1003086002
Level:
C
The equation \( (\sin x + \cos x)^2 = 1.5 \) has two solutions on the interval \( \left(0^{\circ}; 90^{\circ}\right) \). The larger one is:
\( 75^{\circ} \)
\( 15^{\circ} \)
\( 30^{\circ} \)
\( 65^{\circ} \)
Trigonometric equations and inequalities
1003086001
Level:
C
The arithmetic mean of all \( x\in\left[0^{\circ};360^{\circ}\right] \) satisfying the equation \( \sin 2x = \sin x \) is:
\( 180^{\circ} \)
\( 240^{\circ} \)
\( 360^{\circ} \)
\( 270^{\circ} \)
1003086110
Level:
B
Which of the following intervals contains four solutions of the equation \( \cos x = 0.5 \)?
\( [\pi;5\pi] \)
\( [-2\pi;\pi] \)
\( [-4\pi;-2\pi] \)
\( [5\pi;10\pi] \)
1003086109
Level:
B
The solution set of the equation \( \mathrm{tg}\,x + \mathrm{cotg}\,x = 2 \) for \( x\in[-2\pi;2\pi] \) is:
\( \left\{-\frac{7\pi}4;-\frac{3\pi}4;\frac{\pi}4;\frac{5\pi}4 \right\} \)
\( \left\{-\frac{7\pi}4;\frac{\pi}4;\right\} \)
\( \left\{-\frac{5\pi}4;-\frac{\pi}4;\frac{3\pi}4;\frac{7\pi}4 \right\} \)
\( \left\{-\frac{3\pi}4;\frac{\pi}4\right\} \)
1003086108
Level:
C
The solution set of the equation \( 1 + \sin x \cdot \cos 2x = \sin x + \cos 2x \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}2+2k\pi;k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}2+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{2k\pi\right\} \)
1003086107
Level:
C
The solution set of the equation \( 2\mathrm{tg}^2x + 4\cos^2x = 7 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}3+k\pi;\frac{2\pi}3+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}3+2k\pi;\frac{2\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}3+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{2\pi}3+k\pi\right\} \)
1003086106
Level:
B
The solution set of the equation \( \sin 2x = \cos 3x \cdot \sin 2x \) for \( x\in\left[0^{\circ};180^{\circ}\right] \) is:
\( \left\{0^{\circ};90^{\circ};120^{\circ};180^{\circ}\right\} \)
\( \left\{90^{\circ};120^{\circ};180^{\circ}\right\} \)
\( \left\{90^{\circ};180^{\circ}\right\} \)
\( \left\{0^{\circ};90^{\circ}\right\} \)
1003086105
Level:
C
How many solutions does the equation \( 2\cos^2 x - \cos x - 1 = 0 \) have in the interval \( [-\pi;3\pi] \)?
\( 6 \)
\( 4 \)
\( 2 \)
\( 3 \)
1003086104
Level:
A
Which of the following equations has exactly two solutions in the interval \( [0;\pi] \)?
\( 3\sin x - 2 = 0 \)
\( 2\sin x - 3 = 0 \)
\( 3\cos x + 2 = 0 \)
\( 3\sin x + 2 = 0 \)
1003086103
Level:
C
The solution set of the equation \( 2\sin x + \mathrm{tg}\,x = 0 \), \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{2\pi}3+2k\pi;\frac{4\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{2k\pi;\frac{2\pi}3+2k\pi;\frac{4\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{5\pi}6+k\pi;\frac{7\pi}6+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{5\pi}6+2k\pi;\frac{7\pi}6+2k\pi\right\} \)