus of points $z$ satisfying $|z-(1-3\mathrm{i})|=|z+2|$ has Cartesian equation $y=x-1$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The complex number $z=4-4\mathrm{i}$.

   (a)

   Find $|z|$.

   [1 mark]

   • A$8$
   • B$4$
   • C$4\sqrt{2}$
   • D$32$

   (b)

   Find $\arg z$, where $-\pi<\arg z\le\pi$.

   [1 mark]

   • A$-\frac{\pi}{4}$
   • B$\frac{\pi}{4}$
   • C$\frac{3\pi}{4}$
   • D$\frac{7\pi}{4}$

   (c)

   The complex number $w=2\left(\cos\frac{\pi}{3}+\mathrm{i}\sin\frac{\pi}{3}\right)$. Find $zw$ in modulus-argument form.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The complex number $z$ satisfies $|z+5-2\mathrm{i}|=\sqrt{10}$.

   (a)

   Which complex number represents the centre of the locus of $z$ on an Argand diagram?

   [1 mark]

   • A$5-2\mathrm{i}$
   • B$-5-2\mathrm{i}$
   • C$2-5\mathrm{i}$
   • D$-5+2\mathrm{i}$

   (b)

   Which of these complex numbers satisfies the equation $|z+5-2\mathrm{i}|=\sqrt{10}$?

   [1 mark]

   • A$-2+5\mathrm{i}$
   • B$-4+5\mathrm{i}$
   • C$-5+5\mathrm{i}$
   • D$1+2\mathrm{i}$

   (c)

   Find the greatest value of $|z|$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The locus $L$ is given by $\arg(z-2-\mathrm{i})=\frac{\pi}{4}$ and the circle $C$ is given by $|z|=5$.

   (a)

   Show that $L$ lies on the line $y=x-1$, and state the set of values of $x$ on $L$.

   [3 marks]

   (b)

   Find the complex number at which $L$ meets $C$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   In this question $z=x+\mathrm{i}y$, where $x$ and $y$ are real numbers, and $z^*$ is the complex conjugate of $z$.

   (a)

   Given that $2z-3\mathrm{i}z^*=8-7\mathrm{i}$, find $z$.

   [6 marks]

   (b)

   (i) Show that the locus of points satisfying $|z-6|=2|z|$ is a circle, and find its centre and radius. (ii) Find the greatest and least values of $|z|$ for points on this circle.

   [6 marks]

   Total for question 8: 12 marks

End of questions