9000101101
Level:
B
Find the distance from the point \(A = [0,1,-3]\)
to the point \(B = [-1,3,0]\).
\(\sqrt{14}\)
\(3\)
\(4\)
\(\sqrt{26}\)
Analytical Space Geometry
9000101102
Level:
B
Find the distance between the point \(A = [1,0,1]\)
and the line \(p\).
\[
\begin{aligned}p\colon x& = 2, &
\\y & = 3t,
\\z & = 1 - t,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(1\)
\(0\)
\(2\)
\(3\)
9000101103
Level:
B
Find the distance between two parallel lines
\(p\) and
\(q\).
\[
\begin{aligned}p\colon x& = 2, &
\\y & = 3t,
\\z & = 1 - t,\ t\in \mathbb{R}
\\ \end{aligned}\qquad \qquad \begin{aligned}q\colon x& = 3, &
\\y & = 6s,
\\z & = 1 - 2s,\ s\in \mathbb{R}
\\ \end{aligned}
\]
\(1\)
\(2\)
\(3\)
\(4\)
9000101104
Level:
B
Find the distance between the point \(A = [-1,1,0]\)
and the plane \(\alpha \).
\[
\alpha \colon 12y + 5z - 2 = 0
\]
\(\frac{10}
{13}\)
\(\frac{15}
{13}\)
\(\frac{17}
{13}\)
\(\frac{14}
{13}\)
9000101106
Level:
B
In the following list identify a point which does not have zero distance from the line
\(m\).
\[
\begin{aligned}m\colon x& = s, &
\\y & = 8 - s,
\\z & = 1 + 3s,\ s\in \mathbb{R}
\\ \end{aligned}
\]
\([2,6,10]\)
\([0,8,1]\)
\([1,7,4]\)
\([8,0,25]\)
9000101105
Level:
B
Find the distance between two parallel planes
\(\alpha \) and
\(\beta \).
\[ \begin{aligned}
\alpha& \colon x - 1.5y - 3z - 1 = 0,\\ \beta &\colon 2x - 3y - 6z + 3 = 0
\end{aligned}\]
\(\frac{5}
{7}\)
\(\frac{1}
{7}\)
\(\frac{2}
{49}\)
\(\frac{2}
{41}\)
9000101107
Level:
B
Find the distance between the line \(p\)
and the plane \(\alpha \).
\[
\alpha \colon x-3y+2z-4 = 0,\qquad \qquad \begin{aligned}[t] p\colon x& = 1 + t, &
\\y & = -3t,
\\z & = 2,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(0\)
\(\frac{5}
{\sqrt{17}}\)
\(2\)
\(1\)
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
9000101108
Level:
B
Find the distance between the line \(q\)
and the plane \(\beta \).
\[
\beta \colon x+4y+2z-4 = 0,\qquad \qquad \begin{aligned}[t] q\colon x& = 4, &
\\y & = -2t,
\\z & = 1 + 4t,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(\frac{2}
{\sqrt{21}}\)
\(\frac{4}
{\sqrt{21}}\)
\(0\)
\(1\)