9000108801
Level:
B
Find the angle between the vector \(\vec{u} = (1,\sqrt{2})\)
and \(\vec{v} = (3,-1)\).
Round to the nearest degree.
\(73^{\circ }\)
\(42^{\circ }\)
\(57^{\circ }\)
\(64^{\circ }\)
B
9000111803
Level:
B
In the following list identify a point such that the distance from this point to the line
\(p\) is
\(\sqrt{3}\).
\[
\begin{aligned}[t] p\colon x& = 2 - t, &
\\y & = -1 + 2t,
\\z & = t,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\([2,2,0]\)
\([5,-1,-3]\)
\([1,1,1]\)
9000108802
Level:
B
Given the points \(A = [1,2]\),
\(B = [2,6]\) and
\(C = [3,-1]\), find the interior
angles of the triangle \(ABC\).
Round to the nearest degree.
\(22^{\circ }\),
\(26^{\circ }\),
\(132^{\circ }\)
\(26^{\circ }\),
\(45^{\circ }\),
\(109^{\circ }\)
\(22^{\circ }\),
\(48^{\circ }\),
\(110^{\circ }\)
\(17^{\circ }\),
\(31^{\circ }\),
\(132^{\circ }\)
9000111804
Level:
B
In the following list identify a line such that the line is parallel to
\(s\) and the distance
between both lines is \(\sqrt{5}\).
\[
\begin{aligned}[t] s\colon x& = -1 + t,&
\\y & = 2t,
\\z & = 2 - t,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(\begin{aligned}[t] r\colon x& = 3 - 2t,&
\\y & = 3 - 4t,
\\z & = 2t,\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 1, &
\\y & = -1 + 5t,
\\z & = 2 - 2t,\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = -5 - t,&
\\y & = 2 - 2t,
\\z & = 2 + t,\ t\in \mathbb{R}
\\ \end{aligned}\)
9000108803
Level:
B
Consider the vector \(\vec{u} = (\sqrt{3},1)\).
Find the vector \(\vec{w}\)
such that \(\left |\vec{w}\right | = 4\) and the
angle between \(\vec{u}\)
and \(\vec{w}\) is
\(60^{\circ }\). Find
all solutions.
\(\vec{w} = (0,4)\),
\(\vec{w} = (2\sqrt{3},-2)\)
\(\vec{w} = (0,-4)\),
\(\vec{w} = (\sqrt{7},-3)\)
\(\vec{w} = (0,4)\),
\(\vec{w} = (\sqrt{7},3)\)
\(\vec{w} = (\sqrt{5},\sqrt{11})\),
\(\vec{w} = (2\sqrt{3},-2)\)
9000111806
Level:
B
In the following list identify the line such that the angle between this line and the
line \(s\) is
\(60^{\circ }\).
\[
\begin{aligned}[t] s\colon x& = 2 + t, &
\\y & = -1 - 2t,
\\z & = 3 - t,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(\begin{aligned}[t] r\colon x& = t, &
\\y & = -3 + t,
\\z & = 1 + 2t,\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 1, &
\\y & = -1 - t,
\\z & = 3 + 2t,\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = -5 - 2t,&
\\y & = 2 + 4t,
\\z & = 2 + 2t,\ t\in \mathbb{R}
\\ \end{aligned}\)
9000108804
Level:
B
The point \(A = [3,2]\) is rotated
about the center \(B = [1,1]\)
by \(60^{\circ }\). Find
the coordinate of its final position. Consider both clockwise and counterclockwise
direction.
\(\left [2\pm \frac{\sqrt{3}}
{2} , \frac{3}
{2} \mp \sqrt{3}\right ]\)
\(\left [1\pm \frac{\sqrt{3}}
{2} , \frac{1}
{2} \mp \sqrt{3}\right ]\)
\(\left [2\pm \frac{\sqrt{2}}
{2} , \frac{3}
{2} \mp \sqrt{2}\right ]\)
\(\left [1\pm \frac{\sqrt{2}}
{2} , \frac{1}
{2} \mp \sqrt{2}\right ]\)
9000111807
Level:
B
In the following list identify a line such that the angle between this line and the
plane
\[
2x - y + 3z - 5 = 0
\]
is \(30^{\circ }\).
\(\begin{aligned}[t] p\colon x& = 2 + t, &
\\y & = 1 + 3t,
\\z & = -2t,\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] r\colon x& = -2t, &
\\y & = -3 + t,
\\z & = 1 - 3t,\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 2 + 3t, &
\\y & = 3 - 2t,
\\z & = 3 + t,\ t\in \mathbb{R}
\\ \end{aligned}\)
9000108805
Level:
B
Find the angle between \(\vec{u} = (1,-2,3)\)
and \(\vec{v} = (-1,0,2)\).
Round to the nearest degree.
\(53^{\circ }\)
\(27^{\circ }\)
\(60^{\circ }\)
\(46^{\circ }\)
9000111808
Level:
B
In the following list identify a plane such that the angle between this plane and the
plane \(\rho \)
is \(45^{\circ }\).
\[
\rho \colon \begin{aligned}[t] x& = 1 + r - 2s, &
\\y& = 3 - r + 2s,
\\z& = -5 - 4r,\ r,\, s\in \mathbb{R}
\\ \end{aligned}
\]
\(\gamma \colon 3x - 2 = 0\)
\(\beta \colon 2z - 2 = 0\)
\(\alpha \colon x + y - 2 = 0\)