C

9000124501

Level: 
C
Similar triangles can be used to estimate the distance from a distant object of a given width. Consider a door of the width \(85\, \mathrm{cm}\). A man stands in an unknown distance from the door and holds a thin pencil vertically in his arm in the distance \(35\, \mathrm{cm}\) from his face. If he closes the left eye, the right eye, the pencil and the left side of the door are aligned in one line. In a similar way, his left eye, the pencil and the right hand side of the door are also aligned in one line, which is apparent when closing the right eye. Assuming the distance \(6\, \mathrm{cm}\) between his eyes, estimate the distance from the man to the door. Give your answer in meters and round to one decimal place.
\(5.3\, \mathrm{m}\)
\(5.0\, \mathrm{m}\)
\(0.5\, \mathrm{m}\)
\(4.5\, \mathrm{m}\)

9000124505

Level: 
C
The picture shows the virtual image \(y'\) of the object \(y\) as created by a concave lens. The points \(F\) and \(F'\) are focal points of the lens. The distance from the lens to each of the focal points is \(20\, \mathrm{cm}\). The object \(y\) is \(25\, \, \mathrm{cm}\) height and it is in the distance \(50\, \mathrm{cm}\) from the lens. Find the height of the virtual image \(y'\).
\(\frac{50} {7} \, \mathrm{cm}\)
\(10\, \mathrm{cm}\)
\(\frac{50} {3} \, \mathrm{cm}\)
\(\frac{175} {2} \, \mathrm{cm}\)

9000138303

Level: 
C
Two dice are rolled. Find the probability that we get the number \(6\) on just one of the dice and the sum of the numbers on both dice is \(8\).
\(\frac{2} {36}\doteq 0{.}0556\)
\(\frac{5} {36}\doteq 0{.}1389\)
\(\frac{11} {36}\doteq 0{.}3056\)
\(\frac{14} {36}\doteq 0{.}3889\)

9000123107

Level: 
C
In the following list identify a line such that the line has a unique intersection with the hyperbola \[ x^{2} - y^{2} = 5 \] but the line is not the tangent to this hyperbola.
\(p\colon \frac{x} {5} + \frac{y} {5} = 1\)
\(p\colon y = 5x\)
\(p\colon 2x + y = 5\)
\(\begin{aligned}[t] p\colon x& = 1 & \\y & = -1 + t\text{; }t\in \mathbb{R} \\ \end{aligned}\)

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

9000124503

Level: 
C
A tall radio mast is attached by several cables. The length of each cable is \(30\, \mathrm{m}\) and all cables are attached \(2\, \mathrm{m}\) under the top of the mast. The second end of the cable is anchored to the ground. The cable is in the height \(6\, \mathrm{m}\) if measured directly above the point which is in the distance \(8\, \mathrm{m}\) from the point where the cable is anchored to the ground. Find the height of the mast.
\(20\, \mathrm{m}\)
\(24\, \mathrm{m}\)
\(22.5\, \mathrm{m}\)
\(24.5\, \mathrm{m}\)

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 \(y = 1\).