$$ \int_0^{\frac{\pi}{2}} \sqrt{\tan x d x}= $$

If $y=f(x)$ is the solution of the differential equation $\left(1+\cos ^2 x\right) f^{\prime}(x)-4 \sin 2 x-f(x) \sin 2 x=0$ when $f(0)=0$, then $f\left(\frac{\pi}{3}\right)=$

The differential equation corresponding to the family of ellipses $\frac{x^2}{a^2}+\frac{y^2}{4}=1$, where ' $a$ ' is an arbitrary constant is

Match the "Technology" given in List-I with the "Principle of physics" given in List-II.

$$ \begin{array}{l|l|l|l} \hline & \text { List-I (Technology) } & & \text { List-II (Principle of physics) } \\ \hline \text { (A) } & \text { Steam engine } & \text { I } & \begin{array}{l} \text { Magnetic confinement of } \\ \text { plasma } \end{array} \\ \hline \text { (B) } & \text { Electron microscope } & \text { II } & \text { Laws of thermodynamics } \\ \hline \text { (C) } & \text { Non-reflecting coatings } & \text { III } & \text { Wave nature of electrons } \\ \hline \text { (D) } & \text { Tokamak } & \text { IV } & \text { Interference of light } \\ \hline \end{array} $$