When light of wavelength 248 nm falls on a metal of threshold energy 3.0 eV, the de-Broglie wavelength of emitted electrons is _______$$\mathop A\limits^o $$. (Round off to the Nearest Integer).

[ Use : $$\sqrt 3 $$ = 1.73, h = 6.63 $$\times$$ 10^$$-$$34 Js

m_e = 9.1 $$\times$$ 10^$$-$$31 kg; c = 3.0 $$\times$$ 10⁸ ms^$$-$$1; 1eV = 1.6 $$\times$$ 10^$$-$$19 J]

The decomposition of formic acid on gold surface follows first order kinetics. If the rate constant at 300 K is 1.0 $$\times$$ 10^$$-$$3 s^$$-$$1 and the activation energy E_a = 11.488 kJ mol^$$-$$1, the rate constant at 200 K is ____________ $$\times$$ 10^$$-$$5 s^$$-$$1. (Round off to the Nearest Integer). (Given : R = 8.314 J mol^$$-$$1 K^$$-$$1)

Let a vector $$\alpha \widehat i + \beta \widehat j$$ be obtained by rotating the vector $$\sqrt 3 \widehat i + \widehat j$$ by an angle 45$$^\circ$$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices ($$\alpha$$, $$\beta$$), (0, $$\beta$$) and (0, 0) is equal to :

Let $${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)} $$. Then $$\mathop {\lim }\limits_{k \to \infty } {S_k}$$ is equal to :