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 \(y = 1\).

9000123106

Level: 
C
Find the tangent line \(q\) to the parabola \(4(y - 2) = (x + 1)^{2}\), so that the tangent \(q\) is parallel to the line \(p\colon 4x - 5y + 17 = 0.\)
\(q\colon 20x - 25y + 54 = 0\)
\(q\colon 20x - 25y - 27 = 0\)
\(q\colon 4x - 5y + 27 = 0\)
\(q\colon 4x -5y - 17 = 0\)

9000123108

Level: 
C
Find all the tangents to the hyperbola \(x^{2} - 2y^{2} = 8\) such that the angle between each tangent and the \(x\)-axis is \(45^{\circ }\).
\(y = x + 2\text{, }y = x - 2\text{, }y = -x + 2\text{, }y = -x - 2\)
\(y = x + 2\text{, }y = x - 2\)
\(y = x + 2\text{, }y = -x + 2\)
\(y = x + 2\)

9000124502

Level: 
C
A rectangle-shaped land has dimensions \(3\times 5\, \mathrm{cm}\) on a map with scale \(1\colon 2\: 000\). The owner increased the size of his land by buying some land from his neighbor. The new land has dimensions \(4\times 5\, \mathrm{cm}\) on the map. Find the actual increase of the perimeter of the land (i.e. find the increase in the length of the fence required to enclose the whole land). Give your answer in meters.
\(40\, \mathrm{m}\)
\(20\, \mathrm{m}\)
\(80\, \mathrm{m}\)
\(10\, \mathrm{m}\)

9000124504

Level: 
C
A force due to gravity on a body is \(1\: 800\, \mathrm{N}\). This body has to be lifted to the height \(50\, \mathrm{cm}\) using a slope. The maximal force which can be used to lift the body is \(600\, \mathrm{N}\). Neglect the friction and find the minimal length of the slope required to accomplish this task.Hint: The force due to gravity can be decomposed into two directions. The normal force \(F_{1}\) is compensated by the reaction of the slope. The force \(F_{2}\) parallel to the slope is required to overcome if we wish to lift the body (see the picture).
\(\frac{3} {2}\, \mathrm{m}\)
\(\frac{2} {3}\, \mathrm{m}\)
\(\frac{1} {6}\, \mathrm{m}\)
\(\frac{20} {9} \, \mathrm{m}\)

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 \(8\).
\(\frac{2} {36}\doteq 0{.}0556\)
\(\frac{5} {36}\doteq 0{.}1389\)
\(\frac{11} {36}\doteq 0{.}3056\)
\(\frac{14} {36}\doteq 0{.}3889\)