B

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

9000120005

Level: 
B
The executives of a camp organize a holiday game. For this game it is important that the direct distance kitchen - tent - fireplace is equal for all tents in the camp. Is this information enough to determine the curve passing through all the tents in the camp? Is this curve a conic? If yes, determine which conic.
Yes, all the tents are on an ellipse.
Yes, all the tents are on a circle.
Yes, all the tents are on a parabola.
Yes, all the tents are on a hyperbola.
No, we do not have enough information to draw any conclusion.

9000115602

Level: 
B
Complete the following statement: „The number is divisible by three if and only if ...”
the sum of its digits is divisible by three.
the number constituted from the last two digits is divisible by three.
the sum of its digits is odd.
the last digit of this number is \(3,\ 6\) or \(9\).

9000108704

Level: 
B
Consider a pair of vectors \(\vec{u} = (1;0;-1)\) and \(\vec{v} = (2;-1;1)\). Find all the vectors \(\vec{w}\) which are perpendicular to both \(\vec{u}\) and \(\vec{v}\) and satisfy \(\left |\vec{w}\right | = 2\).
\(\vec{w} = \left (\frac{2\sqrt{11}} {11} ; \frac{6\sqrt{11}} {11} ; \frac{2\sqrt{11}} {11} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{11}} {11} ;-\frac{6\sqrt{11}} {11} ;-\frac{2\sqrt{11}} {11} \right )\)
\(\vec{w} = (-1;-3;-1)\), \(\vec{w} = (1;3;1)\)
\(\vec{w} = \left (-\frac{1} {2};-\frac{3} {2};-\frac{1} {2}\right )\), \(\vec{w} = \left (\frac{1} {2}; \frac{3} {2}; \frac{1} {2}\right )\)
\(\vec{w} = \left (\frac{2\sqrt{2}} {3} ; \frac{3\sqrt{2}} {2} ; \frac{2\sqrt{2}} {3} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{2}} {3} ;-\frac{3\sqrt{2}} {2} ;-\frac{2\sqrt{2}} {3} \right )\)

9000115603

Level: 
B
Complete the following statement: „The number is divisible by four if and only if ...”
the number constituted from the last two digits is divisible by four.
the sum of its digits is divisible by four.
the last digit of this number is \(4\).
the last digit of this number is even.