$\mathrm{The} \mathrm{set} \mathrm{of} \mathrm{solutions} \mathrm{of} \mathrm{the} \mathrm{equation} \left(\sqrt{3} - 1\right)\sin \left(\mathrm{\theta }\right) + \left(\sqrt{3} + 1\right)\cos \left(\mathrm{\theta }\right) = 2 \mathrm{is}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{4} + \frac{\mathrm{\pi }}{12} \hspace{0.17em}: \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{2\mathrm{n\pi } \pm \frac{\mathrm{\pi }}{4} - \frac{\mathrm{\pi }}{12} \hspace{0.17em}: \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4} + \frac{\mathrm{\pi }}{12} \hspace{0.17em}: \mathrm{n} \in \mathrm{Z}\right\}$

• $\left\{\mathrm{n\pi } + {\left(- 1\right)}^{\mathrm{n}}\frac{\mathrm{\pi }}{4} - \frac{\mathrm{\pi }}{12} \hspace{0.17em}: \mathrm{n} \in \mathrm{Z}\right\}$

$$\begin{gathered} \mathrm{If} ∆ = {\mathrm{a}}^2 - {\left(\mathrm{b} - \mathrm{c}\right)}^2, \mathrm{is} \mathrm{the} \mathrm{area} \mathrm{of} \mathrm{the} ∆\mathrm{ABC}, \\ \mathrm{then} \tan \left(\mathrm{A}\right) = ? \end{gathered}$$

• $\frac{1}{16}$

• $\frac{8}{15}$

• $\frac{3}{4}$

• $\frac{4}{3}$

$\mathrm{In} \mathrm{a} ∆∆\mathrm{ABC}, \mathrm{C} = 90°. \mathrm{Then}, \frac{{\mathrm{a}}^2 - {\mathrm{b}}^2}{{\mathrm{a}}^2 + {\mathrm{b}}^2} = ?$

• sin(A + B)

• sin(A - B)

• cos(A + B)

• cos(A - B)

The sum of angles of elevation of the top of a tower from two points distant a and b from the base and in the same straight line with it is 90°. Then, the height of the tower is

• ${\mathrm{a}}^2\mathrm{b}$

• ab²

• $\sqrt{\mathrm{ab}}$

• ab

45.

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{1 + 3{\mathrm{x}}^2 - \cos 2\mathrm{x}}{{\mathrm{x}}^2}, \mathrm{for} \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{for} \mathrm{x}= 0\right\} \\ \mathrm{is} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0, \mathrm{then} \mathrm{k} \mathrm{is} \mathrm{equal} \mathrm{to} \end{gathered}$$

• 1

• 5

• 6

• 0

$\mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left(\cos \left(\mathrm{x}\right)\right)\left(\cos \left(2\mathrm{x}\right)\right). . . \left(\cos \left(\mathrm{nx}\right)\right), \mathrm{then} \mathrm{f}\text{'}\left(\mathrm{x}\right) + \sum _{\mathrm{r} = 1}^{\mathrm{n}} \left(\mathrm{rtan}\left(\mathrm{rx}\right)\right)\mathrm{f}\left(\mathrm{x}\right) = ?$

• f(x)

• 0

• - f(x)

• 2f(x)

$$\begin{gathered} \mathrm{If} \mathrm{y} = {\cos }^{-1}\left(\frac{{\mathrm{a}}^2 - {\mathrm{x}}^2}{{\mathrm{a}}^2 + {\mathrm{x}}^2}\right) + {\sin }^{-1}\left(\frac{2\mathrm{ax}}{{\mathrm{a}}^2 + {\mathrm{x}}^2}\right), \\ \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}} = ? \end{gathered}$$

• $\frac{\mathrm{a}}{{\mathrm{x}}^2 + {\mathrm{a}}^2}$

• $\frac{2\mathrm{a}}{{\mathrm{x}}^2 + {\mathrm{a}}^2}$

• $\frac{4\mathrm{a}}{{\mathrm{x}}^2 + {\mathrm{a}}^2}$

• $\frac{{\mathrm{a}}^2}{{\mathrm{x}}^2 + {\mathrm{a}}^2}$

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right), \\ \mathrm{then} \mathrm{f}\left(\frac{\mathrm{\pi }}{4}\right){\mathrm{f}}^{\left(\mathrm{iv}\right)}\left(\frac{\mathrm{\pi }}{4}\right) = ? \end{gathered}$$

• 1

• 2

• 3

• 4

$$\begin{gathered} \mathrm{If} \mathrm{y} = \sin \left({\mathrm{msin}}^{-1}\left(\mathrm{x}\right)\right), \mathrm{then} \left(1 - {\mathrm{x}}^2\right){\mathrm{y}}_2 - {\mathrm{xy}}_1 = ? \\ \left(\mathrm{Here}, {\mathrm{y}}_{\mathrm{n}} \mathrm{denotes} \frac{{\mathrm{d}}^{\mathrm{n}}\mathrm{y}}{{\mathrm{dx}}^{\mathrm{n}}}\right) \end{gathered}$$

• m²y

• - m²y

• 2m²y

• - 2m²y

The height of the cone of maximum volume inscribed in a sphere of radius R is

• $\frac{\mathrm{R}}{3}$

• $\frac{{\displaystyle 2\mathrm{R}}}{{\displaystyle 3}}$

• $\frac{4\mathrm{R}}{3}$

• $\frac{4\mathrm{R}}{\sqrt{3}}$