9000141901
Level:
A
Given the function \(f\),
find \(\lim _{x\to 1}f(x)\).
\[
f(x)=\begin{cases}
x^3+1 & \text{if } x\neq 1,\\
3 & \text{if } x = 1
\end{cases}
\]
\(2\)
\(3\)
\(1\)
Does not exist
A
9000141906
Level:
A
Given the function \(g\) (see the picture),
find \(\lim _{x\to \infty }g(x)\).
\[
g(x)=\begin{cases}
-\frac12(x-1)^2+2 & \text{if } x < 1,\\
\frac2{x^2}+1 & \text{if } x \geq 1
\end{cases}
\]
\(1\)
\(0\)
\(\infty \)
\(-\infty \)
This limit does not exist.
9000141902
Level:
A
Given the function \(f\),
find \(\lim _{x\to \infty }f(x)\).
\[
f(x)=\begin{cases}
x^3+1 & \text{if } x\neq 1,\\
3 & \text{if } x = 1
\end{cases}
\]
\(\infty \)
\(-\infty \)
\(4\)
Does not exist
9000141907
Level:
A
Given the function \(h\) (see the picture),
find \(\lim _{x\to 1^{-}}h(x)\).
\[
h(x)=\begin{cases}
-\frac1{x-1} & \text{if } x< 1,\\
-(x-1)^2+2 & \text{if } x\geq 1
\end{cases}
\]
\(\infty \)
\(1\)
\(2\)
\(-\infty \)
This limit does not exist.
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.
9000140002
Level:
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\).
\[ \frac{x+a} {a} = ax - 1\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a\in\{-1;1\} & \emptyset \\
a\notin\{-1;0;1\} & \left\{\frac{2a}{(a-1)(a+1)}\right\}
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=-1 & \emptyset \\
a\notin\{-1;0\} & \left\{\frac{2a}{(a-1)(a+1)}\right\}
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a\in\{-1;1\} & \mathbb{R} \\
a\notin\{-1;0;1\} & \left\{\frac{2a}{(a-1)(a+1)}\right\}
\\\hline \end{array}\)
9000139504
Level:
A
The average salary of five employees is
\(3\: 000\, \mathrm{Euro}\). This
group of the employees is expanded by one new person. The salary of the new person
is \(2\: 400\, \mathrm{Euro}\).
Find the change in the average salary of this group.
The average salary decreases by \(100\, \mathrm{Euro}\).
The average salary decreases by \(480\, \mathrm{Euro}\).
The average salary increases by \(400\, \mathrm{Euro}\).
The average salary increases by \(480\, \mathrm{Euro}\).
9000140003
Level:
A
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\).
\[ax - \frac{2} {a^{2}} = \frac{4x+1} {a} \]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=-2 & \mathbb{R} \\
a=2 & \emptyset \\
a\notin\{-2;0;2\} & \left\{\frac1{a(a-2)}\right\}
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=-2 & \mathbb{R}\setminus\{1\} \\
a=2 & \emptyset \\
a\notin\{-2;0;2\} & \left\{\frac1{a(a-2)}\right\}
\\\hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline
a=-2 & \emptyset \\
a=2 & \mathbb{R} \\
a\notin\{-2;0;2\} & \left\{\frac1{a(a-2)}\right\}
\\\hline \end{array}\)
9000139506
Level:
A
There are eight mandarins of average mass
\(90\, \mathrm{g}\) in the box. We
got two another mandarins and add them to the box. The new average mass of the mandarins
in the box is \(92\, \mathrm{g}\).
Find the average mass of the two added mandarins.
\(100\, \mathrm{g}\)
\(92\, \mathrm{g}\)
\(96\, \mathrm{g}\)
\(106\, \mathrm{g}\)
9000139303
Level:
A
A DJ's playlist contains \(18\)
songs. In this list there are \(7\)
rap songs, \(5\)
oldies and \(6\)
rock songs. The opening part should consist of one rap song, two oldies and one rock
song. The order of the songs does not matter. Find the number of possible ways how
to put the opening together.
\(420\)
\(120\)
\(320\)
\(520\)