9000038608
Level:
B
Find the algebraic form of the following complex number.
\[
8(\cos \pi + \mathrm{i}\sin \pi )
\]
\(- 8\)
\(8\)
\(8 + 8\mathrm{i}\)
\(8 - 8\mathrm{i}\)
B
9000038902
Level:
B
Consider the function \(f\colon y = A\cdot \sin (B\cdot x + C)\), with
real nonzero parameters \(A\),
\(B\) and
\(C\).
Which of the following operations makes the amplitude of the function five times
bigger?
Decrease \(A\)
by a factor \(5\).
Increase \(A\) by
a factor \(5\).
Increase \(B\)
by a factor \(5\).
Decrease \(B\)
by a factor \(5\).
Increase \(C\)
by a factor \(5\).
Decrease \(C\)
by a factor \(5\).
9000038609
Level:
B
Find the algebraic form of the following complex number.
\[
5\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )
\]
\(-\frac{5\sqrt{2}}
{2} + \mathrm{i}\frac{5\sqrt{2}}
{2} \)
\(\frac{5\sqrt{2}}
{2} -\mathrm{i}\frac{5\sqrt{2}}
{2} \)
\(\frac{5}
{2} + \mathrm{i}\frac{5}
{2}\)
\(\frac{5}
{2} -\mathrm{i}\frac{5}
{2}\)
9000038903
Level:
B
Find the minimal value of the parameter
\(D\) which ensures
that the function \(f\colon y = D + 3\cdot \sin x\)
is nonnegative.
\(D = 3\)
\(D = -3\)
\(D = 6\)
\(D = -6\)
\(D = 1\)
No solution.
9000038610
Level:
B
Find the algebraic form of the following complex number.
\[
2\left (\cos \frac{3\pi }
{4} + \mathrm{i}\sin \frac{3\pi }
{4}\right )
\]
\(-\sqrt{2} + \mathrm{i}\sqrt{2}\)
\(\sqrt{2} -\mathrm{i}\sqrt{2}\)
\(2 + 2\mathrm{i}\)
\(2 - 2\mathrm{i}\)
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 \(f\).
\(g\colon y =\cos \frac{x}
{2} \)
\(k\colon y =\cos \left (\frac{x}
{2} + \frac{\pi }
{2}\right )\)
\(b\colon y =\cos \left (\frac{x}
{2} - \frac{\pi }
{2}\right )\)
\(h\colon y =\cos \left (\frac{x}
{2} -\pi \right )\)
\(m\colon y =\cos 2x\)
9000038904
Level:
B
Consider function \(f\colon y =\sin x\) with domain
restricted to \(\mathop{\mathrm{Dom}}(f) =\mathbb{R}_{ 0}^{+}\). Identify the
function which is defined on \([ - 5;+\infty )\).
\(f(x + 5)\)
\(f(x - 5)\)
\(f(x) + 5\)
\(f(x) - 5\)
\(5\cdot f(x)\)
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
\[z=\frac{1}
{\cos \frac{7\pi }
{6} +\mathrm{i}\sin \frac{7\pi }
{6} }. \]
\(\cos \frac{5\pi }
{6} + \mathrm{i}\sin \frac{5\pi }
{6}\)
\(\cos \left (-\frac{5\pi }
{6}\right ) + \mathrm{i}\sin \left (-\frac{5\pi }
{6}\right )\)
\(\cos \frac{\pi }{6} + \mathrm{i}\sin \frac{\pi }{6}\)
\(\cos \left (-\frac{\pi }{6}\right ) + \mathrm{i}\sin \left (-\frac{\pi }{6}\right )\)