1103079901
Level:
A
The diagram shows in red all the complex numbers \( z \) such that:
\( |z + \mathrm{i}| \geq 2 \)
\( |z - \mathrm{i}| \geq 2 \)
\( |z + 1| \geq 2 \)
\( |z - 1| \geq 2 \)
Complex numbers in algebraic and polar form
1103079902
Level:
A
The diagram shows in red all the complex numbers \( z \) such that:
\( |z + 1 + 2\mathrm{i}| < 1 \)
\( |z - 1 - 2\mathrm{i}| < 1 \)
\( |z + 1 - 2\mathrm{i}| < 1 \)
\( |z - 1 + 2\mathrm{i}| < 1 \)
1103079903
Level:
A
The diagram shows in red all the complex numbers \( z \) such that:
\( |z- 1 + \mathrm{i}| = 2 \)
\( |z- 1 - \mathrm{i}| = 2 \)
\( |z + 1 - \mathrm{i}| = 2 \)
\( |z + 1 + \mathrm{i}| = 2 \)
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 \( z = u^2 - v^2 \).
2010010401
Level:
A
Given complex numbers \(z_{1} = 10 -7\mathrm{i}\)
and \(z_{2} = 5- 4\mathrm{i}\),
find \(z_{1} - z_{2}\).
\(5 -3 \mathrm{i}\)
\(5-11 \mathrm{i}\)
\(3 -5 \mathrm{i}\)
\(3 - \mathrm{i}\)
2010010402
Level:
A
Given complex numbers
\[
z_1 = \sqrt{5} - \sqrt{3}\mathrm{i},\quad z_2 = \sqrt{5} +\sqrt{3}\mathrm{i},
\]
find \(z_1 \cdot z_2\).
\(8\)
\(2\)
\(5-3 \mathrm{i}\)
\(5+3\mathrm{i}\)
2010010403
Level:
A
Given complex numbers
\[
a = \sqrt{2} + \mathrm{i},\ \quad b = {2} -\sqrt{3}\mathrm{i},\
\]
find \(\frac{a}
{b}\).
\(\frac{2\sqrt{2}-\sqrt{3}}
{7} + \mathrm{i}\frac{\sqrt{6}+2}
{7} \)
\(\frac{2\sqrt{2}-\sqrt{3}}
{7} - \mathrm{i}\frac{\sqrt{6}+2}
{7} \)
\(\frac{2\sqrt{2}+\sqrt{3}}
{4} + \mathrm{i}\frac{\sqrt{6}-2}
{4} \)
\(\frac{\sqrt{2}-\sqrt{6}}
{3} + \mathrm{i}\frac{\sqrt{3}+2}
{3} \)
2010010404
Level:
A
Find the complex conjugate of \(z = \mathrm{i}^{10} +3\mathrm{i}^{5}\).
\(-1 -3\mathrm{i}\)
\(-1 +3\mathrm{i}\)
\(-3-\mathrm{i}\)
\(3-\mathrm{i}\)
9000031201
Level:
A
Given complex numbers \(z_{1} = 1 - 2\mathrm{i}\)
and \(z_{2} = 3 + 5\mathrm{i}\),
find \(z_{1}z_{2}\).
\(13 -\mathrm{i}\)
\(13 + \mathrm{i}\)
\(- 7 -\mathrm{i}\)
\(13 + 11\mathrm{i}\)
9000031202
Level:
A
Find the imaginary part of the complex number
\(z = \frac{2+\mathrm{i}}
{1-\mathrm{i}}\).
\(\frac{3}
{2}\)
\(-\frac{3}
{2}\)
\(\frac{1}
{2}\)
\(-\frac{1}
{2}\)