9000063106
Level:
B
Differentiate the following function.
\[
f(x) =\sin x\cos x
\]
\(f'(x) =\cos 2x,\ x\in \mathbb{R}\)
\(f'(x) = 1,\ x\in \mathbb{R}\)
\(f'(x) = -\cos 2x,\ x\in \mathbb{R}\)
\(f'(x) = -\sin x\cos x,\ x\in \mathbb{R}\)
B
9000046605
Level:
B
Identify an inequality which is true for
\(x = \frac{\pi } {6}\).
\(\sin x\cdot \cos x < \frac{\sqrt{2}}
{2} \)
\(\cos 2x > \frac{\sqrt{2}}
{2} \)
\(\mathop{\mathrm{tg}}\nolimits (-x) > 0\)
\(\mathop{\mathrm{cotg}}\nolimits ^{2}x < 0\)
9000062903
Level:
B
A semicircle is a half of the full circle. An infinite spiral is built from
semicircles with an increasing radius. The radius of the first semicircle is
\(3\, \mathrm{cm}\). The
radius of each semicircle in the spiral is bigger than the radius of the previous
semicircle by one third. Find the total length of the spiral.
\(\infty \)
\(9\pi \)
\(9\)
\(3\pi \)
9000063107
Level:
B
Differentiate the following function.
\[
f(x) =\cos x(1 +\sin x)
\]
\(f'(x) =\cos ^{2}x -\sin ^{2}x -\sin x,\ x\in \mathbb{R}\)
\(f'(x) = -\sin x\cos x,\ x\in \mathbb{R}\)
\(f'(x) =\cos x,\ x\in \mathbb{R}\)
\(f'(x) =\sin x +\sin ^{2}x -\cos ^{2}x,\ x\in \mathbb{R}\)
9000046606
Level:
B
Identify an inequality which is true for
\(x = \frac{3\pi }
{4}\).
\(\mathop{\mathrm{cotg}}\nolimits (-x) > 0\)
\(\sin 2x > 0\)
\(\mathop{\mathrm{tg}}\nolimits x\cdot \mathop{\mathrm{cotg}}\nolimits x < \frac{\sqrt{2}}
{2} \)
\(\cos ^{2}x < 0\)
9000062904
Level:
B
A semicircle is a half of the full circle. An infinite spiral is built from
semicircles with a decreasing radius. The radius of the first semicircle is
\(3\, \mathrm{cm}\). The
radius of each semicircle in the spiral is smaller by one third of the radius of the
previous semicircle. Find the total length of the spiral.
\(9\pi \)
\(9\)
\(\frac{9}
{5}\pi \)
\(\infty \)
9000039106
Level:
B
Find the value of the parameter \(a\)
which guarantees that the quadratic equation
\[
x^{2} + 2ax + a = 0
\]
has a pair of complex conjugate solutions with a nonzero imaginary part.
\(a\in (0;1)\)
\(a\in [ 0;1] \)
\(a\in (-\infty ;0)\cup (1;\infty )\)
Such an \(a\)
does not exist
9000039105
Level:
B
Find the quadratic equation with real coefficients such that one of the solutions is the complex
number \(x_{1} = 1 + 2\mathrm{i}\).
\(x^{2} - 2x + 5 = 0\)
\(x^{2} - 2x + 3 = 0\)
\(x^{2} + 2x + 5 = 0\)
\(x^{2} + 2x - 3 = 0\)
9000046501
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.
\[
\sin x\cdot \cos x = 0
\]
\(\sin 2x = 0\)
\(\cos 2x = 0\)
substitution \( \sin x = z\)
\(\sin ^{2}x\cdot \cos ^{2}x = 0\)
9000046401
Level:
B
Consider the rectangle \(ABCD\)
with length and height \(|AB| = 6\, \mathrm{cm}\)
and \(|BC| = 2\sqrt{3}\, \mathrm{cm}\), respectively.
Find the measure of the angle \( CAB\).
\(30^{\circ }\)
\(45^{\circ }\)
\(60^{\circ }\)