If $$I = {2 \over \pi }\int\limits_{ - \pi /4}^{\pi /4} {{{dx} \over {(1 + {e^{\sin x}})(2 - \cos 2x)}}} $$, then 27I² equals .................

Three lines are given by

$$r = \lambda \widehat i,\,\lambda \in R$$,

$$r = \mu (\widehat i + \widehat j),\,\mu \in R$$ and

$$r = v(\widehat i + \widehat j + \widehat k),\,v\, \in R$$

Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the area of the triangle ABC is $$\Delta $$ then the value of (6$$\Delta $$)² equals ..............

Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If $$AP(1;3) \cap AP(2;5) \cap AP(3;7)$$ = AP(a ; d), then a + d equals ..............

In a radioactive sample, $${}_{19}^{40}K$$ nuclei either decay into stable $${}_{20}^{40}Ca$$ nuclei with decay constant 4.5 $$ \times $$ 10^-10 per year or into stable $${}_{18}^{40}Ar$$ nuclei with decay constant 0.5 $$ \times $$ 10^-10 per year. Given that in this sample all the stable $${}_{20}^{40}Ca$$ and $${}_{18}^{40}Ar$$ nuclei are produced by the $${}_{19}^{40}K$$ nuclei only. In time t $$ \times $$ 10⁹ years, if the ratio of the sum of stable $${}_{20}^{40}Ca$$ and $${}_{18}^{40}Ar$$ nuclei to the radioactive $${}_{19}^{40}K$$ nuclei is 99, the value of t will be

[Given : In 10 = 2.3]