Argand diagrams and modulus-argument formEdexcel A-Level Further Maths: Subtopic test

1. 1
   The complex number $z=-1+i\sqrt3$ is given.

   (a)

   Find $|z|$.

   [1 mark]

   • A$2$
   • B$4$
   • C$1+\sqrt3$
   • D$\sqrt2$

   (b)

   What is the principal argument of $z$?

   [1 mark]

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

   (c)

   Find the modulus of $z^2$ and the principal argument of $z^2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $z_1=4\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$ and $z_2=2\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)$.

   (a)

   Find $\arg(z_1z_2)$.

   [1 mark]

   • A$\frac{\pi}{6}$
   • B$\frac{\pi}{18}$
   • C$8$
   • D$\frac{\pi}{2}$

   (b)

   Find $\left|\frac{z_1}{z_2}\right|$.

   [1 mark]

   • A$8$
   • B$2$
   • C$6$
   • D$\frac12$

   (c)

   Find $\frac{z_1}{z_2}$ in the form $x+iy$, giving exact values.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex numbers $z=1+i\sqrt3$ and $w=-\sqrt3+i$ are represented by the points $Z$ and $W$ on an Argand diagram with origin $O$.

   (a)

   Express $z$ and $w$ in modulus-argument form, with each argument in the range $-\pi<\theta\le\pi$.

   [3 marks]

   (b)

   Find the modulus and argument of $\frac{z}{w}$. Hence show that $O$, $Z$, the point representing $z+w$, and $W$ are the vertices of a square.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The complex numbers $z_1=1+i$ and $z_2=\sqrt3-i$ are given.

   (a)

   Express $z_1$ and $z_2$ in modulus-argument form with principal arguments. Hence find $z_1z_2$ and $\frac{z_1}{z_2}$ in modulus-argument form.

   [6 marks]

   (b)

   By also expanding $z_1z_2$ in Cartesian form, show that $\cos\frac{\pi}{12}=\frac{\sqrt6+\sqrt2}{4}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions