The mass density of a spherical galaxy varies as $${K \over r}$$ over a large distance ‘r’ from its centre. In that region, a small star is in a circular orbit of radius R. Then the period of revolution, T depends on R as :

A charged particle (mass m and charge q)
moves along X-axis with velocity V₀. When it
passes through the origin it enters a region having uniform electric field
$$\overrightarrow E = - E\widehat j$$ which extends upto x = d.
Equation of path of electron in the region x > d is

When radiation of wavelength $$\lambda $$ is used to illuminate a metallic surface, the stopping potential is V. When the same surface is illuminated with radiation of wavelength 3$$\lambda $$, the stopping potential is $${V \over 4}$$. If the threshold wavelength for the metallic surface is n$$\lambda $$ then value of n will be __________.

The least count of the main scale of a vernier callipers is 1 mm. Its vernier scale is divided into 10 divisions and coincide with 9 divisions of the main scale. When jaws are touching each other, the 7^th division of vernier scale coincides with a division of main scale and the zero of vernier scale is lying right side of the zero of main scale. When this vernier is used to measure length of a cylinder the zero of the vernier scale between 3.1 cm and 3.2 cm and 4^th VSD coincides with a main scale division. The length of the cylinder is : (VSD is vernier scale division)