C

9000022901

Level: 
C
An arrow has been shot at the angle \(60^{\circ }\) at the speed \(10\, \mathrm{m}\, \mathrm{s}^{-1}\). Find the time when the height equals to the horizontal distance from the take-off point. Hint: The position is given by the equations \(x = v_{0}t\cdot \cos \alpha \), \(y = v_{0}t\cdot \sin \alpha -\frac{1} {2}gt^{2}\). Use \(g = 10\, \mathrm{m}\, \mathrm{s}^{-2}\) as an acceleration of gravity.
\(\left (\sqrt{3} - 1\right )\, \mathrm{s}\)
\(\left (\sqrt{3} + 1\right )\, \mathrm{s}\)
\(\sqrt{3}\, \mathrm{s}\)
\(\left (\sqrt{2} - 1\right )\, \mathrm{s}\)
\(\left (\sqrt{2} + 1\right )\, \mathrm{s}\)

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 \(\mathbb{R}\times \mathbb{R}\). \[ \begin{alignedat}{80} &x^{2} & + &y^{2} & = 2 & & & & & & \\ &x & + &c & = y & & & & & & \\\end{alignedat}\]
\(|c| = 2\)
\(|c| > 2\)
\(|c| < 2\)
\(c = 2\)

9000021801

Level: 
C
Solve the following system of inequalities. \[\begin{aligned} \frac{1} {3}(2x + 5) &\geq 0.5\left (\frac{2 + 3x} {2} + 2\right ) & & \\0.2(3 - 2x) &\leq \frac{1} {3}\left (\frac{4 - 2x} {5} + 2\right ) & & \end{aligned}\]
\(x\in \left [ -\frac{5} {4};2\right ] \)
\(x\in [ 2;\infty )\)
\(x\in \left (-\infty ;-\frac{5} {4}\right ] \)
\(x\in \emptyset \)

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.
\(110\, \mathrm{g}\)
\(100\, \mathrm{g}\)
\(0.01\, \mathrm{kg}\)
\(0.09\, \mathrm{kg}\)

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.
\(\$81\: 000\)
\(\$80\: 000\)
\(\$9\: 000\)
\(\$8\: 000\)

9000010608

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

9000009304

Level: 
C
A tank contains \(1\: 000\) litres of petrol. The petrol escapes at a constant speed \(20\) litres per minute. In what time will there be just \(200\) litres of the petrol in the tank?
\(40\, \mathrm{min}\)
\(10\, \mathrm{min}\)
\(20\, \mathrm{min}\)
\(30\, \mathrm{min}\)