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.
\(k\in \left (-\infty ,-\frac{2} {3}\right )\cup \left (\frac{2} {3},\infty \right )\)
\(k\in \left [ -\frac{2} {3}, \frac{2} {3}\right ] \)
\(k\in \left (-\frac{2} {3}, \frac{2} {3}\right )\)
\(k\in \left (-\infty ,-\frac{2} {3}\right ] \cup \left [ \frac{2} {3},\infty \right )\)

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 \(10^{4}\, \mathrm{km}\).
\(149.58\cdot 10^{6}\, \mathrm{km}\)
\(2.58\cdot 10^{6}\, \mathrm{km}\)
\(299.21\cdot 10^{6}\, \mathrm{km}\)
\(149.61\cdot 10^{6}\, \mathrm{km}\)

9000117703

Level: 
C
For an isothermal process in an ideal gas the product \(pV \) is constant (Boyle's law). In a pressure-volume diagram which shows \(p\) as a function of \(V \) this law describes a hyperbola (called isotherm). Do we have enough information to identify the asymptotes? If so, find these asymptotes.
\(p = 0\), \(V = 0\)
\(p = V \), \(p = -V \)
\(p = 0\), \(p = V \)
It is not possible to draw any conclusion.