C

9000020908

Level: 
C
Assuming that the real parameter \(c\) satisfies \(c > 16\), solve the system and identify a true statement. \[ \begin{alignedat}{80} &y^{2} & - &4x & & = 0 & & & & & & \\8 &x & - &4y & + c & = 0 & & & & & & \\\end{alignedat}\]
The system has no solution.
The system has two solutions.
The system has a unique solution.
The system has infinitely many solutions.

9000020904

Level: 
C
Determine all the values of the parameter \(c\in \mathbb{R}\) so that the following system has two solutions in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &y^{2} & = 2 & & & & & & \\ &x & + &c & = y & & & & & & \\\end{alignedat}\]
\(|c| < 2\)
\(|c| = 2\)
\(|c| > 2\)
\(c = 2\)

9000020905

Level: 
C
Find the condition on the parameter \(c\in \mathbb{R}\) which ensures that the following system has a unique solution in \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &y^{2} & = 2 & & & & & & \\ &x & + &c & = y & & & & & & \\\end{alignedat}\]
\(|c| = 2\)
\(|c| > 2\)
\(|c| < 2\)
\(c = 2\)

9000018010

Level: 
C
The salary of Peter has been increased by \(\$2\: 400\). The salary of Jane increased by \(3\, \%\) and this increase has been bigger than the increase of the Peter's salary. In the following list identify a possible old salary of Jane.
\(\$81\: 000\)
\(\$80\: 000\)
\(\$9\: 000\)
\(\$8\: 000\)

9000018110

Level: 
C
Consider a coil composed from a ferrite core and a wire winding. The mass of the core is \(2\, \mathrm{kg}\). The material for the wire is such that the mass of the wire of the length \(30\, \mathrm{m}\) is bigger than the mass of the wire of the length \(10\, \mathrm{m}\) together with the ferrite core. In the following list identify a possible mass of one meter of the wire.
\(110\, \mathrm{g}\)
\(100\, \mathrm{g}\)
\(0.01\, \mathrm{kg}\)
\(0.09\, \mathrm{kg}\)

9000009901

Level: 
C
The picture shows parts of the graphs of the functions \[ \text{$f(x)= \frac{k_{1}} {x} $ and $g(x) = \frac{k_{2}} {x} $.} \] Find the relationship between \(k_{1}\) and \(k_{2}\)?
\(k_{1} > k_{2}\)
\(k_{1} < k_{2}\)
\(k_{1} = k_{2}\)
No conclusion is possible, more of the above possibilities may occur.

9000010609

Level: 
C
Identify a function which is a possible inverse to the function graphed in the picture.
\(y = x^{-1}\), \(x\in (0;\infty )\)
\(y = x\), \(x\in (0;\infty )\)
\(y = -x\), \(x\in (0;\infty )\)
\(y = -x^{-1}\), \(x\in (0;\infty )\)
\(y = x^{2}\), \(x\in (0;\infty )\)
\(y = -x^{2}\), \(x\in (0;\infty )\)

9000009909

Level: 
C
Consider the system \[\begin{aligned} y & = \frac{k} {x}, & & \\y & = a, & & \end{aligned}\] where \(a\), \(k\) are real parameters and \(x\), \(y\) are real variables. Determine the conditions for \(a\) and \(k\) so that the system has a unique solution in \(\mathbb{R}^{-}\times \mathbb{R}^{-}\).
\(a < 0\) and \(k > 0\)
\(a < 0\) and \(k < 0\)
\(a > 0\) and \(k < 0\)
\(a > 0\) and \(k > 0\)

9000010610

Level: 
C
Identify a function which is a possible inverse to the function graphed in the picture.
\(y = x^{2}\), \(x\in (-\infty ;0] \)
\(y = x^{-2}\), \(x\in (-\infty ;0] \)
\(y = -x^{2}\), \(x\in [ 0;\infty )\)
\(y = x^{\frac{1} {2} }\), \(x\in [ 0;\infty )\)
\(y = -x^{\frac{1} {2} }\), \(x\in [ 0;\infty )\)
\(y = -2x\), \(x\in (-\infty ;0] \)

9000010608

Level: 
C
Identify a function which is a possible inverse to the function graphed in the picture.
\(y = x^{3}\), \(x\in (-\infty ;\infty )\)
\(y = x^{-3}\), \(x\in (-2;2)\)
\(y = x^{\frac{1} {3} }\), \(x\in (0;\infty )\)
\(y = -x^{\frac{1} {3} }\), \(x\in (-\infty ;\infty )\)
\(y = 8x\), \(x\in (-\infty ;\infty )\)
\(y = -4x\), \(x\in (-\infty ;\infty )\)