1003085706
Level:
A
The sum of all \( \theta \), where \( 0^{\circ} < \theta < 360^{\circ} \), satisfying the equation \( \sin\!\left(\theta + 10^{\circ}\right) = 0.5 \) is:
\( 160^{\circ} \)
\( 140^{\circ} \)
\( 300^{\circ} \)
\( 200^{\circ} \)
A
1003085705
Level:
A
Solving the equation \( 2\sin\!\left(x + \frac{\pi}4 \right) = \sqrt3 \) for \( x \), where \( x\in (0; \pi) \), you get:
\( x\in\left\{ \frac{\pi}{12};\frac{5\pi}{12} \right\} \)
\( x\in\left\{ \frac{\pi}{12} \right\} \)
\( x\in\left\{ \frac{3\pi}{12};\frac{5\pi}{12} \right\} \)
\( x\in\left\{ \frac{13\pi}{12};\frac{5\pi}{12} \right\} \)
1003085704
Level:
A
The solution set of the equation \( \cos\!\left(2x - \frac{\pi}3 \right) = - 0.5 \), where \( 0 < x < 2\pi \), is:
\( \left\{ \frac{\pi}2; \frac{3\pi}2; \frac{5\pi}6; \frac{11\pi}6 \right\} \)
\( \left\{ \frac{\pi}2; \frac{3\pi}2 \right\} \)
\( \left\{ \frac{5\pi}6; \frac{11\pi}6 \right\} \)
\( \left\{ \frac{3\pi}2; \frac{5\pi}6; \frac{11\pi}6; \pi \right\} \)
1003085703
Level:
A
The solution set of the equation \( 2\sin\!\left(x - \frac{\pi}6 \right) = 1 \), where \( x\in[0; \pi] \), is:
\( \left\{\frac{\pi}3; \pi \right\} \)
\( \left\{\frac{\pi}6 \right\} \)
\( \left\{\frac{\pi}3 \right\} \)
\( \left\{\frac{\pi}6; \frac{\pi}2 \right\} \)
1003085702
Level:
A
Solve \( 2\sin^2x = \sqrt2 \sin x \) for \( x \), where \( x\in\mathbb{R} \).
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \{k\pi\}\cup\left\{ \frac{\pi}4+2k\pi \right\}\cup\left\{ \frac{3\pi}4+2k\pi\right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \left\{ \frac{\pi}4+2k\pi \right\}\cup\left\{ \frac{3\pi}4+2k\pi\right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \left\{ \frac{\pi}4+k\pi \right\}\cup\left\{ \frac{3\pi}4+k\pi\right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \{2k\pi\}\cup\left\{ \frac{\pi}4+k\pi \right\}\cup\left\{ \frac{3\pi}4+k\pi\right\} \right] \)
1003085701
Level:
A
Find all \( x \), \( x\in\mathbb{R} \), such that \( \mathrm{cotg}^2x = - \mathrm{cotg}\,x \).
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left[ \left\{\frac{3\pi}4+k\pi \right\}\cup\left\{\frac{\pi}2+k\pi \right\} \right] \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+2k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{3\pi}4+k\pi \right\} \)
\( x\in\bigcup\limits_{k\in\mathbb{Z}}\left\{\frac{\pi}2+k\pi \right\} \)
1003099509
Level:
A
Given the numbers \( x = 4+2\sqrt5 \) and \( y=6-2\sqrt5 \), the fraction \( \frac xy \) can be written in the form:
\( \frac{11+5\sqrt5}4 \)
\( \frac{7\sqrt5-9}4 \)
\( \frac{-5\sqrt5}2 \)
\( 8\sqrt5 \)
1003099508
Level:
A
Evaluate the expression \( \frac{2-x}{x-2} \) for \( x=2-\sqrt2 \).
\( -1 \)
\( \sqrt2 - 2 \)
\( 2 - \sqrt2 \)
\( 1 \)
1003099505
Level:
A
Rewrite \( \frac{2-\sqrt3}{2+\sqrt3} \) by rationalizing the denominator.
\( 7-4\sqrt3 \)
\( \left(2-\sqrt3\right)\left(2+\sqrt3\right) \)
\( \frac{7-4\sqrt3}5 \)
\( \frac{7-4\sqrt3}7 \)
1003099504
Level:
A
Rationalizing the denominator \( \frac1{\sqrt5+\sqrt7} \) we get:
\( \frac{\sqrt7-\sqrt5}2 \)
\( \frac{\sqrt5+\sqrt7}2 \)
\( \frac{\sqrt5-\sqrt7}2 \)
\( \frac{-\sqrt7-\sqrt5}2 \)