C

9000034303

Level: 
C
Find the solution set of the following equation in the set of complex numbers. \[ x^{3} + \mathrm{i} = 0 \]
\(\{\mathrm{i},\ \frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i},\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)
\(\{ - 1,\ -\frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i},\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)
\(\{ - 1,\ \frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i},\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)
\(\{\mathrm{i},\ -\frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i},\ -\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\}\)

9000034308

Level: 
C
Two solutions of the equation \[ x^{3} + 1 + \mathrm{i} = 0 \] are \[ \begin{aligned}x_{1}& = \root{6}\of{2}\left (\cos \frac{5} {12}\pi + \mathrm{i}\sin \frac{5} {12}\pi \right ),& \\x_{2}& = \root{6}\of{2}\left (\cos \frac{13} {12}\pi + \mathrm{i}\sin \frac{13} {12}\pi \right ). \\ \end{aligned} \] Find the third solution.
\(x_{3} = \root{6}\of{2}\left (\cos \frac{21} {12}\pi + \mathrm{i}\sin \frac{21} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{9} {12}\pi + \mathrm{i}\sin \frac{9} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{17} {12}\pi + \mathrm{i}\sin \frac{17} {12}\pi \right )\)
\(x_{3} = \root{6}\of{2}\left (\cos \frac{19} {12}\pi + \mathrm{i}\sin \frac{19} {12}\pi \right )\)

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.
\((b^{2} - 4ac\geq 0\text{ and }a\not = 0)\text{ or }(a = 0\text{ and }b\not = 0)\text{ or }(a = b = c = 0)\)
\(a\not = 0\text{ and }b^{2} - 4ac\geq 0\)
\(b^{2} - 4ac\leq 0\)
\((b^{2} - 4ac\geq 0\text{ and }a\not = 0)\text{ or }(a = 0)\text{ or }(b = 0)\)