C

1103077202

Level: 
C
Let \( ABCDEF \) be a regular hexagon. Six circles of equal radii are drawn touching each other with their centres at the hexagon vertices (see the picture). Calculate the area of the coloured region inside the hexagon if you know that the perimeter of the hexagon \( ABCDEF \) is \( 36\,\mathrm{cm} \). Round the result to two decimal places.
\( 36.98\,\mathrm{cm}^2 \)
\( 93.53\,\mathrm{cm}^2 \)
\( 65.26\,\mathrm{cm}^2 \)
\( 25.37\,\mathrm{cm}^2 \)

1103021907

Level: 
C
A plane is flying at a speed of \( 900\,\mathrm{km}\cdot\mathrm{h}^{-1} \) and according to the compass the axis of the airplane is still heading westbound. What angle will the flight path of the airplane make with the east-west direction if the south wind starts to blow at a speed of \( 10\,\mathrm{m}\cdot\mathrm{s}^{-1} \)? Round the result to two decimal places.
\( 2.29^{\circ} \)
\( 0.64^{\circ} \)
\( 0.01^{\circ} \)
\( 87.71^{\circ} \)

1103021906

Level: 
C
The distance between the places \( A \) and \( C \) on a straight road is \( 300\,\mathrm{m} \). There is a balloon \( B \) above the road between the places \( A \) and \( C \). See the picture. The angles of elevation from the places \( A \) and \( C \) to the balloon \( B \) are \( 20^{\circ} \) and \( 40^{\circ} \) respectively. What is the height \( h \) of the balloon?
\( 76\,\mathrm{m} \)
\( 168\,\mathrm{m} \)
\( 488\,\mathrm{m} \)
\( 523\,\mathrm{m} \)

1103021904

Level: 
C
From the highest window of Orava Castle, the angles of depression to the banks of the Orava river are \( 60^{\circ} \) and \( 20^{\circ} \). The height of the window above the river is \( 50\,\mathrm{m} \). What is the width of the river?
\( 108.5\,\mathrm{m} \)
\( 137.4\,\mathrm{m} \)
\( 100.5\,\mathrm{m} \)
\( 125.4\,\mathrm{m} \)

1103021903

Level: 
C
An observer was watching an approaching plane flying at a height of \( 3000\,\mathrm{m} \) in a straight line with constant velocity. At the first moment of measurement the observer saw the plane to be at an angle of elevation of \( 25^{\circ} \). After \( 10 \) seconds the angle of elevation changed to \( 35^{\circ} \). What was the speed of the plane? Round the result to ones.
\( 215\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 2149\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 6576\,\mathrm{m}\cdot\mathrm{s}^{-1} \)
\( 658\,\mathrm{m}\cdot\mathrm{s}^{-1} \)

1103256903

Level: 
C
In isosceles triangle \( ABC \), \( |AB| = 8\,\mathrm{cm} \), \( |BC|=|AC| = 6\,\mathrm{cm} \). Determine what percentage of the triangle area is a circle that is inscribed in it. Round the result to full percentages.
\( 56\,\% \)
\( 48\,\% \)
\( 62\,\% \)
\( 64\,\% \)

1103256901

Level: 
C
The farmer tied two goats on the meadow. The distance of the stakes \( K_1 \), \( K_2 \) to which the goats are tied is \( 5\,\mathrm{m} \) and the ropes have lengths of \( 3\,\mathrm{m} \) and \( 4\,\mathrm{m} \). What is the area of the grassland which is common for both goats? Round the result to two decimal places.
\( 6.64\,\mathrm{m}^2 \)
\( 0.57\,\mathrm{m}^2 \)
\( 0.35\,\mathrm{m}^2 \)
\( 1.52\,\mathrm{m}^2 \)

1003085910

Level: 
C
The solution set of the inequality \( \mathrm{tg}^3x + \mathrm{tg}^2x - \mathrm{tg}\,x - 1 < 0 \) for \( x\in\left(-\frac{\pi}2;\frac{\pi}2\right) \) is:
\( \left(-\frac{\pi}2;\frac{\pi}4\right) \)
\( \left(\frac{\pi}2;\frac{7\pi}4\right) \)
\( \left(-\frac{\pi}2;\frac{3\pi}4\right) \)
\( \left(-\frac{\pi}2;\frac{\pi}2\right) \)

1003085909

Level: 
C
The solution set of the inequality \( |\mathrm{tg}\,x| < 1 \) for \( x\in\left(-\frac{\pi}2;\frac{\pi}2\right) \) is:
\( \left( -\frac{\pi}4;\frac{\pi}4 \right) \)
\( \left( -\frac{\pi}2;\frac{\pi}2 \right) \)
\( \left( 0;\frac{\pi}4 \right) \)
\( \left( -\frac{\pi}2;-\frac{\pi}4 \right) \)

1003085908

Level: 
C
The solution set of the inequality \( \mathrm{cotg}\left(3x -\frac{\pi}4 \right) \geq -1 \) for \( x\in\left[0;\frac{\pi}2\right] \) is:
\( \{0\}\cup\left( \frac{\pi}{12};\frac{\pi}3 \right]\cup\left(\frac{5\pi}{12};\frac{\pi}2\right] \)
\( \left[ 0;\frac{\pi}3\right] \)
\( \left( \frac{\pi}{12};\frac{\pi}3\right] \)
\( \left( 0;\frac{\pi}2\right] \)