Positional properties

2000006512

Level: 
B
Let \(ABCDV\) be a rectangle based-pyramid, where \(V\) is its apex and \(L\), \(N\) are the midpoints of its edges \(BC\), \(CV\) respectively. What is the cross-section of the pyramid if we slice it with a plane \(ALN\)?
a quadrilateral \(ALNR\) with point \(R\) lying on the edge \(DV\)
a triangle \(ALN\)
a quadrilateral \(ALNR\) with point \(R\) lying on the edge \(AV\)
a quadrilateral \(ALNR\) with point \(R\) lying on the edge \(BV\)

2000006511

Level: 
B
Let \(ABCDV\) be a rectangle based-pyramid, where \(V\) is its apex and \(K\), \(M\) are the midpoints of its edges \(AD\), \(BV\) respectively. What is the cross-section of the pyramid if we slice it with a plane \(KCM\)?
a quadrilateral \(KCMP\) with point \(P\) lying on the edge \(AV\)
a triangle \(KCM\)
a quadrilateral \(KCMP\) with point \(P\) lying on the edge \(DV\)
a quadrilateral \(KCMP\) with point \(P\) lying on the median \(KV\) if the triangle \(ADV\)

2000006510

Level: 
B
The bases of the prism shown in the figure are regular hexagons \(ABCDEF\) and \(A'B'C'D'E'F'\). The side edges are perpendicular to the bases. Let \(k\) be a line through the points \(A\) and \(C\) (see the picture). How many diagonals of the prism are parallel to the line \(k\)?
\(3\)
\(1\)
\(2\)
\(0\)

2000006509

Level: 
B
The bases of the prism shown in the figure are regular hexagons \(ABCDEF\) and \(A'B'C'D'E'F'\). The lateral edges are perpendicular to the bases. Let \(k\) be a line through the points \(A\) and \(C\) (see the picture). How many lateral faces of the prism are perpendicular to the line \(k\)?
\(2\)
\(4\)
\(1\)
\(0\)

2000006508

Level: 
B
The bases of the prism shown in the figure are regular hexagons \(ABCDEF\) and \(A'B'C'D'E'F'\). The lateral edges are perpendicular to the bases. Let \(\pi\) be a plane through the points \(B\), \(D\), \(D'\), \(B'\) (see the picture). How many lateral faces of the prism are perpendicular to the plane \(\pi\)?
\(2\)
\(1\)
\(4\)
\(0\)

2000006507

Level: 
B
The bases of the prism shown in the figure are regular hexagons \(ABCDEF\) and \(A'B'C'D'E'F'\). The lateral edges are perpendicular to the bases. Let \(\pi\) be a plane through the points \(B\), \(D\), \(D'\), \(B'\) (see the picture). How many diagonals of the prism are perpendicular to the plane \(\pi\)?
\(2\)
\(4\)
\(3\)
\(1\)

2000006506

Level: 
B
Let \( ABCDEFGH \) be a cube with \( K \) and \( L \) being the midpoints of edges \( AB \) and \( BC \) respectively, and let \( M \) be the centre of its lateral face \( ADHE \). 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 \( CG \), \( DH \), and \( AE \) respectively
a triangle \( KLM \)
a pentagon \( KLPQM \) with points \( P \) and \( Q \) lying on edges \( CG \) and \( DH \) respectively
a quadrilateral \( KLMR \) with point \( R \) lying on the edge \( AE \)

2000006505

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