Trigonometric equations and inequalities

2010010705

Level: 
A
The solution set of the equation \( \cos\!\left(2\varphi + \frac{\pi}6\right) = - 1\) for \( \varphi \in [ 0;2\pi]\) is:
\(\left\{ \frac{5\pi}{12}; \frac{17\pi}{12}\right\}\)
\(\left\{ \frac{5\pi}{12}; \frac{11\pi}{12}\right\}\)
\(\left\{ \frac{7\pi}{12}; \frac{13\pi}{12}\right\}\)
\(\left\{ \frac{7\pi}{12}; \frac{17\pi}{12}\right\}\)

2010010704

Level: 
A
Identify the equation which arises from the following equation using an optimal substitution. \[ \mathop{\mathrm{tg}}\nolimits x + \frac{2\sqrt{3}}{3}=\mathop{\mathrm{cotg}}\nolimits x \]
\(\sqrt{3}t^{2} +2t -\sqrt{3}= 0\)
\(t^{2} +2\sqrt{3}t-1= 0\)
\(3t^{2} -2\sqrt{3}t +{3}= 0\)
\(\sqrt{3}t^{2} +t +2\sqrt{3}= 0\)

2010010702

Level: 
A
The solution set of the equation \( \mathrm{cotg}\, x =\sqrt{3} \) for \( x\in (-\pi;\pi )\) is:
\( \left\{ -\frac{5\pi}6;\frac{\pi}6\right\} \)
\( \left\{ -\frac{\pi}6;\frac{\pi}6\right\} \)
\( \left\{ -\frac{\pi}3;\frac{\pi}3\right\} \)
\( \left\{ -\frac{2\pi}3;\frac{\pi}3\right\} \)

2010010701

Level: 
A
The solution set of the equation \( \cos x =-0.5 \) for \( x\in[ 0;2\pi ]\) is:
\( \left\{ \frac{2\pi}3;\frac{4\pi}3\right\} \)
\( \left\{ \frac{2\pi}3;\frac{5\pi}3\right\} \)
\( \left\{ \frac{4\pi}3;\frac{5\pi}3\right\} \)
\( \left\{ \frac{4\pi}3;\frac{7\pi}3\right\} \)

2010009805

Level: 
C
The solution set of the inequality \( |\cos x| \leq \frac12 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}3+k\pi;\frac{2\pi}3+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[-\frac{\pi}3+k\pi;\frac{\pi}3+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}3+k\pi; \infty\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}3+k\pi;\frac{4\pi}3+k\pi\right] \)

2010009804

Level: 
C
The solution set of the equation \( \mathrm{tg}\, x - \mathrm{cotg}\,x = 0 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}4+k\pi;\frac{3\pi}4+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{k\pi;\frac{\pi}4+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}4+k\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+k\pi\right\} \)