A

9000117402

Level: 
A
Determine whether the following planes \(\rho \) and \(\sigma \) are parallel, identical or intersecting. \[ \begin{aligned}[t] \rho \colon &x = 2 + u - v, & \\&y = 1 + 2u + 4v, \\&z = -1 + 3u + 3v;\ u,v\in \mathbb{R}, \\ \end{aligned}\qquad \begin{aligned}[t] \sigma \colon &x = 2 + r - s, & \\&y = 7 + 2r + 4s, \\&z = 5 + 3r + 3s;\ s,t\in \mathbb{R}. \\ \end{aligned} \]
identical
parallel, not identical
intersecting

9000117404

Level: 
A
Determine whether the following planes are parallel, identical or intersecting. \[\begin{aligned} \rho \colon \frac{3} {8}x + \frac{1} {2}y -\frac{2} {3}z - 1 = 0,\qquad \sigma \colon \frac{3} {4}x + y -\frac{4} {3}z - 2 = 0 & & \end{aligned}\]
identical
parallel, not identical
intersecting

9000120303

Level: 
A
Identify a valid relation involving the angle \(\alpha \) defined as an angle between a solid diagonal and a face diagonal through the same vertex in a cube.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{\sqrt{2}} {2} \)
\(\sin \alpha = \frac{\sqrt{3}} {2} \)
\(\cos \alpha = \frac{\sqrt{5}} {3} \)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \sqrt{3}\)
\(\alpha = 45^{\circ }\)

9000117406

Level: 
A
Determine whether the following planes are parallel, identical or intersecting. \[\begin{aligned} \rho \colon \frac{3} {2}x -\frac{1} {4}y + \frac{2} {3}z -\frac{2} {5} = 0,\qquad \sigma \colon \frac{2} {3}x - 4y + \frac{3} {2}z -\frac{5} {2} = 0 & & \end{aligned}\]
intersecting
identical
parallel, not identical

9000121004

Level: 
A
In the cube \(ABCDEFGH\) find the angle between the lines \(S_{AE}S_{HC}\) and \(S_{HC}S_{BF}\), where \(S_{AE}\), \(S_{HC}\) and \(S_{BF}\) are the centers of the segments \(AE\), \(HC\) and \(BF\), respectively.
\(53.13^{\circ }\)
\(26.57^{\circ }\)
\(60^{\circ }\)
\(36.87^{\circ }\)