1003082402
Level:
B
Find the argument of the complex number \( 4\left(\cos\frac{\pi}3+\mathrm{i}\cdot\sin\frac{\pi}3\right) \).
\( \frac{\pi}3 \)
\( \frac{4\pi}3 \)
\( 4 \)
\( \frac{5\pi}3 \)
Complex numbers in algebraic and polar form
1003082401
Level:
B
Find the argument of the complex number \(\sqrt2\left(\cos160^{\circ}+\mathrm{i}\cdot\sin160^{\circ}\right) \).
\( 160^{\circ} \)
\( 200^{\circ} \)
\( \sqrt{2} \)
\( 340^{\circ} \)
1103079904
Level:
A
Given the complex numbers \( u = 1 + 2\mathrm{i} \) and \( v = 2 -\mathrm{i} \), choose the diagram which shows the complex number \( z \), such that \( z = u^2 - v^2 \).
1103079903
Level:
A
The diagram shows in red all the complex numbers \( z \) such that:
\( |z- 1 + \mathrm{i}| = 2 \)
\( |z- 1 - \mathrm{i}| = 2 \)
\( |z + 1 - \mathrm{i}| = 2 \)
\( |z + 1 + \mathrm{i}| = 2 \)
1103079902
Level:
A
The diagram shows in red all the complex numbers \( z \) such that:
\( |z + 1 + 2\mathrm{i}| < 1 \)
\( |z - 1 - 2\mathrm{i}| < 1 \)
\( |z + 1 - 2\mathrm{i}| < 1 \)
\( |z - 1 + 2\mathrm{i}| < 1 \)
1103079901
Level:
A
The diagram shows in red all the complex numbers \( z \) such that:
\( |z + \mathrm{i}| \geq 2 \)
\( |z - \mathrm{i}| \geq 2 \)
\( |z + 1| \geq 2 \)
\( |z - 1| \geq 2 \)
1103067710
Level:
A
Assuming \( |z - 1| \leq 2 \), choose a diagram, which shows in red all the complex numbers \( z \).
1103067709
Level:
A
Choose a diagram, which shows in red all the complex numbers whose absolute value is three.
1003067708
Level:
A
Let \( z=\frac{1-\mathrm{i}}{1+\mathrm{i}}+\frac{1+\mathrm{i}}{1-\mathrm{i}} \). If \( z \) is a complex number, find its absolute value.
\( 0 \)
\( 2 \)
\( 4 \)
Number \( z \) is not a complex number.
1003067707
Level:
A
Find the complex conjugate of the following complex number.
\[ \frac{2-\mathrm{i}}{2+\mathrm{i}}-\frac{2+\mathrm{i}}{2-\mathrm{i}} \]
\( \frac85\mathrm{i} \)
\( -\frac85\mathrm{i} \)
\( \frac83\mathrm{i} \)
\( -\frac83\mathrm{i} \)