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 \(5\).
\(c\in \{ - 27;23\}\)
\(c\in \{25\}\)
\(c\in \{5;25\}\)
\(c\in \{ - 25;25\}\)
B
9000150102
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int 2\sin 2x\, \mathrm{d}x
\]
\(-\cos 2x + c,\ c\in \mathbb{R}\)
\(\cos 2x + c,\ c\in \mathbb{R}\)
\(- 4\cos 2x + c,\ c\in \mathbb{R}\)
\(4\cos 2x + c,\ c\in \mathbb{R}\)
9000149406
Level:
B
Given points \(A = [2;-5]\),
\(B = [2;3]\) and
\(C = [-4;-1]\), find the length of the altitude of the triangle \(ABC\) through the point \(C\). Hint: The altitude through the point
\(C\) of a triangle
\(ABC\) is the perpendicular line segment drawn from the vertex \(C\) to the line containing the side \(AB\).
\(6\)
\(\sqrt{2}\)
\(\frac{3}
{2}\)
The points \(A\),
\(B\),
\(C\) do
not define a triangle.
9000149402
Level:
B
Find the distance from the origin (i.e. the point
\([0.0]\)) to the
line \(p\colon x + 2y + 5 = 0\).
\(\sqrt{5}\)
\(1\)
The origin is on the line \(p\).
\(8\)
9000149306
Level:
B
Given a translation of a plane, find the property of a line obtained by translating a line
\(r\). The
line \(r\) is
neither parallel not perpendicular to the translation vector.
The resulting line is parallel to the line
\(r\).
The resulting line is perpendicular to the translation vector.
The resulting line is perpendicular to the line
\(r\).
The resulting line is the line \(r\).
(The line \(r\)
is mapped into itself.)
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 \(p\) )
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
9000141504
Level:
B
Let \(A\) be a
set with \(n\)
mutually different elements. If we add one element to the set
\(A\), the number of
\(3\)-combinations
of a set \(A\) is
increased by \(21\).
Find \(n\).
\(7\)
\(6\)
\(43\)
9000146207
Level:
B
Factor the following expression.
\[
4a^{2} -\left (a - 1\right )^{2}
\]
\(\left (a + 1\right )\left (3a - 1\right )\)
\(\left (a - 1\right )\left (3a - 1\right )\)
\(\left (a + 1\right )\left (3a + 1\right )\)
\(\left (a - 1\right )\left (3a + 1\right )\)
9000146208
Level:
B
Factor the following expression.
\[
\left (2x - 1\right )^{2} -\left (x + 3\right )^{2}
\]
\(\left (x - 4\right )\left (3x + 2\right )\)
\(\left (x - 4\right )\left (3x - 2\right )\)
\(\left (x + 4\right )\left (3x + 2\right )\)
\(\left (x + 4\right )\left (3x - 2\right )\)