1103021901
Level:
C
What is the slope of a staircase if the height of a stair is \( 15\,\mathrm{cm} \) and the depth of a stair is \( 30\,\mathrm{cm} \)?
\( 26.57^{\circ} \)
\( 63.43^{\circ} \)
\( 30^{\circ} \)
\( 60^{\circ} \)
C
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.
\( 56\,\% \)
\( 48\,\% \)
\( 62\,\% \)
\( 64\,\% \)
1103256902
Level:
C
The cucumber field has the shape of an isosceles right triangle. The length of its legs is \( 12\,\mathrm{m} \). Rotary sprinklers placed in its vertices have a reach of \( 6\,\mathrm{m} \). Find the area of the field that is not sprinkled with water. Round the result to two decimal places.
\( 15.45\,\mathrm{m}^2 \)
\( 41.10\,\mathrm{m}^2 \)
\( 16.29\,\mathrm{m}^2 \)
\( 15.25\,\mathrm{m}^2 \)
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:
\( \{0\}\cup\left( \frac{\pi}{12};\frac{\pi}3 \right]\cup\left(\frac{5\pi}{12};\frac{\pi}2\right] \)
\( \left[ 0;\frac{\pi}3\right] \)
\( \left( \frac{\pi}{12};\frac{\pi}3\right] \)
\( \left( 0;\frac{\pi}2\right] \)
1003085907
Level:
C
The solution set of the inequality \( |\sin x| \geq 0.5 \) for \( x\in[0;2\pi] \) is:
\( \left[\frac{\pi}6;\frac{5\pi}6\right]\cup\left[\frac{7\pi}6;\frac{11\pi}6\right] \)
\( \left[0;\frac{5\pi}6\right]\cup\left[\frac{7\pi}6;\frac{11\pi}6\right] \)
\( \left[\frac{\pi}6;\frac{11\pi}6\right] \)
\( \left[0;\frac{\pi}6\right]\cup\left[\frac{5\pi}6;\frac{11\pi}6\right] \)
1003085906
Level:
C
The solution set of the inequality \( \cos 3x > \frac{{\sqrt3}}2 \) for \( x\in(0^{\circ};180^{\circ}) \) is:
\( \left(0^{\circ};10^{\circ} \right)\cup\left(110^{\circ};130^{\circ}\right) \)
\( \left(330^{\circ};360^{\circ}\right) \)
\(\left(110^{\circ};120^{\circ}\right) \)
\( \left(0^{\circ};30^{\circ} \right) \)
1003085905
Level:
C
The solution set of the inequality \( |\mathrm{cotg} x |\leq \sqrt3 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}6+k\pi;\frac{5\pi}6+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}6+k\pi;\frac{\pi}2+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}2+k\pi;\frac{5\pi}6+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[ k\pi;(k+1)\pi\right] \)