The angle of elevation of a cloud $\mathrm{C}$ from a point $\mathrm{P},200\mathrm{m}$ above a still take is ${30}^{\circ }$ . If the angle of depression of the image of $C$ in the lake from the point $P$ is ${60}^{\circ }$ , then $PC$ (in $\mathrm{m}$ ) is equal to

  $50tan\left(3ta{n}^{-1}\left(\frac{1}{2}\right)+2co{s}^{-1}\left(\frac{1}{\sqrt[]{5}}\right)\right)+4\sqrt[]{2}tan\left(\frac{1}{2}tan\left(2\sqrt[]{2}\right)\right)$ is equal to……………..

The number of solutions of the equation 2θ - cos²θ + $\sqrt[]{2}$ = 0 in R is equal to………..

Let $f\left(x\right)=|x-2|$ and $g\left(x\right)=f\left(f\right(x\left)\right),x\in \left[\mathrm{0,4}\right]$ . Then ${\int }_0^3?\left(g\right(x)-f(x\left)\right)dx$ is equal to:

The statement $\left(p\wedge q\right)\Rightarrow \left(p\wedge r\right)$ is equivalent to:

The minimum value of $2^{\mathrm{s}\mathrm{i}\mathrm{n}x}+2^{\mathrm{c}\mathrm{o}\mathrm{s}x}$ is: