B

9000106308

Level: 
B
In the following list identify a pair of planes such that the distance of planes from the plane $\alpha$ is the same as the distance between the point $A=[0;0;1]$ and the plane \(\alpha \). \[ \alpha \colon 2x + y - z - 5 = 0 \]
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 11& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 10& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 12& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z + 1& = 0& \\2x + y - z - 9& = 0 \\ \end{aligned}\)

9000106301

Level: 
B
Find the line $k$ which is perpendicular to the plane \(\alpha \) \[ \alpha \colon 2x + y - z - 5 = 0 \] and passes through the point \(A = [0;0;1]\).
\(\begin{aligned}[t] x& =\phantom{ 1 -} 2t, & \\y& =\phantom{ 1 -}\ t, \\z& = 1 - t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2 + 2m, & \\y& =\phantom{ -}1 +\phantom{ 2}m, \\z& = -1 -\phantom{ 2}m;\ m\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2k, & \\y& =\phantom{ -2}k, \\z& = -\phantom{2}k;\ k\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] x& =\phantom{ -}2, & \\y& =\phantom{ -}1, \\z& = -1 + u;\ u\in \mathbb{R} \\ \end{aligned}\)

9000106302

Level: 
B
The plane \(\alpha \) has equation \[ \alpha : 2x + y - z - 5 = 0. \] The line \(k\) passes through the point \(A = [0;0;1]\) and is perpendicular to \(\alpha \). Find the intersection \(S\) of the line \(k\) and the plane \(\alpha \).
\(S = [2;1;0]\)
\(S = [2;0;1]\)
\(S = [-2;1;0]\)
\(S = [-2;0;1]\)

9000106305

Level: 
B
Find the area of the triangle \(ABS\). Only first two coordinates of the point $B=[2;0;?]$ are given and $B$ lies in the plane $\alpha$ defined by the equation \[ \alpha \colon 2x + y - z - 5 = 0. \] The point \(S\) is the intersection point of the plane \(\alpha \) and the line \(k\) which is perpendicular to \(\alpha \) and passes through the point \(A = [0;0;1]\).
\(\sqrt{3}\)
\(2\)
\(4\)
\(\sqrt{6}\)