C

9000150505

Level: 
C
The iron support has the shape of the right triangle \(ABC\) with the side \(AB\) of the length \(30\, \mathrm{cm}\) and the hypotenuse \(AC\) of the length \(50\, \mathrm{cm}\) (see the picture). The maximal allowed force \(F_{1}\) on \(AB\) is \(270\, \mathrm{N}\). Find the maximal force \(G\) allowed at the point \(A\). Hint: The load \(G\) at the point \(A\) can be decomposed to the direction of the hypotenuse and the other side of the triangle as shown in the picture.
\(360\, \mathrm{N}\)
\(450\, \mathrm{N}\)
\(540\, \mathrm{N}\)
\(162\, \mathrm{N}\)

9000150503

Level: 
C
A pendulum constituted of a rope of the length \(l\) and a body is displaced from it's equilibrium. The force due to gravity on the body \(F_{g} = 20\, \mathrm{N}\). The body is higher by \(h = 10\, \mathrm{cm}\) in the displaced position (comparing to the equilibrium position). The tension in the rope in the displaced position is \(F_{1} = 12\, \mathrm{N}\). Find the length of the rope \(l\). Hint: Using a parallelogram, the force of gravity on the body can be decomposed into a force \(F_{1}\) in the direction of the rope and \(F_{2}\) in the perpendicular direction.
\(25\, \mathrm{cm}\)
\(25\, \mathrm{m}\)
\(6\, \mathrm{cm}\)
\(16\frac{2} {3}\, \mathrm{cm}\)

9000150502

Level: 
C
Two hotels and a lake are in a satellite photo. The distance between the hotels is \(400\, \mathrm{m}\) which is \(4\, \mathrm{cm}\) in the photo. The area of the lake in the photo is \(30\, \mathrm{cm}^{2}\). Find the real area of the lake.
\(3\cdot 10^{5}\, \mathrm{m}^{2}\)
\(3\cdot 10^{1}\, \mathrm{m}^{2}\)
\(3\cdot 10^{3}\, \mathrm{m}^{2}\)
There is not enough information to solve this problem.

9000150504

Level: 
C
The object \(y\) is projected using a lens with foci at \(F\) and \(F'\). The focal length of the lens (the distance from the focus to the lens) \(f = 20\, \mathrm{cm}\). The distance from the object \(y\) to the lens \(a = 60\, \mathrm{cm}\). Find the distance from the lens to the image \(y'\).
\(30\, \mathrm{cm}\)
\(600\, \mathrm{cm}\)
\(\frac{20} {3} \, \mathrm{cm}\)
\(25\, \mathrm{cm}\)

9000150104

Level: 
C
Evaluate the following integral on \(\mathbb{R}\). \[ \int \cos x\cdot \left (-3 +\sin x\right )^{5}\, \mathrm{d}x \]
\(\frac{\left (-3+\sin x\right )^{6}} {6} + c\text{, }c\in \mathbb{R}\)
\(6\left (-3 +\sin x\right )^{6} + c,\ c\in \mathbb{R}\)
\(\frac{\left (-3+\cos x\right )^{6}} {6} + c,\ c\in \mathbb{R}\)
\(6\left (-3 +\cos x\right )^{6} + c,\ c\in \mathbb{R}\)

9000146710

Level: 
C
Divide the following two polynomials using long division. \[ \left (x^{3} + 3x^{2} - x + 4\right ) : \left (x^{2} - x + 1\right ) \]
\(x + 4 + \frac{2x} {x^{2}-x+1}\)
\(x + 4 + \frac{2x+8} {x^{2}-x+1}\)
\(x + 2 + \frac{6-2x} {x^{2}-x+1}\)
\(x + 2 + \frac{2x+2} {x^{2}-x+1}\)