2000006502
Level:
A
How many pairs of intersecting planes are created by sides of a cuboid?
\(12\)
\(9\)
\(6\)
\(3\)
Positional properties
2000006503
Level:
A
A plane and a point are given. The point is not lying in the plane. How many lines perpendicular to this plane pass through the given point?
\(1\)
\(2\)
\(0\)
infinitely many
2000006504
Level:
A
A plane and a point are given. The point is not lying in the plane. How many lines parallel to this plane pass through the given point?
infinitely many
\(0\)
\(2\)
\(1\)
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 \(k\)?
\(3\)
\(1\)
\(2\)
\(0\)
1103059502
Level:
C
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:
C
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:
C
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