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
9000065904
Level:
B
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int \frac{x^{3} + 2x}
{x^{2}} \, \text{d}x
\]
\(\frac{1}
{2}x^{2} + 2\ln |x| + c,\ c\in \mathbb{R}\)
\(x +\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{1}
{4}x^{4} + 4x^{2} +\ln |x^{2}| + c,\ c\in \mathbb{R}\)
\(2x^{2} + 2 +\ln |x^{2}| + c,\ c\in \mathbb{R}\)
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 \(\mathbb{R}\).
\[
\int x\mathrm{e}^{x}\, \mathrm{d}x
\]
\(x\mathrm{e}^{x} -\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(x^{2}\mathrm{e}^{x} - 2x\mathrm{e}^{x} + 2\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(2x^{3}\mathrm{e}^{x} - x\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(\frac{1}
{2}x^{2}\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
9000065901
Level:
B
Evaluate the following integral on the interval \((-1;+\infty)\).
\[
\int \frac{1}
{x + 1}\, \text{d}x
\]
\(\ln |x + 1| + c,\ c\in \mathbb{R}\)
\(\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{1}
{x} + c,\ c\in \mathbb{R}\)
\(-\frac{1}
{2}(x + 1)^{-2} + c,\ c\in \mathbb{R}\)
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.
\(x = 2\)
\(x = 1\)
\(x =\log 2\)
\(x = 10\)
\(x = 100\)
9000065903
Level:
B
Evaluate the following integral on the interval \((-6;+\infty)\).
\[
\int \frac{1}
{6x + 36}\, \text{d}x
\]
\(\frac{1}
{6}\ln |x + 6| + c,\ c\in \mathbb{R}\)
\(-\frac{1}
{2}(6x + 36)^{-2} + c,\ c\in \mathbb{R}\)
\(6\ln |x + 6| + c,\ c\in \mathbb{R}\)
\(12x^{2} + 36x + c,\ c\in \mathbb{R}\)
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 \(\frac{z_{1}}
{z_{2}} \).
\(- 2\mathrm{i}\)
\(4\mathrm{i}\)
\(\mathrm{i}\)
\(-\frac{1}
{2}\mathrm{i}\)
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 $n$.
\(n = 13\)
\(n = 14\)
\(n = 15\)
\(n = 16\)
9000065504
Level:
B
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int (1 -\sqrt{x})(1 + \sqrt{x})\, \mathrm{d}x
\]
\(x -\frac{1}
{2}x^{2} + c,\ c\in \mathbb{R}\)
\((x -\frac{1}
{2}x^{2})(x + \frac{1}
{2}x^{2}) + c,\ c\in \mathbb{R}\)
\(x -\frac{1}
{2}x^{\frac{1}
{2} } + c,\ c\in \mathbb{R}\)
\((x -\frac{1}
{2}x^{-\frac{1}
{2} })(x + \frac{1} {2}x^{-\frac{1}
{2} }) + c,\ c\in \mathbb{R}\)