2000014109
Level:
B
Identify which of the following relations is correct.
\( \log_3 10 >2\)
\( \log_2 7 >3\)
\( \log_2 3 < \log_3 2\)
\( \log_4 15 >2\)
B
2000014106
Level:
B
Find the value of \( \log_2 5 + \log_2 20\) if \(\log_2 5=a\) and \( \log_2 20=b\).
\( 2a+2\)
\( 2^a+2^b\)
\( ab\)
\( 5a\)
2000014102
Level:
B
Make a true statement: The number \((\log_63)^2+(\log_62)^2+\log_64\cdot \log_63\) is
positive.
smaller than 1.
negative.
irrational.
2000014101
Level:
B
Find the domain of the function \(f(x)=\log_{2015}\left(\log_{\frac{1}{2015}}(\log_{2015}x)\right)\).
\((1;2015)\)
\((2015;\infty)\)
\((0;\infty)\)
\((0;2015)\)
2010012902
Level:
B
Derive the formula for calculating the area of an equilateral triangle inscribed in a circle of radius \( r \).
\(\frac{3\sqrt{3}r^2}{4} \)
\(\frac{3\sqrt{3}r}{4} \)
\(\frac{3\sqrt{3}r^2}{2} \)
\(\frac{\sqrt{3}r^2}{4} \)
2010012809
Level:
B
Which of the following formulas expresses the area of a regular pentagon inscribed in a circle of radius \( r \)? (See the picture.)
\( \frac{5r^2\sin72^{\circ}}2\)
\( \frac{10r^2\sin72^{\circ}}2\)
\( \frac{5r^2\sin36^{\circ}}2\)
\( \frac{5r \sin36^{\circ}}2\)
2010012606
Level:
B
Part of the graph of the function \(f(x) = \frac{1}
{x^2}\)
is shown in the picture. Consider the region bounded by
\(x\)-axis,
graph of \(f\)
and
lines \(x = 1\) and
\(x = 2\). Find
the volume of the solid of revolution obtained by revolving this region about
\(x\)-axis.
\(\frac{7}
{24} \pi \)
\(\frac{\pi}
{2}\)
\(\frac{9}
{24} \pi \)
\(\frac{7}
{8} \pi \)
2010012605
Level:
B
The function \(f(x) = \frac12 x +2\)
is graphed in the picture. Consider the region between the graph of the function \(f\), the
\(x\)-axis and
the lines \(x = -2\)
and \(x = 1\).
Find the volume of the solid of revolution obtained by revolving this region about
\(x\)-axis.
\(\frac{39}
{4} \pi \)
\(\frac{55}
{4} \pi \)
\(3\pi \)
\(\frac{10}
{3} \pi \)
2110012504
Level:
B
Choose the graph of a function $f$ that satisfies
\begin{gather*}
f'(1) \text{ does not exist}; \\
f''(x) < 0 \text{ if } x < 1 ; \\
f''(x) < 0 \text{ if } x > 2; \\
f''(x) > 0 \text{ if } 1 < x < 2
\end{gather*}
($f'$ is the derivative of a function $f$, $f''$ is the second derivative of a function $f$).
2110012503
Level:
B
Choose the graph of a function $f$ that satisfies
\begin{gather*}
f'(-2)=f'(0)=0; \\
f''(-2) < 0;\ f''(0) > 0
\end{gather*}
($f'$ is the derivative of the function $f$, $f''$ is the second derivative of the function $f$).
