1103018906
Level:
A
In the cube \( ABCDEFGH \), find the angle between two face diagonals \( EG \) and \( EB \).
\( 60^{\circ} \)
\( 90^{\circ} \)
\( 45^{\circ} \)
\( 30^{\circ} \)
A
1103018905
Level:
A
In the cube \( ABCDEFGH \) with \( S_{AC} \) being the midpoint of the diagonal \( AC \), let \( \varphi \) be the angle between the line \( EG \) and the line \( GS_{AC} \). Choose the correct expression for \( \varphi \):
\( \mathrm{tg}\,\varphi = \sqrt2 \)
\( \mathrm{sin}\,\varphi = \frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi = \frac{\sqrt2}2 \)
\( \mathrm{cos}\,\varphi = \frac{\sqrt6}3 \)
1103018904
Level:
A
In the cube \( ABCDEFGH \), let \( S_{FG} \) be the midpoint of the edge \( FG \). Find the angle between the lines \( BS_{FG} \) and \( BF \). Round the result to two decimal places.
\( 26.57^{\circ} \)
\( 22.5^{\circ} \)
\( 45^{\circ} \)
\( 54.74^{\circ} \)
1103018903
Level:
A
In the cube \( ABCDEFGH \) with \( S_{AC} \) being the midpoint of the diagonal \( AC \), let \( \varphi \) be the angle between the line \( ES_{AC} \) and the bottom face \( ABCD \). Choose the correct expression for \( \varphi \).
\( \mathrm{tg}\,\varphi = \sqrt2 \)
\( \mathrm{sin}\,\varphi = \frac{\sqrt2}3 \)
\( \mathrm{tg}\,\varphi = \frac{\sqrt2}2 \)
\( \mathrm{cos}\,\varphi = \frac{\sqrt6}3 \)
1103018902
Level:
A
Let \( \varphi \) bet the angle between a space diagonal of a cube and its face diagonal. Choose the correct expression for \( \varphi \).
\( \mathrm{tg}\,\varphi = \frac{\sqrt2}2 \)
\( \mathrm{sin}\,\varphi = \frac{\sqrt2}2 \)
\( \mathrm{tg}\,\varphi = \frac{\sqrt3}3 \)
\( \mathrm{tg}\,\varphi = \frac{\sqrt6}3 \)
1103018901
Level:
A
In the cube \( ABCDEFGH \), find the angle between the space diagonals \( AG \) and \( BH \). Round the result to two decimal places.
\( 70.53^{\circ} \)
\( 45^{\circ} \)
\( 54.74^{\circ} \)
\( 35.27^{\circ} \)
1103037402
Level:
A
The graph of the function \(-0.3^x+4\) is below. What is the range of \( f \)?
\( (-\infty;4) \)
\( (4;\infty) \)
\( (-\infty;4] \)
\( (-\infty;\infty) \)
1103037401
Level:
A
The graph of the function \(3^x-9\) is below. What is the domain of \( f \)?
\( (-\infty;\infty ) \)
\( (-9;\infty ) \)
\( (-\infty;9) \)
\( [-9;\infty ) \)
1103024310
Level:
A
The picture shows the triangle \( KLM \) with indicated vectors \( \overrightarrow{a} \), \( \overrightarrow{b} \), \( \overrightarrow{c} \) in a coordinate system. What are the vector coordinates \( \overrightarrow{b} \)? Express \( \overrightarrow{b} \) as a linear combination of \( \overrightarrow{a} \) and \( \overrightarrow{c} \).
\( \overrightarrow{b} = \left(1;3;4.5\right);\ \overrightarrow{b} = \frac12\overrightarrow{a} + \frac12\overrightarrow{c} \)
\( \overrightarrow{b} = \left(3;1;4.5\right);\ \overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c} \)
\( \overrightarrow{b} = \left(1;3;4.5\right);\ \overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c} \)
\( \overrightarrow{b} = \left(3;1;4.5\right);\ \overrightarrow{b} = \frac12\overrightarrow{a} + \frac12\overrightarrow{c} \)
1103024309
Level:
A
Given the vectors \( \overrightarrow{a} \), \( \overrightarrow{b} \), \( \overrightarrow{c} \) shown in the picture, express a vector \( \overrightarrow{b} \) as a linear combination of vectors \( \overrightarrow{a} \) and \( \overrightarrow{c} \).
\( \overrightarrow{b} = 2\overrightarrow{a} + \overrightarrow{c} \)
\( \overrightarrow{b} = 2\overrightarrow{a} - \overrightarrow{c} \)
\( \overrightarrow{b} = -2\overrightarrow{a} + \overrightarrow{c} \)
\( \overrightarrow{b} = -2\overrightarrow{a} - \overrightarrow{c} \)