9000060604
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} = x\),
\(a_{2} = x + 2\) and
\(a_{3} = 2x\) are
three consecutive terms of an arithmetic sequence.
\(x = 4\)
\(x = 0\)
\(x = 2\)
\(x = 6\)
\(x = 8\)
B
9000062908
Level:
B
A quarter circle is an arc formed by one quarter of the full circle. An infinite spiral is
built from quarter circles with an increasing radius. The radius of the first quarter circle
is \(4\, \mathrm{cm}\).
The radius of each quarter circle in the spiral is one half of the radius of the previous
quarter circle. Find the total length of the spiral.
\(4\pi \)
\(8\)
\(\frac{8}
{3}\)
\(\infty \)
9000060608
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} =\log x\),
\(a_{2} =\log(2x)\) and
\(a_{3} = 1\) are
three consecutive terms of an arithmetic sequence.
\(x = 2.5\)
\(x = 1\)
\(x =\log 2\)
\(x = 2\)
\(x = 1\)
9000060601
Level:
B
Identify the real number \(x\) which
ensures that the numbers \(a_{1} = 10^{2}\),
\(a_{2} = 10^{3}\) and
\(a_{3} = x\) are
three consecutive terms of an arithmetic sequence.
\(x = 1\: 900\)
\(x = 1\: 000\)
\(x = 10\: 000\)
\(x = 1\: 990\)
\(x = 100\: 000\)
9000062909
Level:
B
Consider the square of the side \(4\, \mathrm{cm}\).
The second square is inscribed into this first square by joining the centers of all sides.
In a similar way, the third square is inscribed into the second square by joining the
centers of the sides of the second square and this process continues up to infinity.
Find the sum of the perimeters of all squares.
\(32 + 16\sqrt{2}\)
\(32 - 16\sqrt{2}\)
\(32\)
\(\infty \)
9000063108
Level:
B
Differentiate the following function.
\[
f(x) = x^{5}\mathrm{e}^{x}
\]
\(f'(x) = x^{4}\mathrm{e}^{x}(5 + x),\ x\in \mathbb{R}\)
\(f'(x) = 5x^{4}\mathrm{e}^{x},\ x\in \mathbb{R}\)
\(f'(x) = x^{4}\mathrm{e}^{x}(x - 5),\ x\in \mathbb{R}\)
\(f'(x) = x^{4}\mathrm{e}^{x}(5 + x^{2}),\ x\in \mathbb{R}\)
9000045702
Level:
B
Given a right triangle \(ABC\) (see the picture). Find the
valid relation between the angle \(\alpha \)
and the sides of the triangle.
\(\mathop{\mathrm{tg}}\nolimits \alpha = \frac{a}
{c}\)
\(\sin \alpha = \frac{a}
{c}\)
\(\cos \alpha = \frac{b}
{a}\)
\(\mathop{\mathrm{cotg}}\nolimits \alpha = \frac{b}
{a}\)
9000045710
Level:
B
Find the length \(l\)
of a latitude at \(50^{\circ }\)
N. (Use \(R\)
for the radius of the Earth.)
\(l = 2\pi R\cos 50^{\circ }\)
\(l = 2\pi R\sin 50^{\circ }\)
\(l = 2\pi R\mathop{\mathrm{tg}}\nolimits 50^{\circ }\)
\(l = 2\pi R\mathop{\mathrm{cotg}}\nolimits 50^{\circ }\)
9000045703
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 \(\alpha \)
and the lengths in the triangle.
\(\sin \alpha = \frac{v}
{b}\)
\(\sin \alpha = \frac{v}
{c}\)
\(\sin \alpha = \frac{a}
{v}\)
\(\sin \alpha = \frac{c}
{a}\)
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.
\(x\in \mathbb{R}\)