1103040103
Level:
A
The diagram of an ellipse in the rectangular coordinate system is shown in the picture.What is the semi-major axis of this ellipse?
The segment \( SN \)
The segment \( SK \)
The line \( EF \)
The line \( KL \)
A
1103040102
Level:
A
The diagram of an ellipse in the rectangular coordinate system is shown in the picture. Find the standard form equation of this ellipse:
\( \frac{(x-3)^2}4+\frac{(y-3)^2}9=1 \)
\( \frac{(x-3)^2}9+\frac{(y-3)^2}4=1 \)
\( \frac{(x+3)^2}4+\frac{(y+3)^2}9=1 \)
\( \frac{(x+3)^2}9+\frac{(y+3)^2}4=1 \)
1103040101
Level:
A
The equation of an ellipse is given by \( \frac{(x-2)^2}4+y^2=1 \).
Which of the pictures below shows the given ellipse in the rectangular coordinate system?
1103018906
Level:
A
In the cube \( ABCDEFGH \), find the angle between two face diagonals \( EG \) and \( EB \).
\( 60^{\circ} \)
\( 90^{\circ} \)
\( 45^{\circ} \)
\( 30^{\circ} \)
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 \( f \)?
\( (-\infty;4) \)
\( (4;\infty) \)
\( (-\infty;4] \)
\( (-\infty;\infty) \)