B

2000006704

Level: 
B
Let \(X\) and \(Y \) be intersections of the graph of the function \(f(x) = \frac{3x-5} {2+x}\) with \(x\)- and \(y\)-axis, respectively. Find these points.
\(X = \left[\frac{5}{3},0\right]\), \(Y = \left[0,-\frac{5}{2}\right]\)
\(X = \left[-\frac{5}{2},0\right]\), \(Y = \left[0,\frac{5}{3}\right]\)
\(X = \left[0,\frac{5}{3}\right]\), \(Y = \left[-\frac{5}{2},0\right]\)
\(X = \left[\frac{5}{2},0\right]\), \(Y = \left[0,-\frac{5}{3}\right]\)

2000006701

Level: 
B
A part of the graph of the function \( f(x)=-\frac2x \) is shown in the picture. Identify which of the following statements is true.
The function \( g \) defined by \( g(x)=-\left|f(x)\right| \) is an even function.
The function \( g \) defined by \( g(x)=-\left|f(x)\right| \) is bounded below.
The function $f$ is decreasing on \( (-\infty,0)\).
The function $m$ defined by \( m(x)=f(x)-3 \) is bounded.

2000006604

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \mathrm{cotg}\,{x} \geq -\frac{\sqrt{3}}{3}\] \[ x \in (-\pi ,\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{cotg}\,{x} \geq \frac{1}{2} \] \[ x \in (-\pi ,\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{cotg}\,{x} \geq \frac{\sqrt{3}}{2}\] \[ x \in (-\pi ,\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{cotg}\,{x} \leq \frac{\sqrt{3}}{3}\] \[ x \in (-\pi ,\pi ) \setminus \left\{ 0\right\}\]

2000006603

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \mathrm{cotg}\,{x} \leq 1 \] \[ x \in (-\pi ,\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{cotg}\,{x} \geq 1 \] \[ x \in (-\pi ,\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{tg}\,{x} \leq 1\] \[ x \in (-\pi ,\pi ) \setminus \left\{ 0\right\}\]
\[ \mathrm{tg}\,{x} \geq 1\] \[ x \in (-\pi ,\pi ) \setminus \left\{ 0\right\}\]

2000006602

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \mathrm{tg}\,{x} \leq -\sqrt{3} \] \[ x \in [ -\pi ,\pi ] \setminus \left\{ -\frac{\pi}{2},\frac{\pi}{2} \right\}\]
\[ \mathrm{tg}\,{x} \geq -\sqrt{3} \] \[ x \in [ -\pi ,\pi ] \setminus \left\{ -\frac{\pi}{2},\frac{\pi}{2} \right\}\]
\[ \mathrm{cotg}\,{x} \leq -\sqrt{3} \] \[ x \in [ -\pi ,\pi ] \setminus \left\{ -\frac{\pi}{2},\frac{\pi}{2} \right\}\]
\[ \mathrm{cotg}\,{x} \geq -\sqrt{3} \] \[ x \in [ -\pi ,\pi ] \setminus \left\{ -\frac{\pi}{2},\frac{\pi}{2} \right\}\]

2000006601

Level: 
B
An inequality is solved graphically as shown in the picture. The solution is marked in red. Choose the corresponding inequality.
\[ \mathrm{tg}\,{x} \geq \frac{\sqrt{3}}{3} \] \[ x \in [ 0 ,\pi ] \setminus \left\{ \frac{\pi}{2} \right\}\]
\[ \mathrm{tg}\,{x} \geq \frac{\sqrt{3}}{2} \] \[ x \in [ 0 ,\pi ] \setminus \left\{ \frac{\pi}{2} \right\}\]
\[ \mathrm{cotg}\,{x} \geq \frac{\sqrt{3}}{2} \] \[ x \in [ 0 ,\pi ] \setminus \left\{ \frac{\pi}{2} \right\}\]
\[ \mathrm{cotg}\,{x} \geq \frac{\sqrt{3}}{3} \] \[ x \in [ 0 ,\pi ] \setminus \left\{ \frac{\pi}{2} \right\}\]

2000006510

Level: 
B
The bases of the prism shown in the figure are regular hexagons \(ABCDEF\) and \(A'B'C'D'E'F'\). The side edges are perpendicular to the bases. Let \(k\) be a line through the points \(A\) and \(C\) (see the picture). How many diagonals of the prism are parallel to the line \(k\)?
\(3\)
\(1\)
\(2\)
\(0\)

2000006509

Level: 
B
The bases of the prism shown in the figure are regular hexagons \(ABCDEF\) and \(A'B'C'D'E'F'\). The lateral edges are perpendicular to the bases. Let \(k\) be a line through the points \(A\) and \(C\) (see the picture). How many lateral faces of the prism are perpendicular to the line \(k\)?
\(2\)
\(4\)
\(1\)
\(0\)

2000006508

Level: 
B
The bases of the prism shown in the figure are regular hexagons \(ABCDEF\) and \(A'B'C'D'E'F'\). The lateral edges are perpendicular to the bases. Let \(\pi\) be a plane through the points \(B\), \(D\), \(D'\), \(B'\) (see the picture). How many lateral faces of the prism are perpendicular to the plane \(\pi\)?
\(2\)
\(1\)
\(4\)
\(0\)

2000006507

Level: 
B
The bases of the prism shown in the figure are regular hexagons \(ABCDEF\) and \(A'B'C'D'E'F'\). The lateral edges are perpendicular to the bases. Let \(\pi\) be a plane through the points \(B\), \(D\), \(D'\), \(B'\) (see the picture). How many diagonals of the prism are perpendicular to the plane \(\pi\)?
\(2\)
\(4\)
\(3\)
\(1\)