Progressive Complex numbers practice questions
Use Clevolab for progressive practice in complex numbers. This page shows what the topic covers, what skills the current set targets, and a few real examples from the reviewed question bank.
About this topic
Clevolab treats complex numbers as repeated practice with explanation, not just answer checking. This page is designed to make the topic legible before you open the app.
The current Progressive set covers the imaginary unit, algebra of complex numbers, modulus, argument, and related forms. 148 reviewed questions currently published for this page.
What you can practise
- Real and imaginary parts
- Arithmetic with complex numbers
- Modulus and argument
- Conjugates
- Standard algebraic manipulation
Real sample questions from the current set
These examples come from the reviewed questions currently stored for this topic. They are here so the page shows the actual flavour of Clevolab, not just a summary.
Sample question
The imaginary part of $(3-2i)+(-1+7i)$ is
- A$5$Correct answer
- B$-5$Answer option
- C$7$Answer option
- D$2$Answer option
Why this answer is right
Add imaginary coefficients: $-2+7=5$. The imaginary part of the sum is therefore $5$.
Add componentwise: $(3-2i)+(-1+7i)=(3-1)+(-2+7)i=2+5i$. The imaginary part is the coefficient of $i$. So $$\operatorname{Im}(2+5i)=5.$$
Sample question
Compute $\frac{z_1}{z_2}$ for $z_1=2(\cos 80^\circ+i\sin 80^\circ)$ and $z_2=4(\cos 20^\circ+i\sin 20^\circ)$.
- A$2(\cos 60^\circ+i\sin 60^\circ)$Answer option
- B$0.5(\cos 100^\circ+i\sin 100^\circ)$Answer option
- C$0.5(\cos 60^\circ+i\sin 60^\circ)$Correct answer
- D$0.5(\cos 20^\circ+i\sin 20^\circ)$Answer option
Why this answer is right
Divide moduli and subtract arguments: $r=\frac{2}{4}=0.5$, $\theta=80^\circ-20^\circ=60^\circ$. So $0.5(\cos60^\circ+i\sin60^\circ)$.
In polar form, division scales by $\frac{r_1}{r_2}$ and rotates by $\theta_1-\theta_2$. Here, $$r=\frac{2}{4}=0.5,$$ $$\theta=80^\circ-20^\circ=60^\circ.$$ Thus $$\frac{z_1}{z_2}=0.5(\cos60^\circ+i\sin60^\circ).$$ Adding angles would correspond to multiplication, not division.
Sample question
On $|z|=1$, solve $z+\dfrac{1}{z}=\sqrt{3}$. Which set is correct?
- A$\{e^{i\pi/3},e^{-i\pi/3}\}$Answer option
- B$\{e^{i\pi/4},e^{-i\pi/4}\}$Answer option
- C$\{e^{i\pi/6},e^{-i\pi/6}\}$Correct answer
- D$\{e^{i\pi/2},e^{-i\pi/2}\}$Answer option
Why this answer is right
Write $z=e^{i\theta}$. Then $z+z^{-1}=2\cos\theta=\sqrt{3}$ gives $\cos\theta=\sqrt{3}/2$, so $\theta=\pm\pi/6$.
On the unit circle, set $z=e^{i\theta}$. Then $z+\frac{1}{z}=e^{i\theta}+e^{-i\theta}=2\cos\theta$. Solving $2\cos\theta=\sqrt{3}$ gives $\cos\theta=\frac{\sqrt{3}}{2}$. Hence $\theta=\pm\frac{\pi}{6}$ modulo $2\pi$, so $z=e^{\pm i\pi/6}$.
How this page fits into Clevolab
Clevolab is broader than any one exam mode. GCSE and A-level pages are useful entry points, while the wider project is about sharpening understanding through repeated topic practice.