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

Two identical glass rods $S_1$ and $S_2$ (refractive index $=1.5$) have one convex end of radius of curvature $10 \mathrm{cm}$.

They are placed with the curved surfaces at a distance $d$ as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light $P$ is placed inside rod $S_1$ on its axis at a distance of $50 \mathrm{cm}$ from the curved face, the light rays emanating from it are found to be parallel to the axis inside $S_2$. The distance $d$ is:

Two identical thin plano-convex lenses (refractive index $(1.5)$) each having radius of curvature of $20 \mathrm{cm}$ are placed with their convex surfaces in contact at the centre. The intervening space is filled with the oil of refractive index $1.7$. The focal length of the combination is:

A small object is placed $50 \mathrm{cm}$ to the left of thin convex lens of focal length $30 \mathrm{cm}$. A convex spherical mirror of radius of curvature $100 \mathrm{cm}$ is placed to the right of the lens at a distance of $50 \mathrm{cm}$. The mirror is tilted such that the axis of the mirror is at an angle$\theta ={30}^{\circ }$  to the axis of the lens, as shown in the figure. If the origin of the coordinate system is taken to be at the centre of the lens, the coordinates (in cm) of the point ($x, y$) at which the image is formed are:

A convex lens is put $10 \mathrm{cm}$ from a light source and it makes a sharp image on a screen, kept $10 \mathrm{cm}$ from the lens. Now a glass block (refractive index $1.5$) of $1.5 \mathrm{cm}$ thickness is placed in contact with the light source. To get the sharp image again, the screen is shifted by a distance $d$. Then $d$ is: