The refractive index of the material of a prism is $$\sqrt 2 $$ and the angle of the prism is 30°. One of the two refracting surfaces of the prism is made a mirror inwards, by silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if its angle of incidence on the prism is

An object is placed at a distance of 40 cm from a concave mirror of focal length 15 cm. If the object is displaced through a distance of 20 cm towards the mirror, the displacement of the image will be

An astronomical refracting telescope will have large angular magnification and high angular resolution, when it has an objective lens of

A beam of light from a source L is incident normally on a plane mirror fixed at a certain distance x from the source. The beam is reflected back as a spot on a scale placed just above the source L. When the mirror is rotated through a small angle $$\theta $$, the spot of the light is found to move through a distance y on the scale. The angle $$\theta $$ is given by