B

9000100008

Level: 
B
Part of the graph of the function \(f(x) = \frac{1} {x}\) is shown in the picture. Complete the following sentence: „Formula \[ V =\pi \int _{ 1}^{2}x^{-2}\, \mathrm{d}x \] determines the volume of the solid of revolution obtained by revolving region bounded by ...”
\(x\)-axis, graph of \(f\) on \([ 1;\, 2] \) and lines \(x = 1\), \(x = 2\) about \(x\)-axis.
\(y\)-axis, graph of \(f\) on \([ 1;\, 2] \) and lines \(y = 1\), \(y = \frac{1} {2}\) about \(x\)-axis.
\(x\)-axis, graph of \(f^{2}\) on \([ 1;\, 2] \) and lines \(x = 1\), \(x = 2\) about \(x\)-axis.
\(y\)-axis, graph of \(f^{2}\) on \([ 1;\, 2] \) and lines \(y = 1\), \(y = \frac{1} {2}\) about \(x\)-axis.

9000100003

Level: 
B
The function \(f(x) = x^{2} + 2\) is graphed in the picture. Consider the region between the graph of the function on the interval \([ 0;\, 1] \), both axes and the line \(x = 1\). Find the formula for the volume of the solid of revolution obtained by revolving this region about \(y\)-axis.
\(V =\pi \int _{ 0}^{3}1\, \mathrm{d}y -\pi \int _{2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)
\(V =\pi \int _{ 0}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)
\(V =\pi \int _{ 2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y -\pi \int _{0}^{3}1\, \mathrm{d}y\)
\(V =\pi \int _{ 2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)

9000100002

Level: 
B
The function \(f(x) = 3 - 2x\) is graphed in the picture. Consider the region between the graph of the function \(f\), the \(x\)-axis and the lines \(x = 1\) and \(x = -1\). Find the volume of the solid of revolution obtained by revolving this region about \(x\)-axis.
\(\frac{62} {3} \pi \)
\(6\pi \)
\(12\pi \)
\(\frac{8} {3}\pi \)

9000100009

Level: 
B
Part of the graph of the function \(f(x) = \frac{1} {x}\) is shown in the picture. Consider the region bounded by \(x\)-axis, graph of \(f\) and lines \(x = 1\) and \(x = 4\). Find the volume of the solid of revolution obtained by revolving this region about \(x\)-axis.
\(\frac{3} {4}\pi \)
\(\frac{5} {4}\pi \)
\(\frac{5} {3}\pi \)
\(\frac{4} {3}\pi \)

9000085310

Level: 
B
The pad in the form of (not necessarily regular) octagon is manufactured from a square. The side of the square is \(4\, \mathrm{cm}\). An isosceles right triangle of the hypotenuse \(1\, \mathrm{cm}\) is removed from each corner of the square which results in the desired octagon. How much of the material of the initial square goes to the waste?
\(12.5\, \%\)
\(10\, \%\)
\(15\, \%\)
\(20\, \%\)

9000086608

Level: 
B
Determine the truth values of statements \(a\) and \(b\) if you know that the compound statement \[ \neg a \iff (a \wedge b) \] is true.
The statement \(a\) is true, \(b\) is false.
Both statements are true.
The statement \(a\) is false, \(b\) is true.
Both statements are false.