An engine is supposed to operate between two reservoirs at temperatures $727^{\circ}C$ and $227^{\circ}C$. The maximum possible efficiency of such an engine is:

$A$ small body of mass $m$ slides down from the top of a hemisphere of radius $r$. The surfaces of the block and the hemisphere are frictionless. The height at which the body loses contact with the surface of the sphere is

If $y$ is a function of $x$ and $\log(x+y) = 2xy$,then the value of $y'(0)$ is

$A$ circular disc of radius $R$ is removed from a bigger circular disc of radius $2R$ such that the circumferences of the discs touch. The centre of mass of the new disc is at a distance $\alpha R$ from the centre of the bigger disc. The value of $\alpha$ is

The equation of the straight line perpendicular to $5x - 2y = 7$ and passing through the point of intersection of the lines $2x + 3y = 1$ and $3x + 4y = 6$ is

The plant group that lives in an amphibious state between algae and pteridophytes is known as ......