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

1. 1
   The complex numbers $z_1=3+2\mathrm{i}$ and $z_2=1-4\mathrm{i}$.

   (a)

   Find $z_1z_2$.

   [1 mark]

   • A$11-14\mathrm{i}$
   • B$-5-10\mathrm{i}$
   • C$3-8\mathrm{i}$
   • D$11-10\mathrm{i}$

   (b)

   Find the imaginary part of $\dfrac{z_1}{z_2}$.

   [1 mark]

   • A$\frac{14}{17}$
   • B$-\frac{5}{17}$
   • C$14$
   • D$\frac{10}{17}$

   (c)

   Given that $z_1+\lambda z_2$ is purely imaginary, where $\lambda$ is a real constant, find $\lambda$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex numbers $w_1=4+\mathrm{i}$ and $w_2=1+3\mathrm{i}$ are represented by the points $A$ and $B$ on an Argand diagram. The origin is $O$ and $OACB$ is a parallelogram.

   (a)

   Find the complex number represented by the point $C$.

   [1 mark]

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

   (b)

   Find $\mathrm{i}w_1$.

   [1 mark]

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

   (c)

   Find the exact length of the diagonal $AB$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $z=1+2\mathrm{i}$.

   (a)

   Find $z^2$ and $z^3$, each in the form $a+b\mathrm{i}$.

   [3 marks]

   (b)

   Hence, or otherwise, find $\dfrac{z^2}{z^*}$ in the form $p+q\mathrm{i}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The complex number $z_1=4+3\mathrm{i}$ and the complex number $z_2=\mathrm{i}z_1$ are represented by the points $A$ and $B$ on an Argand diagram. The origin is $O$ and the point $C$ represents $z_1+z_2$.

   (a)

   (i) Find $z_2$ in the form $a+b\mathrm{i}$.
   (ii) Show that $OACB$ is a square.

   [6 marks]

   (b)

   (i) Find the area of $OACB$.
   (ii) Find $\dfrac{z_1}{z_1+z_2}$ in the form $p+q\mathrm{i}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions