9000149403
Level:
B
Find the distance from the point \(M = [1;1]\)
to the line \(p\).
\[
\begin{aligned}p\colon x& = 3 + t, &
\\y & = 1 + t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(\sqrt{2}\)
\(2\)
\(1\)
\(0\) (the point is
on the line \(p\) )
B
9000149307
Level:
B
The rotation through an angle \(\alpha = 180^{\circ }\)
is equivalent to some other geometric mapping. Which one?
reflection through the point
reflection through the line
translation
9000149404
Level:
B
Given points \(A = [-3;13]\),
\(K = [0;4]\),
\(L = [-5;-6]\), find the distance
from the point \(A\)
to the line \(KL\).
\(3\sqrt{5}\)
\(3\)
\(5\)
\(\sqrt{5}\)
9000149308
Level:
B
Consider a rotation through an angle either
\(\alpha = 180^{\circ }\) or
\(\alpha = 360^{\circ }\). How
many lines which are mapped into itself exists for this rotation?
infinitely many
none
one
two
9000149407
Level:
B
Find the distance between the lines \(p\colon 3x - 4y + 1 = 0\)
and \(q\colon 3x - 4y + 4 = 0\).
\(\frac{3}
{5}\)
\(1\)
\(4\)
\(0\) (the
lines have an intersection)
9000149410
Level:
B
Find all lines passing through the point
\(A = [-2;-6]\) such that the distance
from the point \([0.0]\)
to these lines is \(2\sqrt{2}\).
\(p_{1}\colon 7x + y + 20 = 0\),
\(p_{2}\colon x - y - 4 = 0\)
\(p\colon 7x - y = 0\)
\(p\colon x + y + 2\sqrt{2} = 0\)
\(p_{1}\colon x - y + 2\sqrt{2} = 0\),
\(p_{2}\colon x + y - 2\sqrt{2} = 0\)
9000149706
Level:
B
Find the center of the following hyperbola.
\[
4x^{2} - 3y^{2} + 8x - 30y - 49 = 0
\]
\([-1;-5]\)
\([-1;5]\)
\([1;-5]\)
\([1;5]\)
9000146202
Level:
B
Expand the following expression.
\[
\left (a^{2} + \sqrt{3}b\right )^{3}
\]
\(a^{6} + 3\sqrt{3}a^{4}b + 9a^{2}b^{2} + 3\sqrt{3}b^{3}\)
\(a^{6} + \sqrt{3}a^{4}b + 3a^{2}b^{2} + 3\sqrt{3}b^{3}\)
\(a^{5} + 3\sqrt{3}a^{4}b + 9a^{2}b^{2} + 3\sqrt{3}b^{3}\)
\(a^{5} + \sqrt{3}a^{4}b + 3a^{2}b^{2} + 3\sqrt{3}b^{3}\)
9000141501
Level:
B
Let \(A\) be set with
\(n\) mutually different
elements. If \(n\) is increased
by \(2\), then number
of \(3\)-permutations
is increased by \(384\).
Find \(n\). (The term
„\(k\)-permutation” stands for
an ordered arrangement of \(k\)
objects from a set of \(n\)
objects.)
\(8\)
\(64\)
\(32\)
9000141510
Level:
B
Assuming \(x\in \mathbb{N}\),
\(n\geq 2\), find the solution set of the following inequality.
\[
\left({ x\above 0.0pt
x - 2}\right)\cdot \left({x\above 0.0pt
2}\right) - 11\cdot \left({x\above 0.0pt
2}\right) + 28 < 0
\]
\(\{4\}\)
\(\{5;6\}\)
\((4;7)\)