C

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 \( x=2 \)

2010008707

Level: 
C
Let \(ABCDEFGH\) be a cube with an edge length of \(2\) units placed in the rectangular coordinate system. In the cube a regular tetrahedron \(BDEG\) is highlighted (see the picture). Find the angle between its faces and round the number to the nearest minute.
\(70^{\circ}32'\)
\(45^{\circ}0'\)
\(51^{\circ}4'\)
\(54^{\circ}44'\)

2010008706

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 4 \) units is placed in a coordinate system (see the picture). Find an angle \( \psi \) between the plane \( \rho \) passing through the points \( B \), \( D \) and \( H \) and the straight line \( CF \). Hint: An angle between a line and a plane is an angle between the line and its orthogonal projection into this plane.
\( \psi = \frac{\pi}6 \)
\( \psi = \frac{\pi}{12} \)
\( \psi = \frac{\pi}4 \)
\( \psi = \frac{\pi}3 \)

2010008705

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 4 \) units is placed in a coordinate system (see the picture). Find the distance of parallel lines \( p=PQ\) and \( r=RS \), where points \( P \), \( Q \), \( R\) and \( S \) are midpoints of edges \(BF\), \(BC\), \(EH\) and \(DH\) respectively.
\( |pr|=2\sqrt6 \)
\( |pr|=4\sqrt3 \)
\( |pr|=6\sqrt2 \)
\( |pr|=4\sqrt2 \)

2010008704

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 3 \) is placed in a coordinate system (see the picture). Find the distance between parallel planes \( \rho \) and \( \sigma \), where \( \rho \) is passing through \( D \), \( E \) and \( G \) and \( \sigma \) is passing through \( A \), \( C \) and \( F \).
\( |\rho\sigma|=\sqrt3 \)
\( |\rho\sigma|=\frac{2\sqrt3}3 \)
\( |\rho\sigma|=\frac{3\sqrt3}2 \)
\( |\rho\sigma|=\frac{4\sqrt3}3 \)