Lines and planes: intersecting, perpendicular, parallel

1103059502

Level: 
B
Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex and \( K \), \( L \), and \( M \) are the midpoints of its edges \( AD \), \( BC \), and \( CV \) respectively. What is the mutual position of planes \( BVK \) and \( DLM \)?
distinct parallel planes
identical planes
intersecting planes

1103059503

Level: 
B
Let ABCDEFGH be a cube. What is the mutual position of planes \( ECG \), \( BDF \), and \( ABH \)?
three mutually intersecting planes which share one common point
three mutually intersecting planes which share one common line
two planes are parallel with the third intersecting them in distinct parallel lines

1103059504

Level: 
B
Let \( ABCDEFGH \) be a cube with \( K \), and \( L \) being the midpoints of its edges \( AE \) and \( CG \), and let \( M \) be the centre of its face \( ABFE \). What is the mutual position of planes \( BCE \), \( ADF \), and \( KLM \)?
three mutually intersecting planes which share one common line
three mutually intersecting planes which share one common point
two planes are parallel with the third intersecting them in distinct parallel lines

1103059505

Level: 
B
Let \( ABCDEFGH \) be a cube and \( X \) be the midpoint of its edge \( AE \). What is the cross-section of the cube if we slice it with a plane \( BGX \)?
a quadrilateral \( BGPX \) with \( P \) being the midpoint of the edge \( EH \)
a quadrilateral \( BGHX \)
a triangle \( BGX \)
a quadrilateral \( BGPX \) with \( P \) being the midpoint of the edge \( DH \)

1103059506

Level: 
B
Let \( ABCDEFGH \) be a cube and let \( X \), \( Y \), and \( Z \) be the midpoints of edges \( AB \), \( AE \), and \( CG \) respectively. What is the cross-section of the cube if we slice it with a plane \( XYZ \)?
a hexagon \( XLZKMY \) with points \( L \), \( K \), and \( M \) lying on edges \( BC \), \( GH \), and \( EH \) respectively
a pentagon \( XLZKY \) with points \( L \) and \( K \) lying on edges \( BC \) and \( GH \) respectively
a triangle \( XYZ \)
a quadrilateral \( XZKY \) with \( K \) being the midpoint of the edge \( GH \)

1103059507

Level: 
B
Let \( ABCDEFGH \) be a cube with \( K \) and \( L \) being the midpoints of edges \( AE \) and \( AB \) respectively, and let \( M \) be the midpoint of the face diagonal \( EG \). What is the cross-section of the cube if we slice it with a plane \( KLM \)?
a pentagon \( KLPQR \) with points \( P \), \( Q \), and \( R \) lying on edges \( BC \), \( FG \), and \( EH \) respectively
a triangle \( KLM \)
a pentagon \( KLPQM \) with points \( P \) and \( Q \) lying on edges \( BC \) and \( FG \) respectively
a quadrilateral \( KLMR \) with point \( R \) lying on the edge \( EH \)

1103059601

Level: 
B
Let \( ABCDV \) be a rectangle based-pyramid and \( V \) its apex. The pyramid is sliced by a plane \( EFG \) which is defined by: \begin{align*} E&\in BC\ \wedge\ |BE|=2|CE|, \\ F&\in AV\ \wedge\ |AF|=2|VF|, \\ G&\in DV\ \wedge\ |DG|=2|VG| \end{align*} (see the picture). What is the cross-section of the pyramid if we slice it with the plane \( EFG \)?
a trapezium \( BCGF \)
a triangle \( EFG \)
a triangle \( AEV \)
a pentagon \( ABEGF \)

1103059602

Level: 
B
Let \( ABCDV \) be a rectangle based-pyramid and \( V \) its apex. The pyramid is sliced by a plane \( XYZ \) which is defined by: \begin{align*} X&\text{ is the midpoint of the edge }AD,\\ Y&\in CD\ \wedge\ |DY|=3|CY|,\\ Z&\in BV\ \wedge\ |BZ|=3|VZ| \end{align*} (see the picture). What is the cross-section of the pyramid if we slice it with the plane \( XYZ \)?
a pentagon \( XYKZL \) with points \( K \) and \( L \) lying on the edges \( CV \) and \( AV \)
a triangle \( XYZ \)
a quadrilateral \( XYZL \) with point \( L \) lying on the edge \( AV \)
a quadrilateral \( XYKZ \) with point \( K \) lying on the edge \( CV \)

1103059603

Level: 
B
Let \( ABCDEFGH \) be a cube and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }BC\text{ and }|BX|=1.5|BC|,\\ Y&\text{ lays on a ray }HE\text{ and }|HY|=1.5|HE| \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the cube lay:
on the sides \( ABFE \) and \( DCGH \)
on the side \( ABFE \) and the edge \( CG \)
on the edges \( AE \) and \( CG \)
on the sides \( ADHE \) and \( BCGF \)