C

9000123102

Level: 
C
Find a true statement about the ellipse \[ x^{2} + 4y^{2} - 8y = 0. \]
The tangent to the ellipse can pass through any point on the line \(y = -1\).
The tangent to the ellipse can pass through any point on the line \(x = 1\).
The tangent to the ellipse can pass through the point \([-1,1]\).
The tangent to the ellipse can pass through any point on the line \(y = 1\).

9000123103

Level: 
C
The ellipse \[ 5x^{2} + 9y^{2} = 45 \] has tangent \(2x + 3y = 9\). Find the values of the real parameter \(k\) which ensure that the line \(y = kx + 3\) is a secant for the ellipse.
\(k\in \left (-\infty ,-\frac{2} {3}\right )\cup \left (\frac{2} {3},\infty \right )\)
\(k\in \left [ -\frac{2} {3}, \frac{2} {3}\right ] \)
\(k\in \left (-\frac{2} {3}, \frac{2} {3}\right )\)
\(k\in \left (-\infty ,-\frac{2} {3}\right ] \cup \left [ \frac{2} {3},\infty \right )\)

9000117706

Level: 
C
Satellites travel along approximately circular paths. Consider a satellite in the height \(h\) measured from the Earth surface. Further, consider the coordinate system with origin on the Earth surface directly below the satellite and the \(y\)-axis oriented up (to the satellite). The \(x\)-axis is perpendicular to \(y\)-axis and it is in the plane defined by the trajectory of the satellite. Neglect the Earth's rotation and find the equation which describes the path of the satellite. The Earth radius is \(R\).
\(x^{2} + (y + R)^{2} = (R + h)^{2}\)
\(x^{2} + y^{2} = (R + h)^{2}\)
\(x^{2} + (y + R)^{2} = h^{2}\)
\(x^{2} + y^{2} = h^{2}\)

9000120308

Level: 
C
The height \(v\) of a regular hexagonal prism is a double of its side \(a\). The volume of the prism is \(648\sqrt{3}\, \mathrm{cm}^{3}\). Use this information to find the length of the longest solid diagonal in the prism.
\(12\sqrt{2}\, \mathrm{cm}\)
\(10\sqrt{6}\, \mathrm{cm}\)
\(12\sqrt{6}\, \mathrm{cm}\)
\(6\sqrt{10}\, \mathrm{cm}\)
\(\sqrt{432}\, \mathrm{cm}\)

9000120304

Level: 
C
The side of a regular hexagonal prism \(ABCDEFA'B'C'D'E'F'\) is \(a = 3\, \mathrm{cm}\) and the height \(v = 8\, \mathrm{cm}\). Find the length of the diagonal \(AD'\).
\(10\, \mathrm{cm}\)
\(\sqrt{73}\, \mathrm{cm}\)
\(\sqrt{82}\, \mathrm{cm}\)
\(2\sqrt{8}\, \mathrm{cm}\)
\(2\sqrt{6}\, \mathrm{cm}\)

9000120305

Level: 
C
The side of a regular hexagonal prism \(ABCDEFA'B'C'D'E'F'\) shown in the picture is \(a = 3\, \mathrm{cm}\) and the height is \(v = 8\, \mathrm{cm}\). Find the angle between the diagonal \(AD'\) and the base plane \(ABC\) (round your result to the nearest degree).
\(53^{\circ }\)
\(37^{\circ }\)
\(45^{\circ }\)
\(61^{\circ }\)
\(72^{\circ }\)

9000117701

Level: 
C
A body is thrown at the initial angle \(\alpha = 30^{\circ }\) and the initial velocity \(v_{0} = 20\, \mathrm{m}/\mathrm{s}\). The trajectory of the body is a part of parabola. Find the directrix of this parabola. Hint: The coordinates of the moving body as functions of time are \[ \begin{aligned}x& = v_{0}t\cdot \cos \alpha , & \\y& = v_{0}t\cdot \sin \alpha -\frac{1} {2}gt^{2}. \\ \end{aligned} \] Consider the standard acceleration due to gravity \(g = 10\, \mathrm{m}/\mathrm{s}^{2}\).
\(y = 20\)
\(y = 5\)
\(y = 15\)
\(y = 10\)

9000117702

Level: 
C
The Earth travels around the Sun on an elliptical orbit. The Sun is in the focus of this ellipse. The maximal distance from Earth to the Sun is \(152.1\cdot 10^{6}\, \mathrm{km}\), the minimal distance from Earth to the Sun is \(147.1\cdot 10^{6}\, \mathrm{km}\). Find the length of the semi-minor axis (one half of the length of the shorter axis) and round your answer to the nearest \(10^{4}\, \mathrm{km}\).
\(149.58\cdot 10^{6}\, \mathrm{km}\)
\(2.58\cdot 10^{6}\, \mathrm{km}\)
\(299.21\cdot 10^{6}\, \mathrm{km}\)
\(149.61\cdot 10^{6}\, \mathrm{km}\)