$\mathrm{If} 3\sin \left(\mathrm{x}\right) + 4\cos \left(\mathrm{x}\right) = 5, \mathrm{then} 6\tan \left(\frac{\mathrm{x}}{2}\right) - 9{\tan }^2\left(\frac{\mathrm{x}}{2}\right) = ?$

• 3

• 2

• 1

• 0

$$\begin{gathered} \mathrm{If} \mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{are} \mathrm{length} \mathrm{of} \mathrm{the} \mathrm{altitudes} \mathrm{of} \mathrm{a} ∆\mathrm{ABC} \mathrm{with} \mathrm{area} ∆, \mathrm{then} \\ \frac{{∆}^2}{{\mathrm{R}}^2}\left(\frac{1}{{\mathrm{\alpha }}^2} + \frac{1}{{\mathrm{\beta }}^2} + \frac{1}{{\mathrm{\gamma }}^2}\right) = ? \end{gathered}$$

• ${\sin }^2\left(\mathrm{A}\right) + {\sin }^2\left(\mathrm{B}\right) + {\sin }^2\left(\mathrm{C}\right)$

• ${\cos }^2\left(\mathrm{A}\right) + {\cos }^2\left(\mathrm{B}\right) + {\cos }^2\left(\mathrm{C}\right)$

• ${\tan }^2\left(\mathrm{A}\right) + {\tan }^2\left(\mathrm{B}\right) + {\tan }^2\left(\mathrm{C}\right)$

• ${\cot }^2\left(\mathrm{A}\right) + {\cot }^2\left(\mathrm{B}\right) + {\cot }^2\left(\mathrm{C}\right)$

In an acute angled triangle, cot(B)cot(C) + cot(A)cot(C) + cot(A)cot(B) = ?

• - 1

• 0

• 1

• 2

$\mathrm{x} = \log \left(\frac{1}{\mathrm{y}} + \sqrt{1 + \frac{1}{{\mathrm{y}}^2}}\right) \Rightarrow \mathrm{y} = ?$

• $\tanh \left(\mathrm{x}\right)$

• $\coth \left(\mathrm{x}\right)$

• $\mathrm{sech}\left(\mathrm{x}\right)$

• $\mathrm{csch}\left(\mathrm{x}\right)$

The period of f(x) = $\cos \left(\frac{\mathrm{x}}{3}\right) + \sin \left(\frac{\mathrm{x}}{2}\right)$ is

• $2\mathrm{\pi }$

• $4\mathrm{\pi }$

• $8\mathrm{\pi }$

• $12\mathrm{\pi }$

If $\sin \left(\mathrm{\theta }\right) + \cos \left(\mathrm{\theta }\right) = \mathrm{p} \mathrm{and} {\sin }^3\left(\mathrm{\theta }\right) + {\cos }^3\left(\mathrm{\theta }\right) = \mathrm{q}$, then p(p² - 3) is equal to

• q

• 2q

• - q

• - 2q

287.

$\mathrm{If} \tan \left(\left(\mathrm{\pi cos}\left(\mathrm{\theta }\right)\right)\right) = \cot \left(\mathrm{\pi sin}\left(\mathrm{\theta }\right)\right), \mathrm{then} \mathrm{a} \mathrm{value} \mathrm{of} \cos \left(\mathrm{\theta } - \frac{\mathrm{\pi }}{4}\right) \mathrm{among} \mathrm{the} \mathrm{following} \mathrm{is}$

• $\frac{1}{2\sqrt{2}}$

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{2}$

• $\frac{1}{4}$

$$\begin{gathered} \mathrm{The} \mathrm{set} \mathrm{of} \mathrm{solutions} \mathrm{of} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations} \\ \mathrm{x} + \mathrm{y} = \frac{2\mathrm{\pi }}{3} \\ \mathrm{and} \cos \left(\mathrm{x}\right) + \cos \left(\mathrm{y}\right) = \frac{3}{2}, \\ \mathrm{where} \mathrm{x}, \mathrm{y} \mathrm{are} \mathrm{real}, \mathrm{is} \end{gathered}$$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\cos \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \frac{1}{2}\right\}$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\sin \left(\frac{\mathrm{x} - \mathrm{y}}{2}\right) = \frac{1}{2}\right\}$

• $\left\{\left(\mathrm{x}, \mathrm{y}\right)\cos \left(\mathrm{x} - \mathrm{y}\right) = \frac{1}{2}\right\}$

• Empty set

If in the angles of a triangle are in the ratio1 : 1 : 4, then the ratio of the perimeter of the triangle to its largest side is

• $\sqrt{2} + 2 : \sqrt{3}$

• 3 : 2

• $\sqrt{3} + 2 . \sqrt{2}$

• $2 + \sqrt{3} : \sqrt{3}$

$\mathrm{If} \cos \left(\mathrm{x}\right) = \tan \left(\mathrm{y}\right), \cot \left(\mathrm{y}\right) = \tan \left(\mathrm{z}\right) \mathrm{and} \cot \left(\mathrm{z}\right) = \tan \left(\mathrm{x}\right) \mathrm{then} \sin \left(\mathrm{x}\right) = ?$

• $\frac{\sqrt{5} + 1}{4}$

• $\frac{\sqrt{5} - 1}{4}$

• $\frac{\sqrt{5} + 1}{2}$

• $\frac{\sqrt{5} - 1}{2}$