1103124403
Level:
B
Let a solid be obtained by rotating the yellow region about the $x$-axis (see the picture). Find its volume.
$85.5\pi$
$85.5$
$27$
$27\pi$
B
1103163505
Level:
B
Choose the graph of a function $f$ that satisfies
\begin{gather*}
f'(0) \text{ does not exist}; \\
f''(x) > 0 \text{ if } x < 0 ; \\
f''(x) > 0 \text{ if } x > 1; \\
f''(x) < 0 \text{ if } 0 < x < 1
\end{gather*}
($f'$ is the derivative of a function $f$, $f''$ is the second derivative of a function $f$).
1103163504
Level:
B
Choose the graph of a function $f$ that satisfies
\begin{gather*}
f'(0)=f'(3)=0; \\
f''(0)=0;\ f''(3) < 0
\end{gather*}
($f'$ is the derivative of a function $f$, $f''$ is the second derivative of a function $f$).
1103163503
Level:
B
Choose the graph of a function $f$ that satisfies
\begin{gather*}
f'(0)=f'(3)=0; \\
f''(0) < 0;\ f''(3) > 0
\end{gather*}
($f'$ is the derivative of the function $f$, $f''$ is the second derivative of the function $f$).
1103163502
Level:
B
Choose the graph of a function $f$ that satisfies
$$ f''(0) > 0;\ f''(3) < 0 $$
($f''$ is the second derivative of the function $f$).
1103163501
Level:
B
Choose the graph of a function $f$ which does not satisfy
$$ f''(0) > 0;\ f''(2) < 0 $$
($f''$ is the second derivative of the function $f$).
1003040210
Level:
B
Given the points $A = [3;3;0]$ and $B = [0;3;3]$, specify all the points $C$ lying on the $y$-axis, such that $|\measuredangle ABC|=\frac{\pi}3$ holds.
$C_1=[0;0;0];\ C_2=[0;6;0]$
$C_1=[0;3;0];\ C_2=[0;9;0]$
$C_1=[0;-3;0];\ C_2=[0;3;0]$
$C_1=[0;-6;0];\ C_2=[0;6;0]$
1103040209
Level:
B
In the picture, there are indicated vectors $\vec{u}$ and $\vec{v}$ in three squares. Find the measure of an angle $\varphi$ between $\vec{u}$ and $\vec{v}$. Round $\varphi$ to the nearest degree.
Hint: Set up a coordinate system conveniently.
$\varphi\doteq 8^{\circ}$
$\varphi\doteq 9^{\circ}$
$\varphi\doteq 10^{\circ}$
$\varphi\doteq 11^{\circ}$
1003068203
Level:
B
What is the volume of the solid that we get by rotating the curve
\[ y=\frac1{x^2} \]
about the \( x \)-axis on the interval \( [1;3] \)?
\( \frac{26}{81}\pi \)
\( \frac{3^5-1}{3^6}\pi \)
\( \frac{28}{81}\pi \)
\( \frac{3^5+1}{3^6}\pi \)
1003068202
Level:
B
The value of the integral
\[ \pi\cdot\int\limits_0^6\left[9-(x-3)^2\right]\,\mathrm{d}x \]
is a number which represents:
the volume of a sphere with the radius of \( 3\,\mathrm{cm} \).
the volume of a sphere with the radius of \( 6\,\mathrm{cm} \).
the volume of a sphere with the diameter of \( 3\,\mathrm{cm} \).
the volume of a semi-sphere with the radius of \( 3\,\mathrm{cm} \).