Space geometry

2010016104

Level: 
C
Find the equations of all the tangent planes to the sphere \((x + 2)^2 + (y - 1)^2 + (z - 4)^2 = 36\) passing through the point \([t_1; -3; 8]\). The passing point belongs to the sphere and its first coordinate \(t_1\) is greater than \(x\) coordinate of the sphere center.
\( x+2y-2z+26=0\)
\( x-2y+2z-22=0\)
\( x-2y+2z-18=0\)
\( x-2y-2z+14=0\)

2010016103

Level: 
C
Find the equations of all the tangent planes to the sphere \((x - 2)^2 + (y + 1)^2 + (z + 4)^2 = 36\) passing through the point \([-2; 3; t_3]\). The passing point belongs to the sphere and its third coordinate \(t_3\) is greater than \(z\) coordinate of the sphere center.
\( 2x-2y-z+8=0\)
\( 2x-2y+z+16=0\)
\( 2x-2y-3z+4=0\)
\( 2x-2y-5z=0\)

2010016102

Level: 
C
If the equation \( x^2+y^2+z^2+2x-8y+z+18=0\) is the equation of a sphere, find its center \(S\) and radius \(r\).
It is not a sphere equation.
\( S= \left[ -1;4;-\frac12\right]\), \(r=\frac34\)
\( S= \left[ 1;-4;\frac12\right]\), \(r=\frac{\sqrt3}2\)
\( S= \left[ -1;4;-\frac12\right]\), \(r=\frac{\sqrt3}2\)
\( S= \left[ 1;-4;\frac12\right]\), \(r=\frac34\)

2010016101

Level: 
C
If the equation \( x^2+y^2+z^2+2x-8y+z+17=0\) is the equation of a sphere, find its center \(S\) and radius \(r\).
\( S= \left[ -1;4;-\frac12\right]\), \(r=\frac12\)
\( S= \left[ -1;4;-\frac12\right]\), \(r=\frac14\)
\( S= \left[ 1;-4;\frac12\right]\), \(r=\frac12\)
\( S= \left[ 1;-4;\frac12\right]\), \(r=\frac14\)
It is not a sphere equation.

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.
\( |pr|=2\sqrt6 \)
\( |pr|=4\sqrt3 \)
\( |pr|=6\sqrt2 \)
\( |pr|=4\sqrt2 \)