2010016808
Level:
C
Simplifying the expression \( \cos 2x + \sin 2x \cdot \mathrm{tg}\, x \) for \( x \in \left(0;\frac{\pi}2\right)\) we get:
\( 1 \)
\( \sin^2x \)
\( \cos^2 x \)
\(2 \sin^2 x \)
C
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:
\( \mathrm{cotg}\,x \)
\( \mathrm{tg}\,x \)
\( \sin x \cdot \cos x \)
\( 2\,\mathrm{tg}\,x \)
2010016806
Level:
C
The domain of the expression \( \frac{\cos^2 x}{1+\sin x} \) is the set:
\( \left\{x\in\mathbb{R}\colon x\neq\frac{3\pi}2 + 2k\pi,\ k\in\mathbb{Z} \right\} \)
\( \mathbb{R}\)
\( \left\{x\in\mathbb{R}\colon x\neq\frac{\pi}2 + 2k\pi,\ k\in\mathbb{Z} \right\} \)
\( \left\{x\in\mathbb{R}\colon x\neq \pi + 2k\pi,\ k\in\mathbb{Z} \right\} \)
2110016708
Level:
C
Let \(ABC\) be a triangle (see the picture). Find the dilated image of the triangle with \( O \) (the origin) being the centre of dilation and the scale factor \( -2 \).
2110016707
Level:
C
Let \( ABC\) be a triangle with the centroid \( T \) (see the picture). Find the dilated image of the triangle with \( T \) being the centre of dilation and the scale factor \( \frac12 \).
2110016706
Level:
C
Let \( ABC \) be a triangle (see the picture). Find the dilated image of the triangle with \( B \) being the centre of dilation and the scale factor \( \frac32 \).
2110016705
Level:
C
Let \( ABCD \) be a square. Find the dilated image of the square with \( S \) being the centre of dilation and scale factor \( \frac12 \). The point \( S \) is also the middle of the square \( ABCD \) (see the picture).
2010016404
Level:
C
Function \( f \) is given completely by the next graph. Identify which of the following statements is true.
\( f(x)=|-\cos x|;\ x\in [ -2\pi;2\pi ]\)
\( f(x)=-|\cos x|;\ x\in [ -2\pi;2\pi ]\)
\( f(x)=|\sin x|;\ x\in [ -2\pi;2\pi ]\)
\( f(x)=-|\sin x|;\ x\in [ -2\pi;2\pi ]\)
2010016401
Level:
C
Identify the function shown in the graph.
\( f(x) = 2\sin x\)
\( f(x) = -2\sin x\)
\( f(x) = |2\sin x|\)
\( f(x) = |\sin x|\)
2010016114
Level:
C
Let a point \(B\) be the intersection point of the sphere \(x^2 + y^2 + z^2 + 4x + 2y - 4z - 8 = 0\) and \(y\)-axis. Find the equations of all the tangent planes to the given sphere at the point \(B\).
\(2x -3y -2z -12 = 0\), \(2x + 3y - 2z -6 = 0\)
\(2x + 3y - 2z +12 = 0\), \(2x -3 y -2z +6 = 0\)
\(2x -3y -2z -12 = 0\), \(2x -3 y -2z +6 = 0\)
\(2x + 3y - 2z +12 = 0\), \(2x + 3y - 2z -6 = 0\)