1003085604
Level:
A
How many solutions does the equation \( \mathrm{tg}^2x = \frac13 \) have for \( 0\leq x\leq\pi \)?
\( 2 \) solutions
\( 1 \) solution
\( 4 \) solutions
\( 3 \) solutions
Trigonometric Equations and Inequalities
1003085603
Level:
A
How many solutions does the equation \( \cos^2x = 0.25 \) have for \( 0\leq x\leq 2\pi \)?
\( 4 \) solutions
\( 1 \) solution
\( 2 \) solutions
\( 3 \) solutions
1003085602
Level:
A
The solution set of the equation \( \mathrm{tg}\, x =- \sqrt3 \) for \( x\in(0;\pi) \) is:
\( \left\{ \frac{2\pi}3 \right\} \)
\( \left\{ -\frac{2\pi}3 \right\} \)
\( \left\{\frac{\pi}3 \right\} \)
\( \left\{\frac{5\pi}3 \right\} \)
1003085601
Level:
A
The solution set of the equation \( \sin x =-0.5 \) for \( x\in[0;2\pi] \) is:
\( \left\{ \frac{7\pi}6;\frac{11\pi}6\right\} \)
\( \left\{ \frac{7\pi}6 \right\} \)
\( \left\{ \frac{11\pi}6 \right\} \)
\( \left\{ \frac{5\pi}6; \frac{7\pi}6 \right\} \)
9000086705
Level:
A
Identify the optimal substitution which simplifies the following equation as much as
possible.
\[
2\cos (3x + 33^{\circ }) = -\sqrt{2}
\]
\(3x + 33^{\circ } = t\)
\(\sin 3x = t\)
\(3x = t\)
\(3x + 33^{\circ } = -\sqrt{2}\)
9000086706
Level:
A
Identify the equation which arises from the following equation using an optimal
substitution.
\[
2\cos ^{2}x - 7\cos x + 3 = 0
\]
\(2t^{2} - 7t + 3 = 0\)
\(2t^{2} = \frac{3}
{7}\)
\(\cos t = 0\)
\(7\cos t = 3\)
9000086707
Level:
A
Identify the equation which arises from the following equation using an optimal
substitution.
\[
\mathop{\mathrm{tg}}\nolimits ^{2}y - 2\mathop{\mathrm{tg}}\nolimits y = 3
\]
\(t^{2} - 2t - 3 = 0\)
\(\mathop{\mathrm{tg}}\nolimits t = \frac{3}
{2}\)
\(t^{2} = \frac{3}
{2}\)
\(\mathop{\mathrm{tg}}\nolimits t = 3\)
9000086708
Level:
A
Identify the equation which arises from the following equation using an optimal
substitution.
\[
\mathop{\mathrm{tg}}\nolimits ^{2}v -\mathop{\mathrm{cotg}}\nolimits ^{-1}v = 2
\]
\(t^{2} - t - 2 = 0\)
Equation is not convenient for a substitution.
\(t^{2} + t = 0\)
\(\mathop{\mathrm{tg}}\nolimits t = 2\)
9000086709
Level:
A
Identify the equation which arises from the following equation using an optimal
substitution.
\[
6\cos ^{2}x +\sin x - 5 = 0
\]
\(6t^{2} - t = 1\)
\(6t^{2} + t - 5 = 0\)
\(6t = 5\)
Equation is not convenient for a substitution.
9000086710
Level:
A
Identify the equation which arises from the following equation using an optimal
substitution.
\[
2\mathop{\mathrm{tg}}\nolimits x + 3\mathop{\mathrm{cotg}}\nolimits x = 5
\]
\(2t^{2} - 5t = -3\)
\(2t^{2} + 3t - 5 = 0\)
\(2t = \frac{3}
{5}\)
\(2t + 3t = 5\)