In a triangle $ABC$,with usual notations,if $\frac{2 \cos A}{a} + \frac{\cos B}{b} + \frac{2 \cos C}{c} = \frac{a}{bc} + \frac{b}{ca}$,then $\angle A = $

$A$ $p-n$ photodiode is made of a material with a band gap of $2.0 \text{ eV}$. The minimum frequency of the radiation that can be absorbed by the material is nearly

Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B,$ the ratio of the lengths of the strings is

In $\triangle ABC$,with usual notation,observe the two statements given below:
$(I)$ $r r_1 r_2 r_3 = \Delta^2$
$(II)$ $r_1 r_2 + r_2 r_3 + r_3 r_1 = s^2$
Which of the following is correct?

If $y$ is a function of $x$ and $\log(x+y) = 2xy$,then the value of $y'(0)$ is

If $\sin^{-1}(\frac{x}{5}) + \csc^{-1}(\frac{5}{4}) = \frac{\pi}{2}$,then $x = $