C

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.

2010015806

Level: 
C
The side of a regular hexagonal prism \(ABCDEFA'B'C'D'E'F'\) shown in the picture is \(a = 3\, \mathrm{cm}\) and the height is \(v = 8\, \mathrm{cm}\). Find the angle between the diagonal \(AC'\) and the base plane \(ABC\) (round your result to the nearest degree).
\(57^{\circ }\)
\(53^{\circ }\)
\(33^{\circ }\)
\(38^{\circ }\)

2010015802

Level: 
C
Let \( ABCDEFV \) be a regular hexagonal pyramid with a base edge length of \( 4\,\mathrm{cm} \) and a height of \( 8\,\mathrm{cm} \). Find the distance between the point \( V \) and the line \( BD \) (see the picture).
\( 2\sqrt{17}\,\mathrm{cm} \)
\( 4\sqrt{3}\,\mathrm{cm} \)
\( 2\sqrt{19}\,\mathrm{cm} \)
\( 2\sqrt{20}\,\mathrm{cm} \)

2010015801

Level: 
C
Let \( ABCDEFA'B'C'D'E'F' \) be a regular hexagonal prism with the base edge length of \( 4\,\mathrm{cm} \) and the height of \( 6\,\mathrm{cm} \). Find the distance between the lines \( FA \) and \( D'C' \) (see the picture).
\( 2\sqrt{21}\,\mathrm{cm} \)
\( 4\sqrt{3}\,\mathrm{cm} \)
\( 10\,\mathrm{cm} \)
\( 2\sqrt{13}\,\mathrm{cm} \)

2010015601

Level: 
C
A regular hexagonal prism \( ABCDEFA'B'C'D'E'F' \) has the side \( a \) of the length \( 3\,\mathrm{cm} \) and the height \( v \) of the length \( 8\,\mathrm{cm} \). Find the angle between the lines \( AD' \) and \( CD' \). Round the result to two decimal places.
\( 31.31^{\circ} \)
\( 58.69^{\circ} \)
\( 16.70^{\circ} \)
\( 20.57^{\circ} \)