Lines and planes: intersecting, perpendicular, parallel

1103059604

Level: 
B
Let \( ABCDEFGH \) be a cube and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }DH\text{ and }|DX|=1.5|DH|,\\ Y&\text{ lays on a ray }DB\text{ and }|DB|=|BY| \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the cube lay:
on the side \( EFGH \) and the edge \( BF \)
on the edges \( EF \) and \( BF \)
on the sides \( EFGH \) and \( ABCD \)
on the edges \( HG \) and \( BF \)

1103059605

Level: 
B
Let \( ABCDEFGH \) be a cube and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }CB\text{ and }|CX|=1.5|BC|,\\ Y&\text{ lays on a ray }EH\text{ and }|EY|=1.5|EH| \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 sides \( EFGH \) and \( ABCD \)
on the side \( ABCD \) and the edge \( HG \)
on the edges \( HG \) and \( AB \)

1103059606

Level: 
B
Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex, and let \( XY \) be a line where: \begin{align*} X&\text{ lays on the side }AV\text{ and }|AX|=|XV|,\\ Y&\text{ lays on a ray }DC\text{ and }|DY|=1.5|DC| \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the pyramid are:
the point \( X \) and a point on the pyramid face \( BCV \)
the point \( X \) and a point on the pyramid face \( DCV \)
the point \( X \) and a point on the pyramid edge \( CV \)
the point \( X \) only

1103059607

Level: 
B
Let \( ABCDV \) be a rectangle based-pyramid, where \( V \) is its apex, and let \( XY \) be a line where: \begin{align*} X&\text{ lays on a ray }BA\text{ and }|BA|=|AX|,\\ Y&\text{ lays on the height }SV\text{ and }|SY|=|YV|,\\ S&\text{ is the centre of the pyramid base} \end{align*} (see the picture). The points of intersection of the line \( XY \) with the surface of the pyramid lay:
on the pyramid faces \( ADV \) and \( BCV \)
on the pyramid faces \( DCV \) and \( ABV \)
on the pyramid face \( ADV \) and its edge \( CV \)
on the pyramid edges \( AV \) and \( CV \)

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

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 \( AE \)

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 \(\pi\)?
\(2\)
\(4\)
\(3\)
\(1\)

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 \(\pi\)?
\(2\)
\(1\)
\(4\)
\(0\)

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 \(k\)?
\(2\)
\(4\)
\(1\)
\(0\)

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 \(k\)?
\(3\)
\(1\)
\(2\)
\(0\)