9000035801
Level:
A
Find the complex conjugate of the following complex number.
\[
\mathrm{i} + 3\mathrm{i}(2 -\mathrm{i})^{2} - 4(1 -\mathrm{i})^{3}
\]
\(20 - 18\mathrm{i}\)
\(20 - 24\mathrm{i}\)
\(20 + 18\mathrm{i}\)
\(- 8 + 26\mathrm{i}\)
A
9000035803
Level:
A
Given the complex number \(z = -1 + 2\mathrm{i}\),
find the imaginary part of the complex number
\(\frac{1}
{z}\).
\(-\frac{2}
{5}\)
\(\frac{1}
{2}\)
\(\frac{2}
{5}\)
\(-\frac{1}
{2}\)
9000035804
Level:
A
Find the algebraic form of the following complex number. By \(\overline{z }\) the complex conjugate of \(z \) is denoted.
\[
\overline{\overline{(2 + \mathrm{i}) }\; \overline{(3 + 2\mathrm{i}) } }
\]
\(4 + 7\mathrm{i}\)
\(8 + 7\mathrm{i}\)
\(8 - 7\mathrm{i}\)
\(4 - 7\mathrm{i}\)
9000034706
Level:
A
Consider the inequality
\[
px^{2} - 2x + 2 > 0
\]
with the real parameter \(p\).
Solve this inequality for \(p = 0\).
\((-\infty ;1)\)
\((-\infty ;-1)\)
\((-1;\infty )\)
\((1;\infty )\)
9000034804
Level:
A
Find the absolute value of the complex number
\(z = 3 -\mathrm{i}\).
\(\sqrt{10}\)
\(2\)
\(2\sqrt{2}\)
\(\sqrt{2}\)
9000034708
Level:
A
Consider the equation
\[
2x^{2} + 5px + 2 = 0
\]
with the real parameter \(p\).
Solve the equation for \(p = -\frac{4}
{5}\).
\(\left \{1\right \}\)
\(\left \{-1\right \}\)
\(\left \{0\right \}\)
\(\emptyset \)
9000034901
Level:
A
Find the domain of the following expression.
\[
\sqrt{\left (2x - 3 \right ) \left (3x + 1 \right )}
\]
\(\left (-\infty ;-\frac{1}
{3}\right ] \cup \left [ \frac{3}
{2};\infty \right )\)
\(\left [ -\frac{1}
{3}; \frac{3}
{2}\right ] \)
\(\left (-\frac{1}
{3}; \frac{3}
{2}\right )\)
\(\left (-\infty ;-\frac{1}
{3}\right )\cup \left (\frac{3}
{2};\infty \right )\)
9000034710
Level:
A
Solve the following equation with a real parameter
\(t\), assuming
\(t\neq - 1\) and
\(t\neq 1\).
\[
x(t^{2} - 1) = t - 1
\]
\(\left \{ \frac{1}
{t+1}\right \}\)
\(\emptyset \)
\(\mathbb{R}\)
\(\left \{0\right \}\)
9000034707
Level:
A
Consider the equation
\[
x^{2}(1 - q) + 2x + 1 + q = 0
\]
with the real parameter \(q\).
Solve the equation for \(q = 3\).
\(\left \{-1;2\right \}\)
\(\left \{1\right \}\)
\(\left \{-2\right \}\)
\(\emptyset \)
9000034903
Level:
A
Find all \(x\in \mathbb{R}\)
for which the following expression is undefined.
\[
\sqrt{\left (3x + 4 \right ) \left (\frac{1}
{5} - x\right )}
\]
\(\left (-\infty ;-\frac{4}
{3}\right )\cup \left (\frac{1}
{5};\infty \right )\)
\(\left [ -\frac{4}
{3}; \frac{1}
{5}\right ] \)
\(\left (-\infty ;-\frac{4}
{3}\right ] \cup \left [ \frac{1}
{5};\infty \right )\)
\(\left (-\frac{4}
{3}; \frac{1}
{5}\right )\)