Space geometry

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∘32′
45∘0′
51∘4′
54∘44′

2010008908

Level: 
C
We are given skew lines a and b. a:x=−1−2t,b:x=1−3s,y=−2+3t,y=2s,z=−4+2t; t∈R,z=2−2s; s∈R. Find parametric equations of a straight line p, that is intersecting both lines a and b and lying in the plane 2x+3y−z−8=0.
p:x=−9+r,y=10+r,z=4+5r; r∈R
p:x=−9−2r,y=10−2r,z=4+10r; r∈R
p:x=−9−10r,y=10+9r,z=4−r; r∈R
p:x=−9+2r,y=10+2r,z=4−2r; r∈R

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;t3]. The passing point belongs to the sphere and its third coordinate t3 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

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 [t1;−3;8]. The passing point belongs to the sphere and its first coordinate t1 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