B

2010014904

Level: 
B
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. \[ 3 \cos^2 x =2\sin x \cos x \]
\(\cos x (3\cos x-2\sin x)=0\)
\(3\cos x=2\sin x\)
\(3(1-\sin^2 x)=2\sin x \cos x\)
\(\frac{3\cos^2 x}{2\sin x \cos x}=1\)

2010014903

Level: 
B
The solution set of the inequality \( \cos\,x < -\frac{\sqrt3}{2} \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{5\pi}6+2k\pi;\frac{7\pi}6+2k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{5\pi}6+k\pi;\frac{7\pi}6+k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(\frac{5\pi}6+2k\pi;\frac{11\pi}6+2k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{5\pi}6+2k\pi;\frac{5\pi}6+2k\pi\right) \)

2010014902

Level: 
B
The solution set of the inequality \( \mathrm{tg}\, x > -\frac{\sqrt3}3 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}6+k\pi;\ \frac{\pi}2+k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}3+k\pi;\ \frac{\pi}2+k\pi\right) \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}6+k\pi;\ \frac{\pi}6+k\pi\right)\)
\( \bigcup\limits_{k\in\mathbb{Z}}\left(-\frac{\pi}6+k\pi;\ \pi+k\pi\right) \)

2010014901

Level: 
B
The solution set of the inequality \( \sin x \geq \frac{\sqrt{2}}2 \) for \( x\in\mathbb{R} \) is:
\( \bigcup\limits_{k\in\mathbb{Z}}\left[\frac{\pi}4+2k\pi;\ \frac{3\pi}4+2k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[ \frac{\pi}4+k\pi;\ \frac{3\pi}4+k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[ -\frac{\pi}4+2k\pi;\ \frac{\pi}4+2k\pi\right] \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left[ -\frac{\pi}4+k\pi;\ \frac{\pi}4+k\pi\right] \)

2010014608

Level: 
B
Find a general form equation of the straight line that passes through the point \( M=[2;-3] \) and is parallel with the line of symmetry of the line segment \( AB \), where \( A=[4;-1] \), and \( B=\left[-3;\frac32\right] \) (see the picture).
\( 14x-5y-43=0 \)
\( 5x-14y-52=0 \)
\( 14x+5y-13=0 \)
\( 5x+14+32=0 \)