9000101110
Level:
B
The points \(A = [0;5;0]\),
\(B = [5;5;0]\),
\(C = [5;0;0]\),
\(D = [0;0;0]\) define a cube
\(ABCDEFGH\). Find the distance
between the point \(A\)
and the point \(F\).
\(\sqrt{50}\)
\(\sqrt{5}\)
\(5\)
\(10\)
Analytical Space Geometry
9000101108
Level:
B
Find the distance between the line \(q\)
and the plane \(\beta \).
\[
\beta \colon x+4y+2z-4 = 0,\qquad \qquad \begin{aligned}[t] q\colon x& = 4, &
\\y & = -2t,
\\z & = 1 + 4t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(\frac{2}
{\sqrt{21}}\)
\(\frac{4}
{\sqrt{21}}\)
\(0\)
\(1\)
9000101009
Level:
A
Determine whether the following two lines are identical, parallel, intersecting or
skew.
\[\begin{aligned}
a\colon x & = t, & &
\\y & = -t, & &
\\z & = 1 - t;\ t\in \mathbb{R} & &
\end{aligned}\]\[\begin{aligned}
b\colon x & = -s, & &
\\y & = s, & &
\\z & = 1 + s;\ s\in \mathbb{R} & &
\end{aligned}\]
identical lines
skew lines
intersecting lines
parallel, not identical lines
9000101010
Level:
A
Determine whether the following two lines are identical, parallel, intersecting or
skew.
\[\begin{aligned}
a\colon x & = t, & &
\\y & = -t, & &
\\z & = 1 - t;\ t\in \mathbb{R} & &
\end{aligned}\]\[\begin{aligned}
b\colon x & = -s, & &
\\y & = s, & &
\\z & = -1 + s;\ s\in \mathbb{R} & &
\end{aligned}\]
parallel, not identical lines
skew lines
intersecting lines
identical lines
9000101106
Level:
B
In the following list identify a point which does not have zero distance from the line
\(m\).
\[
\begin{aligned}m\colon x& = s, &
\\y & = 8 - s,
\\z & = 1 + 3s;\ s\in \mathbb{R}
\\ \end{aligned}
\]
\([2;6;10]\)
\([0;8;1]\)
\([1;7;4]\)
\([8;0;25]\)
9000101003
Level:
A
Find the value of the real parameter \(m\in \mathbb{R}\)
which ensures that the lines \(p\)
and \(q\)
are parallel and not identical.
\[
\begin{aligned}p\colon x& = 1 + t, &
\\y & = 2 - t,
\\z & = 1 - t;\ t\in \mathbb{R}
\\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, &
\\y & = -s,
\\z & = 3 + ms;\ s\in \mathbb{R}.
\\ \end{aligned}
\]
\(m = -1\)
\(m = -2\)
\(m = 0\)
\(m = 1\)
9000101001
Level:
A
Determine whether two lines $p$ and $q$ are identical, parallel, intersecting or skew.
\[\begin{aligned}
p\colon x & = 1 + t, & &
\\y & = 2 - t, & &
\\z & = 1 - t;\ t\in \mathbb{R} & &
\end{aligned}\] \[\begin{aligned}
q\colon x & = 2s, & &
\\y & = -1, & &
\\z & = 2 - 2s;\ s\in \mathbb{R} & &
\end{aligned}\]
intersecting lines
skew lines
identical lines
parallel lines, not identical
9000101005
Level:
A
Find the value of the real parameter \(m\)
which ensures that the lines \(p\)
and \(q\)
are intersecting lines (with a unique common point).
\[
\begin{aligned}p\colon x& = 1 + t, &
\\y & = 2 - t,
\\z & = 1 - t;\ t\in \mathbb{R}
\\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, &
\\y & = 1 + s,
\\z & = 3 + ms;\ s\in \mathbb{R}
\\ \end{aligned}
\]
\(m = -2\)
No solution exists.
The lines are intersecting for every real
\(m\).
\(m = 2\)
9000101002
Level:
A
Find the intersection of the line \(AB\)
and the line \(p\),
where \(A = [0;1;2]\),
\(B = [4;1;-2]\) and
\[
\begin{aligned}p\colon x& = 1 + t, &
\\y & = 2 - t,
\\z & = 1 - t;\ t\in \mathbb{R}.
\\ \end{aligned}
\]
\([2;1;0]\)
\([1;2;1]\)
\([3;0;-1]\)
There is no intersection.
9000101006
Level:
A
Find the value of the real parameter \(m\)
which ensures that the following lines are parallel and not identical lines.
\[
\begin{aligned}p\colon x& = 1 + t, &
\\y & = 2 - t,
\\z & = 1 - t;\ t\in \mathbb{R}
\\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, &
\\y & = 1 + s,
\\z & = 3 + ms;\ s\in \mathbb{R}
\\ \end{aligned}
\]
No solution exists.
The lines are parallel and not identical for every real
\(m\).
\(m = -2\)
\(m = 2\)