$A$ ring of radius $4a$ is rigidly fixed in a vertical position on a table. $A$ small disc of mass $m$ and radius $a$ is released as shown in the figure. When the disc rolls down,without slipping,to the lowest point of the ring,then its speed will be

Bithional is an example of

If $f(x) = \begin{cases} \frac{\sin(1+[x])}{[x]}, & \text{for } [x] \neq 0 \\ 0, & \text{for } [x] = 0 \end{cases}$ where $[x]$ denotes the greatest integer not exceeding $x$,then $\lim_{x \rightarrow 0^{-}} f(x)$ is equal to

Two identical balls $A$ and $B$ are released from the positions shown in the figure. They collide elastically on the horizontal portion $MN$. The ratio of the heights attained by $A$ and $B$ after the collision will be (neglect friction):

One end of a uniform glass capillary tube of radius $r=0.025 \ cm$ is immersed vertically in water to a depth $h=1 \ cm$. The excess pressure in $N/m^2$ required to blow an air bubble out of the tube is: (Surface tension of water $T=7 \times 10^{-2} \ N/m$,Density of water $\rho=10^3 \ kg/m^3$,Acceleration due to gravity $g=10 \ m/s^2$)

An ideal gas follows a process $PT = \text{constant}$. The correct graph between pressure $(P)$ and volume $(V)$ is: