Complex numbers and the Argand diagramEdexcel International A Level Further Maths: Subtopic test

1. 1
   The complex number $z=-3+4\mathrm{i}$.

   (a)

   Find $|z|$.

   [1 mark]

   • A$7$
   • B$5$
   • C$25$
   • D$1$

   (b)

   Find the principal argument of $z$, in radians to 3 significant figures.

   [1 mark]

   • A$-0.927$
   • B$0.927$
   • C$2.21$
   • D$-2.21$

   (c)

   Find $z^*$ and show that $zz^*=|z|^2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The complex numbers $z_1=5-12\mathrm{i}$ and $z_2=1+\sqrt3\,\mathrm{i}$.

   (a)

   Find $|z_1z_2|$.

   [1 mark]

   • A$26$
   • B$15$
   • C$52$
   • D$6.5$

   (b)

   Find the principal argument of $z_2$.

   [1 mark]

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

   (c)

   Find $\arg z_1$ in radians to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Real numbers $x$ and $y$ satisfy the equation $(2x+y)+(x-3y)\mathrm{i}=7+7\mathrm{i}$.

   (a)

   Find the values of $x$ and $y$.

   [3 marks]

   (b)

   Let $w=x+y\mathrm{i}$ using your values from part (a). Find $|w|$ and the principal argument of $w$, in radians to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The complex numbers $z_1=1+\mathrm{i}$ and $z_2=-\sqrt3+\mathrm{i}$ are represented by the points $A$ and $B$ on an Argand diagram. The origin is $O$.

   (a)

   (i) Find $|z_1|$, $|z_2|$, $\arg z_1$ and $\arg z_2$, giving the arguments in terms of $\pi$.
   (ii) Hence find the size of angle $AOB$, in terms of $\pi$.

   [6 marks]

   (b)

   (i) Show that the area of triangle $OAB$ is $\frac{1+\sqrt3}{2}$.
   (ii) Find the exact length of $AB$.

   [6 marks]

   Total for question 4: 12 marks

End of questions