C

9000046505

Level: 
C
Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. \[ \sin x = 1 +\cos x \]
\(\sin ^{2}x = 1 + 2\cos x +\cos ^{2}x\)
\(\sin ^{2}x = 1 +\cos ^{2}x\)
substitution \( 1 +\cos x = z\)
\(\sin x -\cos x = z\)

9000046507

Level: 
C
Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. \[ \sqrt{3}\cos x = 1 -\sin x \]
\(3\cos ^{2}x = (1 -\sin x)^{2}\)
\(3\cos ^{2}x = 1 -\sin ^{2}x\)
substitution \( 1 -\sin x = z\)
substitution \( \cos x = z\)

9000046508

Level: 
C
Identify the optimal first step convenient to solve the following trigonometric equation. Do not consider the step which is possible but does not help to solve the equation. \[ \sqrt{3}\sin x = 2 -\cos x \]
\(3\sin ^{2}x = 4 - 4\cos x +\cos ^{2}x\)
substitution \( 2 -\cos x = z\)
\(3\sin ^{2}x = 4 -\cos ^{2}x\)
\(3\sin ^{2}x = 1 - 2\cos x +\cos ^{2}x\)

9000038704

Level: 
C
The box is on the slope as in the picture. The angle of the slope is \(\alpha \). The forces acting on the box are the force of gravity \(\vec{F_{G}}\) and the friction \(\vec{F_{t}}\). The force of gravity can be replaced by two components \(\vec{F_{1}}\) and \(\vec{F_{n}}\). (The force \(\vec{F_{1}}\) is parallel to the slope and \(\vec{F_{n}}\) is perpendicular to the slope.) For \(F_{1} = 20\, \mathrm{N}\) and \(F_{n} = 55\, \mathrm{N}\) find the corresponding \(\alpha \).
\(\alpha \doteq 20^{\circ }\)
\(\alpha \doteq 21^{\circ }\)
\(\alpha \doteq 69^{\circ }\)
\(\alpha \doteq 70^{\circ }\)
\(\alpha \doteq 30^{\circ }\)
\(\alpha \doteq 29^{\circ }\)

9000038705

Level: 
C
The box is on the slope as in the picture. The angle of the slope is \(\alpha = 45^{\circ }\). The forces acting on the box are the force of gravity \(\vec{F_{G}}\) and the friction \(\vec{F_{t}}\). The force of gravity can be replaced by two components \(\vec{F_{1}}\) and \(\vec{F_{n}}\). (The force \(\vec{F_{1}}\) is parallel to the slope and \(\vec{F_{n}}\) is perpendicular to the slope.) The friction \(F_{t}\) is given by the formula \(F_{t} = fF_{n}\). The coefficient of the friction \(f = 0.5\). Consider the standard acceleration of gravity \(g = 10\, \mathrm{m\, s^{-2}}\). Find the acceleration of the box.
\(a = \frac{5\sqrt{2}} {2} \, \mathrm{m\, s^{-2}}\)
\(a = 5\sqrt{2}\, \mathrm{m\, s^{-2}}\)
\(a = 5\sqrt{3}\, \mathrm{m\, s^{-2}}\)
\(a = 0\, \mathrm{m\, s^{-2}}\)
\(a = 5\, \mathrm{m\, s^{-2}}\)
\(a = \frac{5\sqrt{3}} {2} \, \mathrm{m\, s^{-2}}\)

9000038706

Level: 
C
The box is on the slope as in the picture. The angle of the slope is \(\alpha \). The forces acting on the box are the force of gravity \(\vec{F_{G}}\) and the friction \(\vec{F_{t}}\). The force of gravity can be replaced by two components \(\vec{F_{1}}\) and \(\vec{F_{n}}\). (The force \(\vec{F_{1}}\) is parallel to the slope and \(\vec{F_{n}}\) is perpendicular to the slope.) The friction \(F_{t}\) is given by the formula \(F_{t} = fF_{n}\). The coefficient of the friction is \(f = 0.47\). Consider the standard acceleration of gravity \(g = 10\, \mathrm{m\, s^{-2}}\). Find the angle \(\alpha \) which ensures that the box moves on the slope with zero acceleration.
\(\alpha \doteq 25^{\circ }\)
\(\alpha \doteq 15^{\circ }\)
\(\alpha \doteq 20^{\circ }\)
\(\alpha \doteq 65^{\circ }\)
\(\alpha \doteq 28^{\circ }\)
\(\alpha \doteq 62^{\circ }\)

9000038707

Level: 
C
The box is on the slope as in the picture. The length of the slope is \(l = 2\, \mathrm{m}\) and the height is \(h = 1.2\, \mathrm{m}\). The forces acting on the box are the force of gravity \(\vec{F_{G}}\) and the friction \(\vec{F_{t}}\). The force of gravity can be replaced by two components \(\vec{F_{1}}\) and \(\vec{F_{n}}\). (The force \(\vec{F_{1}}\) is parallel to the slope and \(\vec{F_{n}}\) is perpendicular to the slope.) The friction \(F_{t}\) is given by the formula \(F_{t} = fF_{n}\) where \(f\) is the coefficient of the friction. Consider the standard acceleration of gravity \(g = 10\, \mathrm{m\, s^{-2}}\). Find the minimal value for the coefficient of the friction \(f\) to ensure that the box does not move with an acceleration.
\(f = 0.75\)
\(f = 0.6\)
\(f = 0.65\)
\(f = 0.7\)
\(f = 0.55\)
\(f = 0.8\)

9000036107

Level: 
C
There are three information panels \(A\), \(B\) and \(C\) in the park. The direct distance between \(B\) and \(C\) is \(150\, \mathrm{m}\). The visual angle of this distance from the panel \(A\) is \(55^{\circ }\). The visual angle of the distance \(AC\) from the panel \(B\) is \(39^{\circ }\). Find the direct distance between the panels \(A\) and \(B\) and round your answer to nearest meters.
\(183\, \mathrm{m}\)
\(147\, \mathrm{m}\)
\(195\, \mathrm{m}\)
\(218\, \mathrm{m}\)

9000036108

Level: 
C
The center of a spherical balloon is at a height of \(500\, \mathrm{m}\) height. The visual angle of the balloon is \(1^{\circ }30'\). The elevation angle of the center of the balloon is \(42^{\circ }50'\). Find the diameter of the balloon in meters and round to nearest one decimal.
\(19.3\, \mathrm{m}\)
\(18.2\, \mathrm{m}\)
\(18.9\, \mathrm{m}\)
\(19.5\, \mathrm{m}\)