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

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

   (a)

   Find $|z|$.

   [1 mark]

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

   (b)

   Find the argument of $z$, in the interval $(-\pi,\pi]$.

   [1 mark]

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

   (c)

   Use the modulus-argument form of $z$ to find $z^3$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex numbers $z_1$ and $z_2$ are given by $z_1=\sqrt3+i$ and $z_2=1+i$.

   (a)

   Find $\arg(z_1z_2)$.

   [1 mark]

   • A$\frac{\pi}{12}$
   • B$\frac{\pi^2}{24}$
   • C$\frac{\pi}{4}$
   • D$\frac{5\pi}{12}$

   (b)

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

   [1 mark]

   • A$2-\sqrt2$
   • B$\frac{1}{\sqrt2}$
   • C$\sqrt2$
   • D$2\sqrt2$

   (c)

   Find $\arg\left(\frac{z_1}{z_2}\right)$, in radians.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   On an Argand diagram the points $A$ and $B$ represent the complex numbers $z=3+4i$ and $iz$ respectively, and $O$ is the origin.

   (a)

   Find the complex number represented by $B$ in the form $x+iy$, and show that $OA=OB$.

   [3 marks]

   (b)

   Find $\arg z$ and $\arg(iz)$ in radians to 3 significant figures, and hence show that angle $AOB$ is a right angle.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $z_1=1+i\sqrt3$ and $z_2=-\sqrt3+i$.

   (a)

   (i) Express $z_1$ and $z_2$ in modulus-argument form, giving each argument in radians in the interval $(-\pi,\pi]$.
   (ii) Hence find $z_1z_2$ in modulus-argument form, and show that $z_1z_2=-2\sqrt3-2i$.

   [6 marks]

   (b)

   (i) Find $\frac{z_1}{z_2}$ in modulus-argument form.
   (ii) Hence write $\frac{z_1}{z_2}$ in the form $x+iy$, and explain what this shows about the angle $POQ$, where $P$ and $Q$ are the points representing $z_1$ and $z_2$ and $O$ is the origin.

   [6 marks]

   Total for question 4: 12 marks

End of questions