A

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. \[ \begin{aligned}p\colon x& = 1 + t, & \\y & = 2 - t, \\z & = 1 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, & \\y & = -s, \\z & = 3 + ms;\ s\in \mathbb{R}. \\ \end{aligned} \]
\(m = -1\)
\(m = -2\)
\(m = 0\)
\(m = 1\)

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.

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 \(m\).
\(m = 2\)

9000101004

Level: 
A
Find all the values of the real parameter \(m\) so that the lines \(p\) and \(q\) are skew lines. \[ \begin{aligned}p\colon x& = 1 + t, & \\y & = 2 - t, \\z & = 1 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, & \\y & = 1 + s, \\z & = 3 + ms;\ s\in \mathbb{R} \\ \end{aligned} \]
\(m\in\mathbb{R}\setminus\{-2\}\)
No solution exists.
The lines are skew for every real \(m\).
\(m = -2\)

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 \(m\).
\(m = -2\)
\(m = 2\)

9000101007

Level: 
A
Find the value of the real parameter \(m\) which ensures that the following two lines are identical. \[ \begin{aligned}p\colon x& = 1 + t, & \\y & = 2 - t, \\z & = 1 - t;\ t\in \mathbb{R} \\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = s, & \\y & = 1 + s, \\z & = 3 + ms;\ s\in \mathbb{R} \\ \end{aligned} \]
No solution exists.
The lines are identical for every real \(m\).
\(m = -2\)
\(m = 2\)