Analytic geometry in a space

9000101909

Level: 
B
Given points \(A = [1;0;2]\), \(B = [1;0;0]\) and the plane \(\alpha \), \[ \alpha \colon 2x - 4y = 0, \] find the angle between the line \(AB\) and the plane \(\alpha \). Round your answer to the nearest minute.
\(0^{\circ }\)
\(22^{\circ }48'\)
\(45^{\circ }19'\)
\(90^{\circ }\)

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

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