2000014104
Level:
C
Find the value of \(L\) if \(L=\log_{\sqrt{2}}2 \cdot \log_2 \sqrt{3} \cdot \log_{\sqrt{3}} 4\).
\( L=4\)
\( L=2\)
\( L=3\)
\( L=1\)
C
2000014103
Level:
C
Find the value of \( \log_{ab}x\) if \(\log_a x=2\) and \(\log_b x=3\).
\( \frac65\)
\( \frac16\)
\( 6\)
\( \frac56\)
2110014006
Level:
C
Between the roots of the quadratic equation \(9x^2-35x+24=0\) insert two numbers, so that the numbers with the roots together form four consecutive terms of a geometric sequence. This part of the geometric sequence is shown in one of the graphs. Choose the graph.
2110014005
Level:
C
Between the roots of the quadratic equation \(4x^2-35x+54=0\) insert two numbers, so that the numbers with the roots together form four consecutive terms of a geometric sequence. This part of the geometric sequence is shown in one of the graphs. Choose the graph.
2010012901
Level:
C
Consider a circle \( k \) with radius \( 5\,\mathrm{cm} \). In the circle is inscribed a convex quadrilateral \( ABCD \) so that the diagonal \( AC \) is the diameter of the circle, the length of \( BC \) is \( 8\,\mathrm{cm} \), and the length of \( DC \) is \( 5\,\mathrm{cm} \). Determine the length of side \( AD \). (See the picture.)
\(5 \sqrt{3}\,\mathrm{cm} \)
\( 8\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)
\(3 \sqrt{5}\,\mathrm{cm} \)
2010012604
Level:
C
The gravitational force of the attraction of two particles is
\[
F(x) = \frac{c}
{x^{2}},
\]
where \(x\) is the distance
in meters and \(c\)
a positive constant. Find the work required to increase the distance between the particles
from \(2\, \mathrm{m}\)
to \(5\, \mathrm{m}\).
\(\frac{3}
{10}c\, \mathrm{J}\)
\(\frac{2}
{5}c\, \mathrm{J}\)
\(c\, \mathrm{J}\)
2010012603
Level:
C
The instantaneous velocity of a moving body is proportional to the cube of the time. The velocity at the
time \(t = 3\, \mathrm{s}\)
is \(v = 9\, \mathrm{m\, s}^{-1}\).
What is the distance traveled by the body in the first \(6\) seconds?
\(108\, \mathrm{m}\)
\(54\, \mathrm{m}\)
\(324\, \mathrm{m}\)
2010012502
Level:
C
Identify a true statement about the function
\(f(x) = x^{3} +6x^{2} + 12x -1\).
There is neither local minimum nor maximum of \(f\).
The function \(f\) has a local maximum at the point \(x = -2\).
The function \(f\) has a local minimum at the point \(x = -2\).
The global minimum of \(f\) on \(\mathbb{R}\) is at \(x = -2\).
2010012501
Level:
C
Find the global extrema of the following function on the interval \( [ 0;2 ] \).
\[ f(x)=x^3+3x^2-9x \]
the global minimum at \( x=1 \), the global maximum at \( x=2 \)
the global minimum at \( x=1 \), the global maximum at \( x=-3 \)
the global minimum at \( x=2 \), the global maximum at \( x=1 \)
the global minimum at \( x=0 \), the global maximum at \( x=2 \)
2010008710
Level:
C
Reflect the point \(R=[1; 10; -8]\) across the plane \(\omega \colon 2x-y-3z+12=0\). Hint: The line \(RR'\) is perpendicular to the plane \(\omega\) (see the picture).
\(R'=[-7;14;4]\)
\(R'=[-3;12;-2]\)
\(R'=[-1;-10;8]\)
\(R'=[-5;34;-14]\)