C

9000146210

Level: 
C
Factor the following expression. \[ 64x^{6} + 125 \]
\(\left (4x^{2} + 5\right )\left (16x^{4} - 20x^{2} + 25\right )\)
\(\left (4x^{2} - 5\right )\left (16x^{4} + 20x^{2} + 25\right )\)
\(\left (4x^{2} + 5\right )\left (16x^{3} - 20x^{2} + 25\right )\)
\(\left (4x^{2} - 5\right )\left (16x^{3} + 20x^{2} + 25\right )\)

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.
\(\frac{1} {5}=0{.}2\)
\(\frac{1} {4}=0{.}25\)
\(\frac{6} {36}\doteq 0{.}1667\)
\(\frac{11} {36}\doteq 0{.}3056\)

9000123108

Level: 
C
Find all the tangents to the hyperbola \(x^{2} - 2y^{2} = 8\) such that the angle between each tangent and the \(x\)-axis is \(45^{\circ }\).
\(y = x + 2\text{, }y = x - 2\text{, }y = -x + 2\text{, }y = -x - 2\)
\(y = x + 2\text{, }y = x - 2\)
\(y = x + 2\text{, }y = -x + 2\)
\(y = x + 2\)