Two blocks are connected by a spring. The combination is suspended,at rest,from a string attached to the ceiling,as shown in the figure. The string breaks suddenly. Immediately after the string breaks,what is the initial downward acceleration of the upper block of mass $2\,m$?

$A$ $U$-tube has a left limb with a cross-sectional area that is one-quarter of the right limb. The tube contains mercury (density $13.6 \ g/cm^3$). The level of mercury in the narrow left limb is at a distance of $36 \ cm$ from the upper end of the tube. What will be the rise in the level of mercury in the right limb if the left limb is filled to the top with water (in $cm$)? (Density of water = $1 \ g/cm^3$)

The image of the point $(3, 2, 1)$ in the plane $2x - y + 3z = 7$ is

The inverse of the point $(1, 2)$ with respect to the circle $x^2 + y^2 - 4x - 6y + 9 = 0$ is

Two stars of masses $M$ and $2M$ are at a distance $d$ apart and are revolving around their common center of mass. The angular velocity of the system of two stars is ($G$ is the universal gravitational constant):

For a first order reaction at $27^{\circ} C$,the ratio of time required for $75 \%$ completion to $25 \%$ completion of reaction is