B

9000038909

Level: 
B
Consider the function \(f\colon y =\sin \left (\frac{x} {2} + \frac{\pi } {2}\right )\). In the following list identify the function which has the same graph as the graph of the function \(f\).
\(g\colon y =\cos \frac{x} {2} \)
\(k\colon y =\cos \left (\frac{x} {2} + \frac{\pi } {2}\right )\)
\(b\colon y =\cos \left (\frac{x} {2} - \frac{\pi } {2}\right )\)
\(h\colon y =\cos \left (\frac{x} {2} -\pi \right )\)
\(m\colon y =\cos 2x\)

9000037408

Level: 
B
Find the polar form of the following complex number \[z=\frac{1} {\cos \frac{2\pi } {3} +\mathrm{i}\sin \frac{2\pi } {3} }. \]
\(\cos \frac{4\pi } {3} + \mathrm{i}\sin \frac{4\pi } {3}\)
\(\cos \left (-\frac{4\pi } {3}\right ) + \mathrm{i}\sin \left (-\frac{4\pi } {3}\right )\)
\(\cos \frac{3} {2\pi } + \mathrm{i}\sin \frac{3} {2\pi }\)
\(\cos \frac{3} {2\pi } -\mathrm{i}\sin \frac{3} {2\pi }\)

9000038906

Level: 
B
Consider the function \(f\colon y =\mathop{\mathrm{tg}}\nolimits x\). In the following list identify the nonnegative function.
None of the given functions is nonnegative.
\(A\cdot f(x)\), where \(A\in (-\infty ;0)\)
\(A\cdot f(x)\), where \(A\in (0;+\infty )\)
\(f(B\cdot x)\), where \(B\in (0;+\infty )\)
\(f(x + C)\), where \(C\in (-\infty ;0)\)

9000037409

Level: 
B
Find the polar form of the complex number \[z=\frac{1} {\cos \frac{7\pi } {6} +\mathrm{i}\sin \frac{7\pi } {6} }. \]
\(\cos \frac{5\pi } {6} + \mathrm{i}\sin \frac{5\pi } {6}\)
\(\cos \left (-\frac{5\pi } {6}\right ) + \mathrm{i}\sin \left (-\frac{5\pi } {6}\right )\)
\(\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\)
\(\cos \left (-\frac{\pi }{6}\right ) + \mathrm{i}\sin \left (-\frac{\pi }{6}\right )\)

9000038908

Level: 
B
Consider the function \(f\colon y =\mathop{\mathrm{tg}}\nolimits x\) with domain restricted to the interval \(\mathop{\mathrm{Dom}}(f) = \left ( \frac{\pi }{2}; \frac{3\pi } {2}\right )\). In the following list identify the function with domain \((0;\pi )\).
\(f\left (x + \frac{\pi } {2}\right )\)
\(\left ( \frac{\pi }{2}\right )\cdot f(x)\)
\(f\left (x - \frac{\pi } {2}\right )\)
\(f(x) + \frac{\pi } {2}\)
\(f(x) - \frac{\pi } {2}\)

9000038905

Level: 
B
Identify the transformation which transforms the graph of the function \(g(x) =\sin 3x\) to the graph of the function \(f(x) =\sin (3x + 5)\).
Shift of graph of \(g\) by \(\frac{5} {3}\) of a unit to the left.
Shift of graph of \(g\) by \(5\) units to the right.
Shift of graph of \(g\) by \(5\) units to the left.
Shift of graph of \(g\) by \(3\) units to the right.
Shift of graph of \(g\) by \(3\) units to the left.
Shift of graph of \(g\) by \(\frac{5} {3}\) of a unit to the right.

9000037509

Level: 
B
Given complex numbers \[ a = 3\left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right ),\quad b = \sqrt{2}\left (\cos \frac{2\pi } {3} + \mathrm{i}\sin \frac{2\pi } {3}\right ) \] find the product \(ab\).
\(- 3\sqrt{2}\)
\(3\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)
\(3\sqrt{2}\left (\cos \frac{\pi }{2} -\mathrm{i}\sin \frac{\pi }{2}\right )\)
\(- 3\sqrt{2}\left (\cos \frac{\pi }{2} + \mathrm{i}\sin \frac{\pi }{2}\right )\)