B

9000115608

Level: 
B
Complete the following statement: „The number is divisible by ten if and only if ...”
the last digit of this number is \(0\).
the sum of its digits is divisible by ten.
the number constituted from the last two digits is divisible by five.
the last digit of this number is even.

9000117401

Level: 
B
Find the intersection of the planes \(\rho \) and \(\sigma \). \[\begin{aligned} \rho \colon 2x - 5y + 4z - 10 = 0,\qquad \sigma \colon x - y - z - 2 = 0 & & \end{aligned}\]
\(\begin{aligned}[t] p\colon x& = 3t, & \\y & = -2 + 2t, \\z & = t;\ t\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] q\colon x& = 2s - 10,& \\y & = 5s - 10, \\z & = s;\ s\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] a\colon x& = 2u - 4,& \\y & = 2u - 4, \\z & = u;\ u\in \mathbb{R} \\ \end{aligned}\)
\(\begin{aligned}[t] b\colon x& = 3v + 1,& \\y & = v - 2, \\z & = v;\ v\in \mathbb{R} \\ \end{aligned}\)

9000117408

Level: 
B
In the following list find the plane perpendicular to the plane \(\rho \). \[\begin{aligned} \rho \colon 2x - 3y + 7z - 2 = 0 & & \end{aligned}\]
\(\omega \colon x + 3y + z + 7 = 0\)
\(\tau \colon - 2x + 3y - 7z + 2 = 0\)
\(\nu \colon - 2x - 3y + 7z + 2 = 0\)
\(\sigma \colon 7x - 3y + 2z - 2 = 0\)

9000117409

Level: 
B
Find the plane parallel to \(\rho \) passing through the point \(M\). \[\begin{aligned} \rho \colon x - 2y + 5z - 3 = 0,\qquad M = [3;-1;1] & & \end{aligned}\]
\(\tau \colon x - 2y + 5z - 10 = 0\)
\(\sigma \colon 3x - y + z - 3 = 0\)
\(\nu \colon x - 2y + 5z + 1 = 0\)
\(\omega \colon 3x - y + z - 11 = 0\)

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