1.13 Complex numbers: polar and exponential formsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The complex number $z=-1+\sqrt{3}\,i$.

   (a)

   Find $|z|$.

   [1 mark]

   • A$4$
   • B$2$
   • C$\sqrt{2}$
   • D$\sqrt{3}$

   (b)

   Find the principal argument of $z$.

   [1 mark]

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

   (c)

   Write $z$ in the form $re^{i\theta}$, with $\theta$ in radians.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $z_1=3e^{\frac{\pi}{4}i}$ and $z_2=2e^{\frac{\pi}{12}i}$.

   (a)

   Find $|z_1z_2|$.

   [1 mark]

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

   (b)

   Find the argument of $\frac{z_1}{z_2}$.

   [1 mark]

   • A$\frac{\pi}{3}$
   • B$\frac{\pi^2}{48}$
   • C$\frac{\pi}{6}$
   • D$3$

   (c)

   Describe fully the single geometrical transformation of the Argand diagram that maps $z_1$ to $z_1z_2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The complex number $w=1+i$.

   (a)

   Write $w$ in the form $re^{i\theta}$ and hence find $w^{10}$ in the form $a+bi$.

   [3 marks]

   (b)

   Find the smallest positive integer $n$ for which $w^n$ is real, and write down the value of $w^n$ for this $n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two alternating voltage sources have voltages $V_1=10\cos(40t)$ and $V_2=10\cos\left(40t+\frac{2\pi}{3}\right)$ volts, where $t$ is the time in seconds. Their sum can be written in the form $A\cos(40t+B)$, where $A>0$ and $-\pi<B\leq\pi$.

   (a)

   By writing each voltage as the real part of a complex expression, show that $A=10$ and $B=\frac{\pi}{3}$.

   [6 marks]

   (b)

   A third source with voltage $V_3=15\cos(40t-0.8)$ volts is added to the circuit. Use your GDC to write $V_1+V_2+V_3$ in the form $A\cos(40t+B)$, and state the maximum value of the total voltage.

   [6 marks]

   Total for question 4: 12 marks

End of questions