B

9000033806

Level: 
B
In the following list identify a true statement for the function \(i\colon y =\mathop{\mathrm{tg}}\nolimits x\), \(x\in \left ( \frac{\pi }{2}, \frac{3\pi } {2}\right )\).
The function \(i\) is increasing.
The function \(i\) is decreasing.
The function \(i\) is neither increasing nor decreasing.

9000033807

Level: 
B
In the following list identify a true statement about the function \(f(x) =\cos x\) on the interval \(I = \left (-\frac{\pi }{2}, \frac{\pi } {2}\right )\).
The function \(f\) has a unique maximum and no minimum on \(I\).
The function \(f\) does not have a minimum or maximum on \(I\).
The function \(f\) has a unique maximum and a unique minimum on \(I\).
The function \(f\) has a unique minimum and no maximum on \(I\).

9000033808

Level: 
B
In the following list identify a true statement for the function \(f\colon y =\sin x\) on the interval \(I = \left (-\frac{\pi }{2}, \frac{\pi } {2}\right )\).
The function \(f\) does not have a minimum or maximum on \(I\).
The function \(f\) has a unique maximum and a unique minimum on \(I\).
The function \(f\) has a unique maximum and no minimum on \(I\).
The function \(f\) has a unique minimum and no maximum on \(I\).

9000034807

Level: 
B
Find the polar form of the complex number \(z = 2\mathrm{i}\).
\(2\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\)
\(2\left (\cos 0 + \mathrm{i}\sin 0\right )\)

9000034301

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

9000034302

Level: 
B
Find the solution set of the following equation in the set of complex numbers. \[ x^{3} + 8 = 0 \]
\(\{ - 2,\ 1 + \mathrm{i}\sqrt{3},\ 1 -\mathrm{i}\sqrt{3}\}\)
\(\{ - 2,\ -1 + \mathrm{i}\sqrt{3},\ -1 -\mathrm{i}\sqrt{3}\}\)
\(\{ - 2,\ \frac{1} {2} + \mathrm{i}\frac{\sqrt{3}} {2} ,\ \frac{1} {2} -\mathrm{i}\frac{\sqrt{3}} {2} \}\)
\(\{ - 2,\ -\frac{1} {2} + \mathrm{i}\frac{\sqrt{3}} {2} ,\ -\frac{1} {2} -\mathrm{i}\frac{\sqrt{3}} {2} \}\)

9000034701

Level: 
B
Identify a set of the values of the real parameter \(m\) which ensure that the equation \[ \frac{m} {x} - 8 = \frac{1} {x} -\frac{m + 3} {2} \] has solution \(x = 2\).
\(\left \{7\right \}\)
\(\left \{10\right \}\)
\(\left \{6\right \}\)
\(\left \{\frac{5} {2}\right \}\)

9000034304

Level: 
B
Find the solution set of the following equation in the set of complex numbers. \[ x^{4} - 1 = 0 \]
\(\{1,\ -1,\ \mathrm{i},\ -\mathrm{i}\}\)
\(\{1 -\mathrm{i},\ -1 -\mathrm{i}\}\)
\(\{1 + \mathrm{i},\ -1 + \mathrm{i}\}\)
\(\{1 + \mathrm{i},\ 1 -\mathrm{i},\ -1 + \mathrm{i},\ -1 -\mathrm{i}\}\)