B

9000106304

Level: 
B
Find the third coordinate of the point \(B = [2;0;?]\) using the fact that this point is in the plane \(\alpha \) defined by the equation \[ \alpha \colon 2x + y - z - 5 = 0. \] Use the point \(B\) to find the angle \(\varphi \) between the plane \(\alpha \) and the line \(AB\), where \(A = [0;0;1]\).
\(\varphi = 60^{\circ }\)
\(\varphi = 45^{\circ }\)
\(\varphi = 30^{\circ }\)
\(\varphi = 75^{\circ }\)

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

9000104307

Level: 
B
Assuming \(a\in \left (0;2\right )\), solve the following inequality. \[ a\left (a - 2\right )x > 1 \]
\(\left (-\infty ; \frac{1} {a\left (a-2\right )}\right )\)
\(\left ( \frac{1} {a\left (a-2\right )};\infty \right )\)
\(\emptyset \)
\(\left \{ \frac{1} {a\left (a-2\right )}\right \}\)