If   x² + y² + sin y = 4, then the value of $${{{d^2}y} \over {d{x^2}}}$$ at the point ($$-$$2,0) is :

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 :