The largest value of the non-negative integer a for which $$\mathop {\lim }\limits_{x \to 1} {\left\{ {{{ - ax + \sin (x - 1) + a} \over {x + \sin (x - 1) - 1}}} \right\}^{{{1 - x} \over {1 - \sqrt x }}}} = {1 \over 4}$$ is

Let f : R $$\to$$ R and g : R $$\to$$ R be respectively given by f(x) = | x | + 1 and g(x) = x² + 1. Define h : R $$\to$$ R by $$h(x) = \left\{ {\matrix{ {\max \{ f(x),g(x)\} ,} & {if\,x \le 0.} \cr {\min \{ f(x),g(x)\} ,} & {if\,x > 0.} \cr } } \right.$$

The number of points at which h(x) is not differentiable is

During Searle's experiment, zero of the Vernier scale lies between 3.20 $$ \times $$ 10^-2 m and 3.25 $$ \times $$ 10^-2 m of the main scale. The 20^th division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between 3.20 $$ \times $$ 10^-2 m and 3.25 $$ \times $$ 10^-2 m of the main scale but now the 45^th division of Vernier scale coincides with one of the main scale divisions. The length of the thin metallic wire is 2 m and its cross-sectional area is 8 $$ \times $$ 10^-7 m². The least count of the Vernier scale is 1.0 $$ \times $$ 10^-5 m. The maximum percentage error in the Young's modulus of the wire is

Let $${E_1}\left( r \right),{E_2}\left( r \right)$$ and $${E_3}\left( r \right)$$ be the respective electric field at a distance $$r$$ from a point charge $$Q,$$ an infinitely long wire with constant linear charge density $$\lambda ,$$ and an infinite plane with uniform surface charge density $$\sigma .$$ If $$E{}_1\left( {{r_0}} \right) = {E_2}\left( {{r_0}} \right) = {E_3}\left( {{r_0}} \right)$$ at a given distance $${r_0}.$$ then