C

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 \( F \).
\( |\rho\sigma|=\sqrt3 \)
\( |\rho\sigma|=\frac{2\sqrt3}3 \)
\( |\rho\sigma|=\frac{3\sqrt3}2 \)
\( |\rho\sigma|=\frac{4\sqrt3}3 \)

2010008703

Level: 
C
A straight line \( q \) is given by the points \( K=[6;6;7] \) and \( L=[4;0;2] \) (see the picture). Find the parametric equations of the line \( q' \) that is symmetrical to the line \( q \) in the plane symmetry across the coordinate \( xz \)-plane.
\( \begin{aligned} q'\colon x&=4+2t, \\ y&=-6t, \\ z&=2+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+6t, \\ y&=6t, \\ z&=2+7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+2t, \\ y&=6t, \\ z&=2+5t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} q'\colon x&=4+6t, \\ y&=-6t, \\ z&=2+7t;\ t\in\mathbb{R} \end{aligned} \)