Diameter of a plano-convex lens is $$6$$ $$cm$$ and thickness at the center is $$3mm$$. If speed of light in material of lens is $$2 \times {10^8}\,m/s,$$ the focal length of the lens is

An object $$2.4$$ $$m$$ in front of a lens forms a sharp image on a film $$12$$ $$cm$$ behind the lens. A glass plate $$1$$ $$cm$$ thick, of refractive index $$1.50$$ is interposed between lens and film with its plane faces parallel to film. At what distance (from lens) should object shifted to be in sharp focus of film?

Let $$x$$-$$z$$ plane be the boundary between two transparent media. Medium $$1$$ in $$z \ge 0$$ has a refractive index of $$\sqrt 2 $$ and medium $$2$$ with $$z < 0$$ has a refractive index of $$\sqrt 3 .$$ A ray of light in medium $$1$$ given by the vector $$\overrightarrow A = 6\sqrt 3 \widehat i + 8\sqrt 3 \widehat j - 10\widehat k$$ is incident on the plane of separation. The angle of refraction in medium $$2$$ is:

A car is fitted with a convex side-view mirror of focal length $$20$$ $$cm$$. A second car $$2.8m$$ behind the first car is overtaking the first car at a relative speed of $$15$$ $$m/s$$. The speed of the image of the second car as seen in the mirror of the first one is :