Space geometry

1003124004

Level: 
A
Find the value of a parameter \( a\in\mathbb{R} \) so that the point \( B=[1;4;5] \) lies on the straight line \( p \) defined by the parametric equations \[\begin{aligned}p\colon x&=-1+m,\\ y&=2+am,\\ z&=3+m;\ m\in\mathbb{R}. \end{aligned}\]
\( a=1 \)
\( a=-1 \)
\( a=2 \)
no such value of \( a \) exists

1003124003

Level: 
A
Find the missing coordinates of the point \( B=[x_B; y_B;-3] \) lying on a straight line \( p \) defined by the parametric equations \[\begin{aligned} p\colon x&=-1+\frac14m,\\ y&=2+m,\\ z&=5-m;\ m\in\mathbb{R}.\end{aligned} \]
\( B=[1;10;-3] \)
\( B=[-3;-6;-3] \)
\( B=[1;3;-3] \)
\( B=[-3;6;-3] \)

1003124002

Level: 
A
From the given options choose the parametric equations which describe a straight line \( p \) passing through the points \( A=[-2;0;1] \) and \( B=[2;0;-3] \).
\( \begin{aligned} p\colon x&=2-t, \\ y&=0, \\ z&=-3+t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2+4t, \\ y&=0, \\ z&=-3+4t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2, \\ y&=0, \\ z&=-3+t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=2-2t, \\ y&=0, \\ z&=-3+t;\ t\in\mathbb{R} \end{aligned} \)

1003124001

Level: 
A
We are given a straight line \( q=\left\{[3t;2-2t;1+t]\text{, }t\in\mathbb{R}\right\} \) and four points \( A=[-6;6;-1] \), \( B=[-3;0;0] \), \( C=[0;2;1] \) and \( D=[3;0;2] \). Out of the given points select all that lie on the straight line \( q \). (Choose the corresponding option.)
\( A \), \( C \), \( D \)
\( B \), \( C \), \( D \)
\( B \), \( C \)
\( A \), \( B \), \( C \)

1103212905

Level: 
C
A rectangle-based right pyramid \( ABCDV \) with its bottom edge length of \( 6 \) units and the perpendicular height of \( 6 \) units is placed in a coordinate system (see the picture). Find the parametric equations of an intersection line \( p \) of planes \( \alpha \) and \( \beta \), where \( \alpha \) passes through the points \( B \), \( C \) and \( V \), and \( \beta \) passes through the points \( A \), \( D \) and \( V \). What is the measure of an angle \( \varphi \) between the planes \( \alpha \) and \( \beta \). Round \( \varphi \) to the nearest minute.
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 53^{\circ}8'\\ y&=3, &\\ z&=6;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 63^{\circ}8'\\ y&=3, &\\ z&=0;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 53^{\circ}8'\\ y&=3+t, &\\ z&=6+2t;\ t\in\mathbb{R} & \end{aligned}\)
\(\begin{aligned} p\colon x&=3+t, & \varphi\doteq 63^{\circ}8'\\ y&=3, &\\ z&=6;\ t\in\mathbb{R} & \end{aligned}\)

1103212904

Level: 
C
A rectangle-based right pyramid \( ABCDV \) with a bottom edge length of \( 6 \) units and the perpendicular height of \( 6 \) units is placed in a coordinate system (see the picture). Let \( S \) be the midpoint of the edge \( AD \). Find the standard equation of the plane \( \alpha \) passing through the points \( B \), \( V \) and \( C \), and calculate the distance of the point \( S \) from \( \alpha \).
\( \alpha\colon 2y+z-12=0;\ d=|S\alpha|=\frac{12\sqrt5}{5} \)
\( \alpha\colon 2x+z-12=0;\ d=|S\alpha|=\frac{12\sqrt5}{5} \)
\( \alpha\colon 2y+z-12=0;\ d=|S\alpha|=\frac{6\sqrt5}{5} \)
\( \alpha\colon 2x+z-12=0;\ d=|S\alpha|=\frac{6\sqrt5}{5} \)

1103212903

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Find an angle \( \varphi \) between the plane \( \alpha \) passing through the points \( E \), \( D \) and \( C \) and the straight line \( AF \). Hint: An angle between a line and a plane is an angle between the line and its orthogonal projection into this plane.
\( \varphi = 30^{\circ} \)
\( \varphi = 15^{\circ} \)
\( \varphi = 45^{\circ} \)
\( \varphi = 60^{\circ} \)

1103212902

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Let \( S \) be the midpoint of the face \( ABFE \), and let \( K \) and \( L \) be the midpoints of edges \( DH \) and \( CG \) consecutively. Find the standard equation of a plane \( \alpha \) passing through the points \( A \), \( B \) and \( L \), and calculate the distance of the point \( S \) from \( \alpha \).
\( \alpha\colon x+2z-2=0;\ |S\alpha|=\frac{2\sqrt5}{5} \)
\( \alpha\colon x+2z-2=0;\ |S\alpha|=\frac{2\sqrt3}{3} \)
\( \alpha\colon x+2y-2=0;\ |S\alpha|=\frac{2\sqrt5}{5} \)
\( \alpha\colon x+2y-2=0;\ |S\alpha|=\frac{2\sqrt3}{3} \)

1103212901

Level: 
C
A cube \( ABCDEFGH \) with an edge length of \( 2 \) units is placed in a coordinate system (see the picture). Find the distance of parallel lines \( p=KL \) and \( q=MN \), where points \( K \), \( L \), \( M \) and \( N \) are midpoints of edges \( CD \), \( BC \), \( EH \) and \( EF \) respectively.
\( |pq|=\sqrt6 \)
\( |pq|=2\sqrt3 \)
\( |pq|=3\sqrt2 \)
\( |pq|=2\sqrt2 \)