B

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 \(A = [0,0,1]\).
\(\sqrt{3}\)
\(2\)
\(4\)
\(\sqrt{6}\)

9000106306

Level: 
B
Find the general equation of the plane which is perpendicular to the plane \(\alpha \) \[ \alpha \colon 2x + y - z - 5 = 0 \] and contains the line \(AB\), where \(A = [0,0,1]\) and \(B\) is a point in \(\alpha \) defined by it's first two coordinates \[ B = [2,0,?]. \]
\(x - y + z - 1 = 0\)
\(x + y - z + 1 = 0\)
\(2x - y + z - 1 = 0\)
\(- 2x + y - z + 1 = 0\)

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 \(\alpha \). \[ \alpha \colon 2x + y - z - 5 = 0 \]
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 11& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 10& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z +\phantom{ 1}1& = 0& \\2x + y - z - 12& = 0 \\ \end{aligned}\)
\(\begin{aligned}[t] 2x + y - z + 1& = 0& \\2x + y - z - 9& = 0 \\ \end{aligned}\)