(a) (i) A point object is kept in front of a convex spherical surface of radius of curvature R. Draw the ray diagram to show the formation of image and derive the relation between the object and image distance (u and v) in terms of refractive index n of the medium and R. (ii) A convex lens of focal length of $20\text{ cm}$ is used to form the image of an object placed $30\text{ cm}$ away from the lens. Find the position and nature of the image formed. OR (b) (i) Two thin converging lenses of focal length $f_1$ and $f_2$ are placed coaxially in contact. Derive expression for the focal length of the combination. (ii) A beam of coherent light of wavelength $550\text{ nm}$ is incident normal to the plane of a pair of two slits $S_1$ and $S_2$ each of width $1\cdot2 \times 10^{-6}\text{ m}$ separated by $1\cdot1\text{ mm}$. Dark and bright fringes are observed on a screen $2\cdot2\text{ m}$ away from the plane of the slits. Calculate : (I) fringe width. (II) distance of the second dark fringe from the central maximum. (III) what will happen when the entire apparatus is immersed in water.