2000003204
Level:
C
The picture shows a triangle \(ABC\) with a circumscribed circle \(k\) that is centered at \(S\). What is the measure of the angle \(\beta\)?
\( 58^{\circ}\)
\( 32^{\circ}\)
\( 148^{\circ}\)
\( 64^{\circ}\)
C
2000003203
Level:
C
A deltoid is composed of two isosceles triangles that have a common base. See the picture. Find the measures of the deltoids interior angles.
\( \alpha=36^{\circ};~\beta=134^{\circ};~\gamma=56^{\circ};~\delta=134^{\circ}\)
\( \alpha=36^{\circ};~\beta=100^{\circ};~\gamma=56^{\circ};~\delta=100^{\circ}\)
\( \alpha=56^{\circ};~\beta=134^{\circ};~\gamma=56^{\circ};~\delta=134^{\circ}\)
\( \alpha=36^{\circ};~\beta=128^{\circ};~\gamma=56^{\circ};~\delta=128^{\circ}\)
2000003110
Level:
C
Consider functions \(f(x)=-2x+3\) and \(g(x)=3x-2\). What is the value of \(f(g(f(-2)))\)?
\(-35\)
\(13\)
\(-1\)
\(-8\)
2000003109
Level:
C
In the morning at \(7\,\mathrm{a.m.}\) we measured \(3^\circ\mathrm{C}\), at \(10\,\mathrm{a.m.}\) we measured \(12^\circ \mathrm{C}\). How many degrees was at \(9\,\mathrm{a.m.}\), if we assume that the temperature rose linearly?
\(9^\circ\mathrm{C}\)
\(10^\circ\mathrm{C}\)
\(8^\circ\mathrm{C}\)
\(6^\circ\mathrm{C}\)
2010000504
Level:
C
Solve the inequality with the natural variable $x$:
$$\sum\limits_{n=1}^x \left(5-\frac12n\right) \geq x $$
$ x\leq 15;\ x\in\mathbb{N}$
$ x\geq 15;\ x\in\mathbb{N}$
$ x\leq 17;\ x\in\mathbb{N}$
$x\in(-\infty;15]$
there is no solution in $\mathbb{N}$
2010000503
Level:
C
Find the sum of the integers which satisfy the following inequality.
\[
x^{2} + 8x - 153\leq 0
\]
\( -108\)
\( 108\)
\( 104\)
\( -104\)
\( -100\)
2010000307
Level:
C
Evaluate the following integral on \(\mathbb{R}\).
\[
\int \sin^{2}x\cos x\, \mathrm{d}x
\]
\(\frac{\sin^{3}x}
{3} + c,\ c\in \mathbb{R}\)
\(-\frac{\cos ^{3}x\sin x}
{3} + c,\ c\in \mathbb{R}\)
\(2\sin x + c,\ c\in \mathbb{R}\)
2010000306
Level:
C
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int x^{3}\ln x\, \mathrm{d}x
\]
\(\frac{x^4}{4}\ln x -\frac{x^4}
{16}+ c,\ c\in \mathbb{R}\)
\(\frac{x^3}{3}\ln x -\frac{x^3}
{9}+ c,\ c\in \mathbb{R}\)
\(\frac{x^2}{2}\ln x -\frac{x^2}
{4}+ c,\ c\in \mathbb{R}\)
\(x\ln x -x+ c,\ c\in \mathbb{R}\)
2010000305
Level:
C
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int \log_2 x\, \mathrm{d}x
\]
\(x\log_2x -\frac{x} {\ln 2}+ c,\ c\in \mathbb{R}\)
\(\log_2 x -\frac{x} {\ln 2}+ c,\ c\in \mathbb{R}\)
\(x\log_2 x -x+ c,\ c\in \mathbb{R}\)
\(x\log_2 x +\frac{x} {\ln 2}+ c,\ c\in \mathbb{R}\)
2010000304
Level:
C
Solve the indefinite integral
\[ \int\mathrm{e}^{\cos x}\sin x\,\mathrm{d}x \]
of a real-valued function.
\( -\mathrm{e}^{\cos x} +c \), \( c\in\mathbb{R} \)
\(- \mathrm{e}^{\cos x}\cdot\cos x+c \), \( c\in\mathbb{R} \)
\( \mathrm{e}^{\sin x}\cdot\cos x+c \), \( c\in\mathbb{R} \)
\( \mathrm{e}^{\cos x}\cdot\sin x+c \), \( c\in\mathbb{R} \)