9000149709
Level:
B
Find the vertex of the following parabola.
\[
y^{2} - 12x + 4y + 64 = 0
\]
\([5;-2]\)
\([5;2]\)
\([-5;2]\)
\([-5;-2]\)
B
9000149305
Level:
B
Given a translation \(T\)
of a plane, find the lines which are mapped to the same line by
\(T\).
All lines parallel to the translation vector are mapped into itself.
All lines perpendicular to the translation vector are mapped into itself.
There are no lines which are mapped into itself by the translation.
Every line is mapped into itself by the translation.
9000149409
Level:
B
Find all lines which are parallel to \(p\colon x - 3y + 2 = 0\)
and the distance from every of these lines to
\(p\) is
\(\sqrt{10}\).
\(p_{1}\colon x - 3y + 12 = 0\),
\(p_{2}\colon x - 3y - 8 = 0\)
\(p\colon x - 3y = 0\)
\(p\colon x - 3y + \sqrt{10} = 0\)
\(p_{1}\colon x - 3y + \sqrt{10} = 0\),
\(p_{2}\colon x - 3y -\sqrt{10} = 0\)
9000149408
Level:
B
On the \(x\)-axis
find the points such that the distance from these points to the line
\(p\colon x - 2y + 2 = 0\) is
\(\sqrt{5}\).
\([3;0]\),
\([-7;0]\)
\([5;0]\)
\(\left [\sqrt{5};0\right ]\),
\(\left [-\sqrt{5};0\right ]\)
\([3;7]\)
9000149708
Level:
B
Find the vertex of the following parabola.
\[
x^{2} + 8x - 4y + 24 = 0
\]
\([-4;2]\)
\([-4;-2]\)
\([4;2]\)
\([4;-2]\)
9000149401
Level:
B
Find the distance from the point \(P = [-4;2]\)
to the line \(3x - 4y - 5 = 0\).
\(5\)
\(1\)
The line contains \(P\).
\(\sqrt{5}\)
9000149405
Level:
B
Find all the values of the parameter
\(c\) so that the distance from the point \(M = [2;-1]\)
to the line \(3x + 4y + c = 0\) is \(5\).
\(c\in \{ - 27;23\}\)
\(c\in \{25\}\)
\(c\in \{5;25\}\)
\(c\in \{ - 25;25\}\)
9000141509
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following inequality.
\[
2\cdot \left({x - 1\above 0.0pt
x - 3}\right) + x\cdot (x - 9)\leq - 8
\]
\(\{3;4;5\}\)
\(\{1;2;3;4;5\}\)
\([ 1;5] \)
9000141502
Level:
B
Let \(A\) be set with
\(n\) mutually different elements.
The number of \(5\)-permutations
with repetition is \(1024\).
Find \(n\). (The term
„\(k\)-permutation
with repetition” stands for an ordered arrangement of
\(k\) objects from
a set of \(n\)
objects, when each object can be chosen more than once.)
\(4\)
\(5\)
\(2\)
9000141503
Level:
B
Let \(A\) be a
set with \(n\)
mutually different elements. If two elements are removed from
\(A\), the number of all
permutations of set \(A\)
decreases \(20\)-times.
Find \(n\).
\(5\)
\(4\)
\(5\) or
\(- 4\)