1003086104
Level:
A
Which of the following equations has exactly two solutions in the interval \( [0;\pi] \)?
\( 3\sin x - 2 = 0 \)
\( 2\sin x - 3 = 0 \)
\( 3\cos x + 2 = 0 \)
\( 3\sin x + 2 = 0 \)
Trigonometric equations and inequalities
1003086103
Level:
C
The solution set of the equation \( 2\sin x + \mathrm{tg}\,x = 0 \), \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{2\pi}3+2k\pi;\frac{4\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{2k\pi;\frac{2\pi}3+2k\pi;\frac{4\pi}3+2k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{5\pi}6+k\pi;\frac{7\pi}6+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{5\pi}6+2k\pi;\frac{7\pi}6+2k\pi\right\} \)
1003086102
Level:
B
How many solutions does the equation \( \mathrm{cotg}\left(2x - \frac{\pi}4\right) = \frac1{\sqrt3} \) have in the interval \( [0;2\pi] \)?
\( 4 \)
\( 8 \)
\( 2 \)
\( 0 \)
1003086101
Level:
B
Which of the following equations has exactly four solutions in the interval \( [0;\pi] \)?
\( \mathrm{tg}\,4x = 2 \)
\( \mathrm{tg}\,2x = 4 \)
\( \mathrm{tg}\,\frac x4 = 2 \)
\( \mathrm{tg}\,\frac x4 = 4 \)
1003085810
Level:
B
Let \( x\in\left(\frac{\pi}2;\pi\right] \). Which of the following statements is true?
\( \sin x \geq \mathrm{tg}\,x \)
\( \sin x > \mathrm{tg}\,x \)
\( \sin x < \mathrm{tg}x \)
\( \sin x \leq \mathrm{tg}\,x \)
1003085809
Level:
B
Which of the following inequalities has no solution for \( x\in\mathbb{R} \)?
\( 99\cos x >100 \)
\( \cos^2x -\sin^2x \geq 1 \)
\( \sin|x| < 0 \)
\( \sin 2x \leq 2 \)
1003085808
Level:
B
Which of the following inequalities has no solutions for \( x\in\mathbb{R} \)?
\( \cos^2x -\sin^2x > 1 \)
\( 100\sin x > 1 \)
\( \sin x \cdot \cos x \geq \frac12 \)
\( |\sin x| \geq 1 \)
1003085807
Level:
B
For which \( x\in\left(-\frac{\pi}2;\ \frac{\pi}2\right) \) is the inequality \( \mathrm{tg}\,3x < -1 \) true?
\( x\in\left(-\frac{\pi}2;-\frac{5\pi}{12}\right)\cup\left(-\frac{\pi}6;-\frac{\pi}{12}\right)\cup\left(\frac{\pi}6;\frac{\pi}4\right) \)
\( x\in\left(-\frac{\pi}6;-\frac{\pi}{12}\right)\cup\left(\frac{\pi}6;\frac{\pi}4\right) \)
\( x\in\left(-\frac{\pi}2;-\frac{5\pi}{12}\right)\cup\left(-\frac{\pi}6; - \frac{\pi}{12}\right) \)
\( x\in\left(-\frac{\pi}6; -\frac{\pi}{12}\right)\cup\left(\frac{\pi}{12};\frac{\pi}6\right)\cup\left(\frac{\pi}6;\frac{\pi}4\right) \)
1003085806
Level:
B
The solution set of the inequality \( \mathrm{tg}\,x > 1 \) for \( 0 \leq x \leq \pi \) is the interval:
\( \left(\frac{\pi}4;\frac{\pi}2 \right) \)
\( \left(0;\frac{\pi}4 \right) \)
\( \left(0;\frac{\pi}2 \right) \)
\( \left(\frac{\pi}2;\pi \right) \)
1003085805
Level:
B
The solution set of the inequality \( 2\sin x \leq 3 \) for \( x\in\mathbb{R} \) is:
\( \mathbb{R} \)
\( \emptyset \)
\( [0;\pi] \)
\( [ 0;2\pi ] \)