9000141910
Level:
A
Given the function \(h\),
find \(\lim _{x\to -\infty }h(x)\).
\[
h(x)=\begin{cases}
-\frac1{x-1} & \text{if } x< 1,\\
-(x-1)^2+2 & \text{if } x\geq 1
\end{cases}
\]
\(0\)
\(2\)
\(\infty \)
\(-\infty \)
Does not exist
A
9000146701
Level:
A
Expand the following polynomial.
\[
2 - (2x + 1) + x(5 - 2x) - 3(x - 2)
\]
\(- 2x^{2} + 7\)
\(- 2x^{2} + 9\)
\(- 2x^{2} - 3\)
\(- 2x^{2} - 5\)
9000146702
Level:
A
Expand the following expression.
\[
a - 4(2 - a) - a(5a + 1) + 2a(3 - 2a)
\]
\(- 9a^{2} + 10a - 8\)
\(- 9a^{2} + 12a - 8\)
\(- 9a^{2} + 2a - 8\)
\(- 9a^{2} + 4a - 8\)
9000146203
Level:
A
Expand the following expression.
\[
\left (x^{5} -\sqrt{2}y\right )^{2}
\]
\(x^{10} - 2\sqrt{2}x^{5}y + 2y^{2}\)
\(x^{10} -\sqrt{2}x^{5}y + 2y^{2}\)
\(x^{10} - 2\sqrt{2}x^{5}y - 2y^{2}\)
\(x^{10} -\sqrt{2}x^{5}y - 2y^{2}\)
9000139708
Level:
A
The shelf contains \(15\)
books. From this amount, \(9\)
books are in English and \(6\)
books in other languages. Find the number of possibilities how to rearrange the
books on the shelf, if all English books have to be on the left and the other on the
right.
\(9!\, 6!=261\:273\:600\)
\(9^{6}=531\:441\)
\(\frac{9!}
{6!}=504\)
\(\frac{9!}
{6!\, 3!}=84\)
9000140003
Level:
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\).
\[ax - \frac{2} {a^{2}} = \frac{4x+1} {a} \]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=-2 & \mathbb{R} \\
a=2 & \emptyset \\
a\notin\{-2,0,2\} & \left\{\frac1{a(a-2)}\right\}
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=-2 & \mathbb{R}\setminus\{1\} \\
a=2 & \emptyset \\
a\notin\{-2,0,2\} & \left\{\frac1{a(a-2)}\right\}
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=-2 & \emptyset \\
a=2 & \mathbb{R} \\
a\notin\{-2,0,2\} & \left\{\frac1{a(a-2)}\right\}
\\\hline \end{array}\)
9000139507
Level:
A
The average mass of five melons is \(2\: 400\, \mathrm{g}\).
We have to add another melon such that the new average value of all six melons will
be \(2\: 420\, \mathrm{g}\).
Find the mass of the sixth melon.
\(2\: 520\, \mathrm{g}\)
\(2\: 540\, \mathrm{g}\)
\(2\: 480\, \mathrm{g}\)
\(2\: 460\, \mathrm{g}\)
9000139304
Level:
A
Find the number of possibilities how to choose a pair of timekeepers for a sport event if
there are \(50\)
candidates available for this job.
\(\frac{50!}
{2!\, 48!} = 1\:225\)
\(\frac{50}
{2} = 25\)
\(50^{2} = 2\:500\)
\(\frac{50!}
{48!} = 2\:450\)
9000139501
Level:
A
Ten apples in a box have average mass
\(200\, \mathrm{g}\). We remove one
apple of the mass \(200\, \mathrm{g}\)
from the box. What is the change in the average mass of the apples from the box?
The average mass of the apples does not change.
The average mass of the apples decreases by
\(20\, \mathrm{g}\).
The average mass of the apples increases by
\(20\, \mathrm{g}\).
There is not enough information to solve this problem.
9000139305
Level:
A
There are five rooms with three beds and one room with five beds in a hotel. A group of
\(20\)
people booked rooms at this hotel. Determine the number of possible choices for the
people to the five-bed room.
\(\frac{20!}
{5!\, 15!}=15\:504\)
\(20\cdot 3\cdot 5=300\)
\(\frac{20!}
{3!\, 5!}=3\:379\:030\:566\:912\:000\)
\(20^{5}=3\:200\:000\)