9000031201
Level:
A
Given complex numbers \(z_{1} = 1 - 2\mathrm{i}\)
and \(z_{2} = 3 + 5\mathrm{i}\),
find \(z_{1}z_{2}\).
\(13 -\mathrm{i}\)
\(13 + \mathrm{i}\)
\(- 7 -\mathrm{i}\)
\(13 + 11\mathrm{i}\)
A
9000031206
Level:
A
Find the opposite number to the complex number
\(z = \frac{1+\mathrm{i}}
{1-\mathrm{i}}\).
\(-\mathrm{i}\)
\(1\)
\(- 1\)
\(\mathrm{i}\)
9000031203
Level:
A
Find the real part of the complex number
\(z = \frac{2-\mathrm{i}}
{2+\mathrm{i}}\).
\(0.6\)
\(0.8\)
\(- 0.8\)
\(1\)
9000031202
Level:
A
Find the imaginary part of the complex number
\(z = \frac{2+\mathrm{i}}
{1-\mathrm{i}}\).
\(\frac{3}
{2}\)
\(-\frac{3}
{2}\)
\(\frac{1}
{2}\)
\(-\frac{1}
{2}\)
9000031204
Level:
A
Find the absolute value of the complex number
\(z = \frac{2-\mathrm{i}}
{2+\mathrm{i}}\).
\(1\)
\(5\)
\(\frac{\sqrt{7}}
{5} \)
\(\frac{\sqrt{5}}
{5} \)
9000031205
Level:
A
Find the complex conjugate of \(z = \mathrm{i}^{5} - 3\mathrm{i}^{10}\).
\(3 -\mathrm{i}\)
\(- 3 -\mathrm{i}\)
\(- 3 + \mathrm{i}\)
\(3 + \mathrm{i}\)
9000031104
Level:
A
Solve the following system of equations and identify a correct statement.
\[\begin{aligned}
\frac{x}
{y + 1} - \frac{2}
{x + 1} & = 0 & &
\\\frac{y}
{x} + \frac{2}
{x} & = -1 & &
\end{aligned}\]
The system has a unique solution.
The system does not have any solution.
The system has two solutions.
The system has infinitely many solutions.
9000031105
Level:
A
Solve the following system of equations and identify a correct statement.
\[\begin{aligned}
\frac{1}
{x + 1} -\frac{1}
{y} = 0 & &
\\y^{2} = 1 & &
\end{aligned}\]
The system has two solutions.
The system does not have any solution.
The system has a unique solution.
The system has infinitely many solutions.
9000028105
Level:
A
Given graph of the linear function \(g\),
find the solution set of the inequality \(g(x)\leq 0\).
\([ 6;\infty )\)
\(\mathbb{R}\)
\((-\infty ;2.4] \)
\((-\infty ;-2.3] \)
9000028306
Level:
A
Find the sum of all real solutions of the following equation.
\[
\left (3 - x\right )\left (x^{2} - 4\right ) = 0
\]
\(3\)
\(0\)
\(2\)
\(5\)