9000120004
Level:
B
Find a conic section which allows to draw a line through the center of this conic and
this line does not have any common point with the conic.
hyperbola
circle
ellipse
parabola
no solution
B
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.
9000117410
Level:
B
Adjust real parameters \(p\)
and \(q\) to
ensure that \(\rho \)
and \(\sigma \)
are parallel but not identical planes.
\[\begin{aligned}
\rho \colon 2x - 3y + 5z + 6 = 0,\qquad \sigma \colon 4x + py + qz - 2 = 0 & &
\end{aligned}\]
\(p = -6;\ q = 10\)
\(p = 6;\ q = 10\)
\(p = 6;\ q = -10\)
\(p = -6;\ q = -10\)
9000115609
Level:
B
Complete the following statement: „A number is divisible by twelve if and only if ...”
it is divisible by three and four.
the sum of its digits is divisible by both two and three.
the sum of the digits is even and the last digit of this number is odd.
the sum of the digits is odd and the last digit of this number is even.
9000107504
Level:
B
Find the angle between the lines
\[
p\colon 2x - 3y + 1 = 0;\quad q\colon 3x + 2y - 3 = 0.
\]
\(90^{\circ }\)
\(60^{\circ }\)
\(0^{\circ }\)
\(30^{\circ }\)
9000108804
Level:
B
The point \(A = [3;2]\) is rotated
about the center \(B = [1;1]\)
by \(60^{\circ }\). Find
the coordinate of its final position. Consider both clockwise and counterclockwise
direction.
\(\left [2\pm \frac{\sqrt{3}}
{2} ; \frac{3}
{2} \mp \sqrt{3}\right ]\)
\(\left [1\pm \frac{\sqrt{3}}
{2} ; \frac{1}
{2} \mp \sqrt{3}\right ]\)
\(\left [2\pm \frac{\sqrt{2}}
{2} ; \frac{3}
{2} \mp \sqrt{2}\right ]\)
\(\left [1\pm \frac{\sqrt{2}}
{2} ; \frac{1}
{2} \mp \sqrt{2}\right ]\)
9000108703
Level:
B
Given points \(A = [1;3]\),
\(C = [4;3]\),
\(B = [x;2]\), find the value of the
parameter \(x\) which ensures
that the vector \(AB\) is
perpendicular to the vector \(AC\).
\(1\)
\(2\)
\(- 1\)
\(0\)
9000107506
Level:
B
Find \(\cos \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
p\colon y = 2x - 11;\quad q\colon y = \frac{1}
{4}x
\]
\(\frac{6\sqrt{85}}
{85} \)
\(\frac{1}
{\sqrt{22}}\)
\(\frac{\sqrt{6}}
{85} \)
\(\frac{\sqrt{17}}
{30} \)
9000108705
Level:
B
Given vectors \(\vec{u} = (1;y;3)\) and
\(\vec{v} = (1;2;1)\), find the value of
the parameter \(y\)
which ensures that the vectors are perpendicular.
\(- 2\)
\(1\)
\(2\)
\(0\)