9000101902
Level:
B
Given points \(A = [0;1;2]\),
\(B = [1;2;0]\),
\(C = [1;2;3]\), find the angle
between the lines \(AB\)
and \(AC\).
Round your answer to the nearest degree.
\(90^{\circ }\)
\(0^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)
Analytical Space Geometry
9000101904
Level:
B
Find the angle between the \(x\)-axis
and the line \(p\).
\[
\begin{aligned}p\colon x& = 2 - t, &
\\y & = 3t,
\\z & = 1;\ t\in \mathbb{R}
\\ \end{aligned}
\]
Round your answer to the nearest minute.
\(71^{\circ }34'\)
\(0^{\circ }\)
\(69^{\circ }17'\)
\(90^{\circ }\)
9000101905
Level:
B
The points \(A = [0;5;0]\),
\(B = [5;5;0]\),
\(C = [5;0;0]\),
\(D = [0;0;0]\) define the cube
\(ABCDEFGH\). Find the angle
between the lines \(BF\)
and \(AC\).
Round your answer to the nearest minute.
\(90^{\circ }\)
\(0^{\circ }\)
\(45^{\circ }\)
\(73^{\circ }47'\)
9000101906
Level:
B
Find the angle between the planes \(\alpha \)
and \(\beta \).
\[
\alpha \colon 2x - 5y + 3z - 4 = 0,\qquad \beta \colon x - 3 = 0
\]
Round your answer to the nearest minute.
\(71^{\circ }4'\)
\(21^{\circ }42'\)
\(82^{\circ }19'\)
\(85^{\circ }2'\)
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.
\(0^{\circ }\)
\(22^{\circ }48'\)
\(45^{\circ }19'\)
\(90^{\circ }\)
9000101910
Level:
B
The points \(A = [0;5;0]\),
\(B = [5;5;0]\),
\(C = [5;0;0]\) and
\(D = [0;0;0]\) define the cube
\(ABCDEFGH\). Find the angle
between the line \(BF\)
and the plane \(AFE\).
Round your answer to the nearest minute.
\(0^{\circ }\)
\(35^{\circ }16'\)
\(45^{\circ }\)
\(90^{\circ }\)
9000101907
Level:
B
The general plane \(\alpha \)
has the equation
\[
\alpha \colon 3z - 4 = 0
\]
and the plane \(\beta \) has a
normal vector \(\vec{n} = (0;0;1)\). Find
the angle between \(\alpha \)
and \(\beta \)
and round your answer to the nearest degree.
\(0^{\circ }\)
\(30^{\circ }\)
\(45^{\circ }\)
\(90^{\circ }\)
9000101903
Level:
B
Given points \(A = [-1;0;3]\),
\(B = [0;2;0]\), find the angle
between the line \(AB\)
and the line \(m\).
\[
\begin{aligned}m\colon x& = 1 + 2t, &
\\y & = -3t,
\\z & = 1;\ t\in \mathbb{R}
\\ \end{aligned}
\]
Round your answer to the nearest minute.
\(72^{\circ }45'\)
\(0^{\circ }\)
\(48^{\circ }15'\)
\(90^{\circ }\)
9000101908
Level:
B
Find the angle between the line \(p\)
and the plane \(\alpha \).
\[
\alpha \colon x-3z+5 = 0;\qquad \qquad \begin{aligned}[t] p\colon x& = 3, &
\\y & = 3t,
\\z & = 1 - t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
Round your answer to the nearest minute.
\(17^{\circ }27'\)
\(0^{\circ }\)
\(47^{\circ }33'\)
\(90^{\circ }\)
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
\(m\).
\(m = -2\)
\(m = 2\)