If x = $\sec \left(\mathrm{\theta }\right) - \cos \left(\mathrm{\theta }\right), \mathrm{y} = {\sec }^{\mathrm{n}}\left(\mathrm{\theta }\right) - {\cos }^{\mathrm{n}}\left(\mathrm{\theta }\right)$, then $\left({\mathrm{x}}^2 + 4\right){\left(\frac{d\mathrm{y}}{d\mathrm{x}}\right)}^2$ is equal to

• n²(y² - 4)

• n²(4 - y²)

• n²(y² + 4)

• None of these

122.

The two curves y = 3 and y = 5 intersect at an angle

• ${\tan }^{-1}\left(\frac{\log \left(3\right) - \log \left(5\right)}{1 + \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) + \log \left(5\right)}{1 - \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) + \log \left(5\right)}{1 + \log \left(3\right)\log \left(5\right)}\right)$

• ${\tan }^{-1}\left(\frac{\log \left(3\right) - \log \left(5\right)}{1 - \log \left(3\right)\log \left(5\right)}\right)$

The period of ${\sin }^4\left(\mathrm{x}\right) + {\cos }^4\left(\mathrm{x}\right)$ is

• $\frac{{\mathrm{\pi }}^4}{2}$

• $\frac{{\mathrm{\pi }}^2}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

If 3 cos x $\ne$ 2 sin x, then the general solution of ${\sin }^2\left(\mathrm{x}\right) - \cos \left(2\mathrm{x}\right) = 2 - \sin \left(2\mathrm{x}\right)$ is

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• $\frac{\mathrm{n\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• $\left(4\mathrm{n} \pm 1\right)\frac{\mathrm{\pi }}{2}, \mathrm{n} \in \mathrm{Z}$

• (2n - 1)$\mathrm{\pi }, \mathrm{n} \in \mathrm{Z}$

If $\cos \left(\mathrm{x}\right) + {\cos }^2\left(\mathrm{x}\right) = 1$, then the value of ${\sin }^{12}\left(\mathrm{x}\right) + 3{\sin }^{10}\left(\mathrm{x}\right) + 3{\sin }^8\left(\mathrm{x}\right) + {\sin }^6\left(\mathrm{x}\right) - 1$, is equal to :

• 2

• 1

• - 1

• 0

The product of all values of $\left(\cos \left(\mathrm{\alpha }\right) + \mathrm{isin}\left(\mathrm{\alpha }\right)\right)$^3/5 is :

• 1

• $\cos \left(\mathrm{\alpha }\right) + \mathrm{isin}\left(\mathrm{\alpha }\right)$

• $\cos \left(3\mathrm{\alpha }\right) + \mathrm{isin}\left(3\mathrm{\alpha }\right)$

• $\cos \left(5\mathrm{\alpha }\right) + \mathrm{isin}\left(5\mathrm{\alpha }\right)$

$\left(1 + \cos \left(\frac{\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{3\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{5\mathrm{\pi }}{8}\right)\right)\left(1 + \cos \left(\frac{7\mathrm{\pi }}{8}\right)\right)$ is equal to

• $\frac{1}{2}$

• $\frac{1}{8}$

• $\cos \left(\frac{\mathrm{\pi }}{8}\right)$

• $\frac{1}{4}$

If $\sqrt{3}\cos \left(\mathrm{\theta }\right) + \sin \left(\mathrm{\theta }\right) = \sqrt{2}$, then the value of $\mathrm{\theta }$ is

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4}$

• ${\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4} - \frac{\mathrm{\pi }}{3}$

• $\mathrm{n\pi } + \frac{\mathrm{\pi }}{4} - \frac{\mathrm{\pi }}{3}$

• $\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4} - \frac{\mathrm{\pi }}{3}$

If in a AABC, (s - a)(s - b) = s(s - c) then angle C is equal to

• 90°

• 45°

• 30°

• 60°

The length of the shadows of a vertical pole of height h, thrown by the sun's rays at three different moments are h, 2h and 3h. The sum of the angles of elevation of the rays at these three moments is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$