Beats are produced by two waves given by $y_1 = a \sin(2000 \pi t)$ and $y_2 = a \sin(2008 \pi t)$. The number of beats heard per second 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 = $

If the slope of one line is twice the slope of the other in the pair of straight lines $ax^2 + 2hxy + by^2 = 0$,then $8h^2$ is equal to (in $ab$)

In a floral formula,what does the notation $(K)$ denote?

Which of the following has the smallest size?

$A$ child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal floor. If the child starts running towards the other end of the trolley with a speed $u$,the centre of mass of the system (trolley + child) will move with a speed.