The equation $3{\sin }^2\left(\mathrm{x}\right) + 10\cos \left(\mathrm{x}\right) - 6$ is satisfied, if

• $\mathrm{x} = \mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{3}\right)$

• $\mathrm{x} = 2\mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{3}\right)$

• $\mathrm{x} = \mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{6}\right)$

• $\mathrm{x} = 2\mathrm{n\pi } \pm {\cos }^{-1}\left(\frac{1}{6}\right)$

If ${\mathrm{e}}^{\cos \left(\mathrm{x}\right)} - {\mathrm{e}}^{- \cos \left(\mathrm{x}\right)} = 4$, then the value of $\cos \left(\mathrm{x}\right)$ is

• $\log \left(2 + \sqrt{5}\right)$

• $- \log \left(2 + \sqrt{5}\right)$

• $\log \left(- 2 + \sqrt{5}\right)$

• None of these

If ${\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{x}}\right) + {\tan }^{-1}\left(\frac{\mathrm{b}}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to

• $\sqrt{\mathrm{ab}}$

• $\sqrt{2\mathrm{ab}}$

• 2ab

• ab

If $\cos \left(\mathrm{\theta } - \mathrm{\alpha }\right) = \mathrm{a}, \cos \left(\mathrm{\theta } - \mathrm{\beta }\right) = \mathrm{b}$, then

${\sin }^2\left(\mathrm{\alpha } - \mathrm{\beta }\right) + 2\mathrm{abcos}\left(\mathrm{\alpha } - \mathrm{\beta }\right)$ is equal to

• a² + b²

• a² - b²

• b² - a²

• - a² - b²

The most general solutions of the equation

$\sec \left(\mathrm{x}\right) - 1 = \left(\sqrt{2} - 1\right)\tan \left(\mathrm{x}\right)$ are given by

• $\mathrm{n\pi } + \frac{\mathrm{\pi }}{8}$

• $2\mathrm{n\pi }, 2\mathrm{n\pi } + \frac{\mathrm{\pi }}{4}$

• $2\mathrm{n\pi }$

• None of these

If $\mathrm{\alpha } + \mathrm{\beta } - \mathrm{\gamma } = \mathrm{\pi }, \mathrm{then} {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) - {\sin }^2\left(\mathrm{\gamma }\right)$ is equal to

• $2\sin \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\beta }\right)\cos \left(\mathrm{\gamma }\right)$

• $2\cos \left(\mathrm{\alpha }\right)\cos \left(\mathrm{\beta }\right)\cos \left(\mathrm{\gamma }\right)$

• $2\sin \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\beta }\right)\sin \left(\mathrm{\gamma }\right)$

• None of the above

The range of the function f (x) = ${}^{7 - \mathrm{x}}\mathrm{P}_{\mathrm{x} - 3}$ is

• {1, 2, 3}

• {1, 2, 3, 4, 5, 6}

• {1, 2, 3, 4}

• {1, 2, 3, 4, 5}

The principal amplitude of ${\left(\sin \left(40°\right) + \mathrm{icos}\left(40°\right)\right)}^5$ is

• $70°$

• $- 110°$

• $110°$

• $- 70°$

109.

If the relation between direction ratios of two lines are given by a + b + c = 0 and 2ab + 2ac - bc = 0, then the angle between the lines is

• $\mathrm{\pi }$

• $\frac{2\mathrm{\pi }}{3}$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

The value of ${\left[\sqrt{2}\left\{\cos \left(56°15\text{'}\right) +\hspace{0.17em}\mathrm{isin}\left(56°15\text{'}\right)\right\}\right]}^8$

• - 16i

• 16i

• 8i

• 4i