B

9000108704

Level: 
B
Consider a pair of vectors \(\vec{u} = (1,0,-1)\) and \(\vec{v} = (2,-1,1)\). Find all the vectors \(\vec{w}\) which are perpendicular to both \(\vec{u}\) and \(\vec{v}\) and satisfy \(\left |\vec{w}\right | = 2\).
\(\vec{w} = \left (\frac{2\sqrt{11}} {11} , \frac{6\sqrt{11}} {11} , \frac{2\sqrt{11}} {11} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{11}} {11} ,-\frac{6\sqrt{11}} {11} ,-\frac{2\sqrt{11}} {11} \right )\)
\(\vec{w} = (-1,-3,-1)\), \(\vec{w} = (1,3,1)\)
\(\vec{w} = \left (-\frac{1} {2},-\frac{3} {2},-\frac{1} {2}\right )\), \(\vec{w} = \left (\frac{1} {2}, \frac{3} {2}, \frac{1} {2}\right )\)
\(\vec{w} = \left (\frac{2\sqrt{2}} {3} , \frac{3\sqrt{2}} {2} , \frac{2\sqrt{2}} {3} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{2}} {3} ,-\frac{3\sqrt{2}} {2} ,-\frac{2\sqrt{2}} {3} \right )\)

9000107509

Level: 
B
In the following list identify a parametric line such that the angle between this line and the line \(q\) is \(0^{\circ }\). \[ q\colon x - 2y + 11 = 0 \]
\(\begin{aligned}[t] p\colon x& = 1 + 4t, & \\y & = 3 + 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = 1 + 2t, & \\y & = 2 - t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = 2 - t, & \\y & = 3 + 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = t, & \\y & = 1 - 2t,\ t\in \mathbb{R} \\ \end{aligned}\)

9000108706

Level: 
B
Find all vectors which are parallel to the vector \(\vec{u} = (3,-1)\) and have the length equal to \(1\).
\(\left (\frac{3\sqrt{10}} {10} ,-\frac{\sqrt{10}} {10} \right )\), \(\left (-\frac{3\sqrt{10}} {10} , \frac{\sqrt{10}} {10} \right )\)
\((0,-1)\), \((0,1)\)
\((-3,1)\), \((3,-1)\)
\(\left (\frac{3} {4},-\frac{1} {4}\right )\), \(\left (-\frac{3} {4}, \frac{1} {4}\right )\)

9000111802

Level: 
B
In the following list identify a line parallel to the plane \(\rho \) such that the distance between the line and the plane equals \(1\). \[ \begin{aligned}[t] \rho \colon x& = 1 + r, & \\y& = 1 + 2s, \\z& = 1 + r + s,\ r,s\in \mathbb{R} \\ \end{aligned} \]
\(\begin{aligned}[t] o\colon x& = t, & \\y & = 2 + 2t, \\z & = -1 + 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = 1 - 2t, & \\y & = -3 - t, \\z & = 2 + 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 1 - 2t, & \\y & = -3 - t, \\z & = 1 + 2t,\ t\in \mathbb{R} \\ \end{aligned}\)

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 \(\sqrt{5}\). \[ \begin{aligned}[t] s\colon x& = -1 + t,& \\y & = 2t, \\z & = 2 - t,\ t\in \mathbb{R} \\ \end{aligned} \]
\(\begin{aligned}[t] r\colon x& = 3 - 2t,& \\y & = 3 - 4t, \\z & = 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 1, & \\y & = -1 + 5t, \\z & = 2 - 2t,\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = -5 - t,& \\y & = 2 - 2t, \\z & = 2 + t,\ t\in \mathbb{R} \\ \end{aligned}\)