C

9000034309

Level: 
C
Find the angle \(\varphi \) such that the angles in the polar form of any two solutions of the equation \[ x^{5} - 1 + \mathrm{i}\sqrt{3} = 0 \] differ by an integer multiple of \(\varphi \).
\(\varphi = \frac{2} {5}\pi \)
\(\varphi = \frac{3} {5}\pi \)
\(\varphi = \frac{4} {5}\pi \)
\(\varphi =\pi \)

9000033708

Level: 
C
A stone has been thrown vertically up at the velocity \(15\, \mathrm{m}\, \mathrm{s}^{-1}\) from the initial height \(10\, \mathrm{m}\). How long (in seconds) has been the height of the stone at least \(20\, \mathrm{m}\)? Hint: The height \(h\) is given by the expression \(h = s_{0} + v_{0}t -\frac{1} {2}gt^{2}\), the standard acceleration is \(g\mathop{\mathop{\doteq }}\nolimits 10\, \mathrm{m}\, \mathrm{s}^{-2}\).
exactly \(1\, \mathrm{s}\)
less than \(1\, \mathrm{s}\)
more than \(1\, \mathrm{s}\)
The information is not sufficient to give a definite answer.

9000033709

Level: 
C
A square shaped garden with the side \(a\) should be reduced by a length \(x\) to another square garden. The difference between the areas of the gardens should not be bigger than \(25\%\) of the original area. Find the possible values of \(x\).
\(x\leq a -\frac{\sqrt{3}} {2} a\)
\(x\leq \sqrt{3}a\)
\(x\leq \frac{3} {4}a\)
\(x\leq a + \frac{\sqrt{3}} {2} a\)

9000033705

Level: 
C
Find the domain of the following function. \[ f(x) = \sqrt{\log (x^{2 } + 2x + 1)} \]
\(\left (-\infty ;-2] \cup [ 0;\infty \right )\)
\(\mathbb{R}\setminus \left \{-1\right \}\)
\(\left (-1;\infty \right )\)
\(\left (-\infty ;-1\right )\cup \left (1;\infty \right )\)
\(\left (-\infty ;0\right )\cup \left (2;\infty \right )\)

9000028401

Level: 
C
Find the condition which is equivalent to the fact that the equation \(ax^{2} + bx + c = 0\) with \(x\in \mathbb{R}\) and real coefficients \(a\), \(b\), \(c\) has at least one real solution.
\((b^{2} - 4ac\geq 0\text{ and }a\not = 0)\text{ or }(a = 0\text{ and }b\not = 0)\text{ or }(a = b = c = 0)\)
\(a\not = 0\text{ and }b^{2} - 4ac\geq 0\)
\(b^{2} - 4ac\leq 0\)
\((b^{2} - 4ac\geq 0\text{ and }a\not = 0)\text{ or }(a = 0)\text{ or }(b = 0)\)

9000031110

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \left |x - 2\right | & = y & & \\\left |y + 2\right | & = x - 6 & & \end{aligned}\]
The system has no solution.
The system has a unique solution.
The system has two solutions.
The system has more than two solutions.

9000031108

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} 2x^{2} - y = 2 & & \\\left |x\right | + y = 1 & & \end{aligned}\]
The system has two solutions.
The system does not have any solution.
The system has a unique solution.
The system has more than two solutions.

9000031109

Level: 
C
Solve the following system of equations and identify a correct statement. \[\begin{aligned} \left |x\right | = x + y & & \\\left |y\right | = 1 + x & & \end{aligned}\]
The system has a unique solution.
The system does not have any solution.
The system has two solutions.
The system has more than two solutions.