2010016807
Level:
C
The expression \( \frac{\sin x-\sin^3 x}{\cos x-\cos^3 x } \) for $x\in\left(0;\frac{\pi}{2}\right)$ is equal to:
\( \mathrm{cotg}\,x \)
\( \mathrm{tg}\,x \)
\( \sin x \cdot \cos x \)
\( 2\,\mathrm{tg}\,x \)
Sine, cosine, tangent and cotangent
2010016806
Level:
C
The domain of the expression \( \frac{\cos^2 x}{1+\sin x} \) is the set:
\( \left\{x\in\mathbb{R}\colon x\neq\frac{3\pi}2 + 2k\pi,\ k\in\mathbb{Z} \right\} \)
\( \mathbb{R}\)
\( \left\{x\in\mathbb{R}\colon x\neq\frac{\pi}2 + 2k\pi,\ k\in\mathbb{Z} \right\} \)
\( \left\{x\in\mathbb{R}\colon x\neq \pi + 2k\pi,\ k\in\mathbb{Z} \right\} \)
2010016805
Level:
A
The value of the expression \( 3\cos\frac{\pi}4 - 3\sin\frac{\pi}4 + 2\left(\cos\frac{\pi}3 - \sin\frac{\pi}6 \right) \) is:
\( 0\)
\( \sqrt2\)
\( 1\)
\( \frac12\)
2010016804
Level:
B
How many \( x \)-intercepts has the graph of the function \( f(x)=\sin 2x \) on the interval \( [ -\pi; 2\pi ] \)?
\( 7\)
\( 5\)
\( 8\)
\( 6\)
2010016803
Level:
B
The value of \( \cos\left(-\frac{28\pi}3\right) \) is the same as the value of
\( \cos\frac{4\pi}3 \).
\( \cos\frac{\pi}3 \).
\( \cos\left(-\frac{7\pi}3\right) \).
\( \cos\frac{5\pi}3 \).
2010016802
Level:
B
Choose the true statement:
\( \sin 240^{\circ} < \sin 120^{\circ} \)
\( \cos50^{\circ} < \cos130^{\circ} \)
\( \sin 300^{\circ} < \sin 270^{\circ} \)
\( \cos330^{\circ} < \cos150^{\circ} \)
2010016801
Level:
B
To which quadrant does the angle \( \varphi \) belong if \( \cos\varphi=0.8 \) and \( \sin\varphi < 0 \)?
IV.
I.
II.
III.
2010016408
Level:
B
Consider the function \(f(x)=\mathop{\mathrm{cotg}}\nolimits x\) with
domain restricted to the interval \( (0;\pi )\).
In the following list identify the function with domain
\(\left (0; \frac{\pi } {2}\right )\).
\(f(2\cdot x)\)
\(f(x+2)\)
\(f(x-2)\)
\(f(\frac{x}2)\)
2010016407
Level:
B
Identify the transformation which transforms the graph of the function
\(g(x) =\cos (2x)\) to the graph
of the function \(f(x) =\cos (2x -1)\).
Shift of graph of \(g\)
by \(\frac{1}
{2}\) of a
unit to the right.
Shift of graph of \(g\)
by \(\frac{1}
{2}\) of a
unit to the left.
Shift of graph of \(g\)
by \(1\) unit to the left.
Shift of graph of \(g\)
by \(1\) unit to the right.
2010016406
Level:
B
In the following list identify a true statement about the function
\(f(x) =\sin x\) on the interval \(I=\left( -\frac{\pi }{2}; \frac{\pi } {2} \right) \).
The function \(f\) does not have
a minimum or maximum on \(I\).
The function \(f\) has a unique minimum and no maximum on \(I\).
The function \(f\) has a unique maximum and no minimum on \(I\).
The function \(f\) has a unique maximum and a unique minimum on \(I\).