C

9000145401

Level: 
C
Identify a true statement on the function \(f(x) = 2x^{3} + 3x^{2} - 12x - 12\).
The function \(f\) has a local maximum at the point \(x = -2\).
The function \(f\) has a local minimum at the point \(x = -2\)..
The global maximum of \(f\) on \(\mathbb{R}\) is at \(x = -2\).
The global minimum of \(f\) on \(\mathbb{R}\) is at \(x = -2\).

9000140001

Level: 
C
Consider the equation \[ \frac{4a} {x} - \frac{1} {ax} + \frac{2} {a} = 4 \] with unknown \(x\) and a parameter \(a\in \mathbb{R}\setminus \{0\}\). Identify a true statement.
If \(a = \frac{1} {2}\), then the solution is \(x\in \mathbb{R}\setminus \{0\}\).
If \(a = \frac{1} {2}\), then the equation has no solution.
If \(a = \frac{1} {2}\), then the solution is \(x\in \mathbb{R}\).

9000140004

Level: 
C
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\). \[ \frac{a^{2}(x-1)} {ax-3} = 3 \]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a=3 & \mathbb{R}\setminus\{1\} \\ a\notin\{0;3\} & \left\{\frac{a+3}a\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a=3 & \{1\} \\ a\notin\{0;3\} & \left\{\frac{a+3}a\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a\in\{0;3\} & \emptyset \\ a\notin\{0;3\} & \left\{\frac{a+3}a\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=0 & \emptyset \\ a=3 & \mathbb{R} \\ a\notin\{0;3\} & \left\{\frac{a+3}a\right\} \\ \hline \end{array}\)

9000140005

Level: 
C
Solve the following equation with unknown \(x\) and a real parameter \(a\in\mathbb{R}\setminus\{0\}\). \[\frac ax-\frac4{ax}=1-\frac2a\]
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=-2 & \emptyset \\ a=2 & \mathbb{R}\setminus\{0\} \\ a\notin\{-2;0;2\} & \left\{a+2\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \mathbb{R}\setminus\{0\} \\ a\notin\{0;2\} & \left\{a+2\right\} \\ \hline \end{array}\)
\( \begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \mathbb{R} \\ a\notin\{0;2\} & \left\{a+2\right\} \\ \hline \end{array}\)
\(\begin{array}{cc} \hline \text{Parameter} & \text{Solution set}\\ \hline a=2 & \mathbb{R}\setminus\{1\} \\ a\notin\{0;2\} & \left\{a+2\right\} \\ \hline \end{array}\)

9000139710

Level: 
C
The wallet contains nine coins: three \(1\)-Euro coins, three \(2\)-Euro coins and three \(5\)-Euro coins. How many different amounts can be paid if we have to pay the amount exactly and use just three coins for this payment?
\(\frac{5!} {3!\, 2!}=10\)
\(\frac{5!} {3!}=20\)
\(3^{3}=27\)
\(3!=6\)

9000139704

Level: 
C
There are \(5\) different kinds of cakes in a shop. Find the number of possibilities how to buy \(8\) cakes in this shop. (There is more than \(8\) cakes of each kind available.)
\(\frac{12!} {8!\, 4!}=495\)
\(5!\, 8!=4\:838\:400\)
\(5^{8}=390\:625\)
\(\frac{8!} {5!\, 3!}=56\)

9000138305

Level: 
C
Two different dices (a white dice and a black dice) are rolled. We get the sum of the numbers on both dices \(6\). Find the probability that there is an even number on the black dice.
\(\frac{2} {5}=0{.}4\)
\(\frac{5} {36}\doteq 0{.}1389\)
\(\frac{5} {18}\doteq 0{.}2778\)
\(\frac{13} {36}\doteq 0{.}3611\)

9000138308

Level: 
C
Two different dices (a white dice and a black dice) are rolled. The sum of the numbers on both dices is \(8\). Find the probability that there is \(4\) on the black dice.
\(\frac{1} {5}=0{.}2\)
\(\frac{1} {4}=0{.}25\)
\(\frac{6} {36}\doteq 0{.}1667\)
\(\frac{11} {36}\doteq 0{.}3056\)

9000124502

Level: 
C
A rectangle-shaped land has dimensions \(3\times 5\, \mathrm{cm}\) on a map with scale \(1\colon 2\: 000\). The owner increased the size of his land by buying some land from his neighbor. The new land has dimensions \(4\times 5\, \mathrm{cm}\) on the map. Find the actual increase of the perimeter of the land (i.e. find the increase in the length of the fence required to enclose the whole land). Give your answer in meters.
\(40\, \mathrm{m}\)
\(20\, \mathrm{m}\)
\(80\, \mathrm{m}\)
\(10\, \mathrm{m}\)