9000068703
Level:
B
Identify the real number \(x\)
which converts the numbers \(a_{1} = -x\),
\(a_{2} = -5\) and
\(a_{3} = 5\) into
three consecutive numbers of a geometric series.
\(x = -5\)
\(x = 0\)
\(x = 5\)
\(x = -10\)
\(x = 10\)
B
9000066002
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int x\sin x\, \mathrm{d}x
\]
\(- x\cos x +\sin x + c,\ c\in \mathbb{R}\)
\(- x\cos x -\sin x + c,\ c\in \mathbb{R}\)
\(x\cos x +\sin x + c,\ c\in \mathbb{R}\)
\(x\cos x -\sin x + c,\ c\in \mathbb{R}\)
9000066008
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int x\mathrm{e}^{x}\, \mathrm{d}x
\]
\(x\mathrm{e}^{x} -\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(x^{2}\mathrm{e}^{x} - 2x\mathrm{e}^{x} + 2\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(2x^{3}\mathrm{e}^{x} - x\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(\frac{1}
{2}x^{2}\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
9000068708
Level:
B
Identify the real number \(x\)
which converts the numbers \(a_{1} = 2^{x-4}\),
\(a_{2} = 1\) and
\(a_{3} = 2^{x}\) into
three consecutive terms of a geometric series.
\(x = 2\)
\(x = 1\)
\(x =\log 2\)
\(x = 10\)
\(x = 100\)
9000070110
Level:
B
Given \(z_{1} = 4\left (\cos \frac{5}
{3}\pi + \mathrm{i}\sin \frac{5}
{3}\pi \right )\)
and \(z_{2} = 2\left (\cos \frac{1}
{6}\pi + \mathrm{i}\sin \frac{1}
{6}\pi \right )\),
evaluate \(\frac{z_{1}}
{z_{2}} \).
\(- 2\mathrm{i}\)
\(4\mathrm{i}\)
\(\mathrm{i}\)
\(-\frac{1}
{2}\mathrm{i}\)
9000069907
Level:
B
Find the quadratic equation with real valued coefficients and one of the solutions
\(x_{1} = -5 + \mathrm{i}\).
\(x^{2} + 10x + 26 = 0\)
\(x^{2} - 10x + 26 = 0\)
\(x^{2} - 10x - 24 = 0\)
\(x^{2} + 10x + 24 = 0\)
9000069908
Level:
B
One of the solutions of the quadratic equation
\[
2x^{2} + px + 5 = 0
\]
with a real parameter \(p\)
is
\[
x_1 = -1 + \frac{\sqrt{6}}
{2} \mathrm{i}.
\]
Find the value of \(p\).
\(4\)
\(- 4\)
\(8\)
\(- 8\)
9000065506
Level:
B
Evaluate the following integral on the interval \((0,+\infty)\).
\[
\int \frac{x^{2}}
{\sqrt{x}}\, \mathrm{d}x
\]
\(\frac{2}
{5}x^{2}\sqrt{x} + c,\ c\in \mathbb{R}\)
\(\frac{2\sqrt{x}}
{x} + c,\ c\in \mathbb{R}\)
\(\frac{2}
{5}x\sqrt{x} + c,\ c\in \mathbb{R}\)
\(\frac{\sqrt{x}}
{x} + c,\ c\in \mathbb{R}\)
9000065307
Level:
B
In the arithmetic sequence given by the first term
\(a_{1} = 4\) and the common
difference \(d = 2\)
find the sum of the first twelve terms of the sequence.
\(s_{12} = 180\)
\(s_{12} = 72\)
\(s_{12} = 120\)
\(s_{12} = 168\)
9000065308
Level:
B
For the arithmetic sequence with the first term $a_1=3$ holds $a_n=27$ and the sum of the first $n$ terms is $195$. Find $n$.
\(n = 13\)
\(n = 14\)
\(n = 15\)
\(n = 16\)