B

9000034301

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

9000033805

Level: 
B
In the following list identify a true statement for the function \(h\colon y =\mathop{\mathrm{cotg}}\nolimits x\), \(x\in \left (-\frac{\pi }{2};0\right )\cup \left (0; \frac{\pi } {2}\right )\).
The function \(h\) is neither increasing nor decreasing.
The function \(h\) is increasing.
The function \(h\) is decreasing.

9000034302

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

9000033704

Level: 
B
Find the values of real parameter \(p\) which ensure that the following quadratic equation has solutions with nonzero imaginary part. \[ px^{2} + 4x - p + 5 = 0 \]
\(p\in \left (1;4\right )\)
\(p\in [ 1;4] \)
\(p\in \left (-\infty ;1\right )\cup \left (4;\infty \right )\)
\(p\in \left (-\infty ;1\right ] \cup \left [ 4;\infty \right )\)

9000031210

Level: 
B
Given complex numbers \(z_{1} = 2\sqrt{3}\left (\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\right )\) and \(z_{2} = \sqrt{3}\left (\cos \frac{4\pi } {3} + \mathrm{i}\sin \frac{4\pi } {3}\right )\), find the quotient \(\frac{z_{1}} {z_{2}} \).
\(-\sqrt{3} + \mathrm{i}\)
\(\sqrt{3} -\mathrm{i}\)
\(\sqrt{3} + \mathrm{i}\)
\(-\sqrt{3} -\mathrm{i}\)

9000031209

Level: 
B
Given complex numbers \(z_{1} = 2\sqrt{2}\left (\cos \frac{\pi }{4} + \mathrm{i}\sin \frac{\pi }{4}\right )\) and \(z_{2} = \sqrt{2}\left (\cos \frac{7\pi } {4} + \mathrm{i}\sin \frac{7\pi } {4}\right )\), find the product \(z_{1}z_{2}\).
\(4\)
\(4\mathrm{i}\)
\(- 4\mathrm{i}\)
\(- 4\)