9000039004
Level:
B
Find all \(x\) such that the expression \( (x - 1)(x - 7) \) is negative.
\(x\in (1;7)\)
\(x\in (-\infty ;1)\cup (7;+\infty )\)
No such \(x\)
exists.
\(x\in \mathbb{R}\)
B
9000045704
Level:
B
Given a right triangle \(ABC\) with the right angle at $C$ and an altitude $v$ (see the picture). Find the
valid relation between the angle \(\beta \)
and the lengths in the triangle.
\(\sin \beta = \frac{v}
{a}\)
\(\mathop{\mathrm{tg}}\nolimits \beta = \frac{a}
{v}\)
\(\cos \beta = \frac{v}
{a}\)
\(\mathop{\mathrm{tg}}\nolimits \beta = \frac{v}
{a}\)
9000039003
Level:
B
Assuming \(x\in \mathbb{R}\),
find the solution set of the following system.
\[
x + 2 > 2x + 3 > 3x + 5
\]
\((-\infty ;-2)\)
\((-2;+\infty )\)
\((-\infty ;-1)\)
\((-2;-1)\)
9000046403
Level:
B
Consider an isosceles triangle, i.e. the triangle with two
sides of equal length. The length of the third side is
\(4\, \mathrm{cm}\). One of the
interior angles is \(120^{\circ }\).
Find the area of this triangle.
\(\frac{4\sqrt{3}}
{3} \, \mathrm{cm}^{2}\)
\(4\sqrt{3}\, \mathrm{cm}^{2}\)
\(\frac{8\sqrt{3}}
{3} \, \mathrm{cm}^{2}\)
9000046506
Level:
B
Identify the optimal first step convenient to solve the following trigonometric
equation. Do not consider the step which is possible but does not help to solve the
equation.
\[
\sin 2x =\mathop{\mathrm{tg}}\nolimits x
\]
\(2\sin x\cdot \cos x = \frac{\sin x}
{\cos x}\)
substitution \( 2x = z\)
\(\sin x = \frac{\mathop{\mathrm{tg}}\nolimits x}
{2} \)
\(\cos ^{2}x -\sin ^{2}x =\mathop{\mathrm{tg}}\nolimits x\)
9000046404
Level:
B
The parallelogram has sides of the length
\(5\, \mathrm{cm}\) and
\(4\, \mathrm{cm}\) (see the picture). The area of this
parallelogram is \(S = 10\sqrt{2}\, \mathrm{cm}^{2}\).
Find the measure of the smaller of the interior angles.
\(45^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)
9000046509
Level:
B
Identify the optimal first step convenient to solve the following trigonometric
equation. Do not consider the step which is possible but does not help to solve the
equation.
\[
2\cos ^{2}x =\sin x + 1
\]
\(2 - 2\sin ^{2}x =\sin x + 1\)
substitution \( \sin x + 1 = z\)
substitution \( \cos x = z\)
\(2\cos ^{2}x = \sqrt{1 -\sin ^{2 } x} + 1\)
9000046406
Level:
B
Find the area of the regular octagon of the perimeter
\(16\, \mathrm{cm}\).
Round the result to two decimal places. (The regular octagon is a polygon which has
eight sides of equal length, see the picture. The perimeter of the octagon is the sum
of the length of all eight sides.)
\(19.31\, \mathrm{cm}^{2}\)
\(3.31\, \mathrm{cm}^{2}\)
\(20.88\, \mathrm{cm}^{2}\)
9000045701
Level:
B
The right angled triangle \(ABC\) is given (see the picture). Choose the correct statement.
\(\cos \beta = \frac{a}
{c}\)
\(\cos \beta = \frac{b}
{c}\)
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{b}
{a}\)
\(\sin \alpha = \frac{c}
{a}\)
9000038608
Level:
B
Find the algebraic form of the following complex number.
\[
8(\cos \pi + \mathrm{i}\sin \pi )
\]
\(- 8\)
\(8\)
\(8 + 8\mathrm{i}\)
\(8 - 8\mathrm{i}\)