Let S = {($$\lambda $$, $$\mu $$) $$ \in $$ R $$ \times $$ R : f(t) = (|$$\lambda $$| e^|t| $$-$$ $$\mu $$). sin (2|t|), t $$ \in $$ R, is a differentiable function}. Then S is a subset of :

If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :

A planoconvex lens becomes an optical system of $$28$$ $$cm$$ focal length when its plane surface is silvered and illuminated from left to right as shown in Fig-A.

If the same lens is instead silvered on the curved surface and illuminated from other side as in Fig-B, it acts like an optical system of focal length $$10$$ $$cm.$$ The refractive index of the material of lens is :

In a meter bridge, as shown in the figure, it is given that resistance $$Y = 12.5\,\,\Omega $$ and that the balance is obtained at a distance $$39.5$$ $$cm$$ from end $$A$$ (by Jockey J). After interchanging the resistances $$X$$ and $$Y$$, a new balance point is found at a distance $${l_2}$$ from end $$A.$$ What are the value of $$X$$ and $${l_2}$$ ?