C

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 \(y = 1\).

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.
\(40\, \mathrm{m}\)
\(20\, \mathrm{m}\)
\(80\, \mathrm{m}\)
\(10\, \mathrm{m}\)

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

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).
\(\frac{3} {2}\, \mathrm{m}\)
\(\frac{2} {3}\, \mathrm{m}\)
\(\frac{1} {6}\, \mathrm{m}\)
\(\frac{20} {9} \, \mathrm{m}\)