The number of terms common to the two A.P.'s 3, 7, 11, ....., 407 and 2, 9, 16, ....., 709 is ______.

Starting at temperature 300 K, one mole of an
ideal diatomic gas ($$\gamma $$ = 1.4) is first compressed
adiabatically from volume V₁ to V₂ = $${{{V_1}} \over {16}}$$. It is
then allowed to expand isobarically to volume 2V₂. If all the processes are the quasi-static then
the final temperature of the gas (in ^oK) is (to the nearest integer) _____.

An electric field $$\overrightarrow E = 4x\widehat i - \left( {{y^2} + 1} \right)\widehat j$$ N/C
passes through the box shown in figure. The
flux of the electric field through surfaces ABCD
and BCGF are marked as $${\phi _I}$$ and $${\phi _{II}}$$
respectively. The difference between $$\left( {{\phi _I} - {\phi _{II}}} \right)$$ is (in Nm²/C) _______.

In a Young's double slit experiment 15 fringes are observed on a small portion of the screen when light of wavelength 500 nm is used. Ten fringes are observed on the same section of the screen when another light source of wavelength $$\lambda $$ is used. Then the value of $$\lambda $$ is (in nm) __________.