If a circle of radius R passes through the origin O and intersects the coordinates axes at A and B, then the locus of the foot of perpendicular from O on AB is :

A vertical closed cylinder is separated into two parts by a frictionless piston of mass m and of negligible thickness. The piston is free to move along the length of the cylinder. The length of the cylinder above the piston is $$\ell $$₁, and that below the piston is $$\ell $$₂, such that $$\ell $$₁ > $$\ell $$₂. Each part of the cylinder contains n moles of an ideal gas at equal temperature T. If the piston is stationary, its mass, m, will be given by :
(R is universal gas constant and g is the acceleration due to gravity)

In the above circuit, C = $${{\sqrt 3 } \over 2}$$$$\mu $$F, R₂ = 20 $$\Omega $$, L = $${{\sqrt 3 } \over {10}}$$ H and R₁ = 10 $$\Omega $$. Current in L-R₁ path is I₁ and in C-R₂ path it is I₂ . The voltage of A.C. source is given by, V = 200 $${\sqrt 2 }$$ sin (100 t) volts . The phase difference between I₁ and I₂ is :

A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is 5 cm and its rotational speed is 2 rotations per second, then the difference in the heights between the centre and the sides, in cm, will be :