If $R$ is a relation from a finite set $A$ having $m$ elements to a finite set $B$ having $n$ elements,then the number of relations from $A$ to $B$ is

If $f: \mathbb{R} \rightarrow \mathbb{R}$ is such that $f(x+y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$,$f(1) = 7$,and $\sum_{r=1}^n f(r) = 14112$,then $n = $

Four bodies $P, Q, R$ and $S$ are projected with equal velocities having angles of projection $15^o, 30^o, 45^o$ and $60^o$ with the horizontal respectively. The body having the shortest range is

For the reversible reaction $N_{2(g)} + 3H_{2(g)} \rightleftharpoons 2NH_{3(g)}$ at $500 \, ^\circ C$,the value of $K_p$ is $1.44 \times 10^{-5}$. When the partial pressure is measured in atmospheres,the value of $K_c$ in $\text{mol L}^{-1}$ is:

The product of the lengths of the perpendiculars drawn from the point $(-1, 5)$ to the pair of lines $2x^2 - xy - 3y^2 + 6x + y + 4 = 0$ is

$A$ bag $X$ contains $2$ white and $3$ black balls and another bag $Y$ contains $4$ white and $2$ black balls. One bag is selected at random and a ball is drawn from it. Then, the probability for the ball chosen to be white is: