9000150102
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int 2\sin 2x\, \mathrm{d}x
\]
\(-\cos 2x + c,\ c\in \mathbb{R}\)
\(\cos 2x + c,\ c\in \mathbb{R}\)
\(- 4\cos 2x + c,\ c\in \mathbb{R}\)
\(4\cos 2x + c,\ c\in \mathbb{R}\)
B
9000149707
Level:
B
Find the center of the following hyperbola.
\[
5x^{2} - 6y^{2} - 30x + 12y + 9 = 0
\]
\([3,1]\)
\([3,-1]\)
\([-3,1]\)
\([-3,-1]\)
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 \(p\).
\(8\)
9000149708
Level:
B
Find the vertex of the following parabola.
\[
x^{2} + 8x - 4y + 24 = 0
\]
\([-4,2]\)
\([-4,-2]\)
\([4,2]\)
\([4,-2]\)
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)
9000149709
Level:
B
Find the vertex of the following parabola.
\[
y^{2} - 12x + 4y + 64 = 0
\]
\([5,-2]\)
\([5,2]\)
\([-5,2]\)
\([-5,-2]\)
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.)
\(8\)
\(64\)
\(32\)
9000141505
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following equation.
\[
\frac{(x + 2)!}
{x!} = 2\cdot \frac{x!}
{(x - 2)!} + 3!
\]
\(\{4\}\)
\(\{4,1\}\)
\(\{1\}\)
9000141506
Level:
B
Assuming \(x\in \mathbb{N}\),
find the solution set of the following equation.
\[
\left({11\above 0.0pt
4} \right) =\left ({11\above 0.0pt
x} \right)
\]
\(\{4,7\}\)
\(\{4\}\)
\(\{ - 4\}\)