B

9000128801

Level: 
B
The base \(ABCD\) of a square pyramid \(ABCDV \) has side \(6\, \mathrm{cm}\). The height of the pyramid is \(4\, \mathrm{cm}\). The point \(M\) is the middle of the side \(CV \). Find the distance between the point \(M\) and the plane \(ABC\).
\(2\, \mathrm{cm}\)
\(\frac{\sqrt{34}} {2} \, \mathrm{cm}\)
\(\frac{5} {2}\, \mathrm{cm}\)

9000115607

Level: 
B
Complete the following statement: „The number is divisible by nine if and only if ...”
the sum of its digits is divisible by nine.
the number constituted from the last two digits is divisible by nine.
the sum of its digits is odd.
the last digit of this number is \(9\).

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 \(M\). \[\begin{aligned} \rho \colon x - 2y + 5z - 3 = 0,\qquad M = [3;-1;1] & & \end{aligned}\]
\(\tau \colon x - 2y + 5z - 10 = 0\)
\(\sigma \colon 3x - y + z - 3 = 0\)
\(\nu \colon x - 2y + 5z + 1 = 0\)
\(\omega \colon 3x - y + z - 11 = 0\)

9000115610

Level: 
B
Complete the following statement: „A number is divisible by fifteen if and only if ...”
it is divisible by both three and five.
the sum of its digits is divisible by both three and five.
the sum of its digits is odd and divisible by five.
the last digit of this number is either \(5\) or \(0\).

9000120007

Level: 
B
On a map of a city, the town hall is represented by a point and a river through the city by a straight line. There are places in the city with the property that the direct distance from each place to the town hall is equal to the direct distance to the river. In the following list identify a curve which can be used to join all these places.
parabola
circle
ellipse
hyperbola
none of them