B

9000111805

Level: 
B
In the following list identify a plane which is parallel to the plane \(\delta \) and the distance between both planes is \(2\). \[ \delta \colon x - 2y + 2y - 2 = 0 \]
\(\begin{aligned}[t] \beta \colon x& = -4 + 2s, & \\y& = 1 + r + s, \\z& = 1 + r;\ r,s\in \mathbb{R} \\ \end{aligned}\)
\(\gamma \colon - x + 2y - 2z - 2 = 0\)
\(\alpha \colon 2x - 4y + z - 4 = 0\)

9000108807

Level: 
B
Find the angle between the median \(t_{c}\) and side \(c\) in the triangle \(ABC\) for \(A = [1;2]\), \(B = [7;-2]\) and \(C = [6;1]\). Round to the nearest degree. Hint: In geometry, the median \(t_{c}\) of the triangle \(ABC\) is the line segment joining the vertex \(C\) to the midpoint of the opposing side.
\(60^{\circ }\)
\(50^{\circ }\)
\(43^{\circ }\)
\(71^{\circ }\)

9000111806

Level: 
B
In the following list identify the line such that the angle between this line and the line \(s\) is \(60^{\circ }\). \[ \begin{aligned}[t] s\colon x& = 2 + t, & \\y & = -1 - 2t, \\z & = 3 - t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(\begin{aligned}[t] r\colon x& = t, & \\y & = -3 + t, \\z & = 1 + 2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 1, & \\y & = -1 - t, \\z & = 3 + 2t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] p\colon x& = -5 - 2t,& \\y & = 2 + 4t, \\z & = 2 + 2t;\ t\in \mathbb{R} \\ \end{aligned}\)

9000108808

Level: 
B
Find the angle between the altitude \(v_{c}\) and side \(b\) in the triangle \(ABC\) for \(A = [1;2]\), \(B = [7;-2]\) and \(C = [6;1]\). Round to the nearest degree. Hint: In geometry, the altitude \(v_{c}\) of the triangle \(ABC\) is the line segment through the vertex \(C\) and perpendicular to the line containing the opposite side of the triangle.
\(68^{\circ }\)
\(75^{\circ }\)
\(44^{\circ }\)
\(61^{\circ }\)

9000111808

Level: 
B
In the following list identify a plane such that the angle between this plane and the plane \(\rho \) is \(45^{\circ }\). \[ \rho \colon \begin{aligned}[t] x& = 1 + r - 2s, & \\y& = 3 - r + 2s, \\z& = -5 - 4r;\ r,\; s\in \mathbb{R} \\ \end{aligned} \]
\(\gamma \colon 3x - 2 = 0\)
\(\beta \colon 2z - 2 = 0\)
\(\alpha \colon x + y - 2 = 0\)

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

9000107509

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