In a reaction,$A + B \rightleftharpoons C + D$,$40 \%$ of $B$ has reacted at equilibrium,when $1 \ mole$ of $A$ was heated with $1 \ mole$ of $B$ in a $10 \ L$ closed vessel. The value of $K_C$ is

The direction ratios of the normal to the plane passing through $(0,0,1)$, $(0,1,2)$, and $(1,0,3)$ are

$N$ divisions on the main scale of a vernier calipers coincide with $(N + 1)$ divisions of the vernier scale. If each division of main scale is $a$ units, then the least count of the calipers is

The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$. Its equation is

An object of mass $2m$ is projected with a speed of $100 \ m/s$ at an angle $\theta = \sin^{-1}(3/5)$ to the horizontal. At the highest point,the object breaks into two pieces of same mass $m$ and the first one comes to rest. The distance between the point of projection and the point of landing of the bigger piece (in metre) is (Given,$g = 10 \ m/s^2$)

If $A$ and $B$ are events of a random experiment such that $P(A \cup B) = \frac{4}{5}$,$P(\bar{A} \cup \bar{B}) = \frac{7}{10}$ and $P(B) = \frac{2}{5}$,then $P(A)$ equals