Analytic geometry in a space

2010008905

Level: 
A
Determine the relative position of the plane \( \sigma \) with general equation \( x-2y+3z-1=0 \) and the straight line \( p \) with parametric equations: \[ \begin{aligned} x&=4, \\ y&=5+3t, \\ z&=2+2t;\ t\in\mathbb{R}. \end{aligned} \]
\( p\parallel\sigma,\ p\not{\!\!\subset} \sigma \)
\( p \subset \sigma \)
\( p \) is intersecting the plane \( \sigma \)

2010008906

Level: 
A
We are given two intersecting planes \(2x - 3y + 5z - 9 = 0\) and \(3x - y + 2z - 1 = 0\). Find the parametric equations of their line of intersection \(p\).
\( \begin{aligned} p\colon x&=-1-t, \\ y&=-2+ 11t, \\ z&=1+ 7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=-1-11t, \\ y&=-2+ 11t, \\ z&=1+ 7t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=-1+t, \\ y&=-2+ 11t, \\ z&=1- 11t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=-1-11t, \\ y&=-2+ 11t, \\ z&=1- 11t;\ t\in\mathbb{R} \end{aligned} \)

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

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.

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

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

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

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