An object is placed at a distance of $20 \ cm$ from a convex lens of focal length $10 \ cm$. The image is formed on the other side of the lens at a distance of . . . . . . $cm$.

$A$ glass concave lens is placed in a liquid in which it behaves like a convergent lens. If the refractive indices of glass and liquid with respect to air are $_a \mu_g$ and $_a \mu_l$ respectively,then

Write the lens formula for a thin lens.

The same size images are formed by a convex lens when the object is placed at $20\, cm$ or at $10\, cm$ from the lens. The focal length of the convex lens is ............ $cm$.

$A$ rod of length $2 \text{ cm}$ makes an angle of $\frac{2 \pi}{3} \text{ rad}$ with the principal axis of a thin convex lens. The lens has a focal length of $10 \text{ cm}$ and is placed at a distance of $\frac{40}{3} \text{ cm}$ from the object as shown in the figure. The height of the image is $\frac{30 \sqrt{3}}{13} \text{ cm}$ and the angle made by it with respect to the principal axis is $\alpha \text{ rad}$. The value of $\alpha$ is $\frac{\pi}{n} \text{ rad}$,where $n$ is:

$A$ plano-convex lens is made of refractive index $1.6$. The radius of curvature of the curved surface is $60 \ cm$. The focal length of the lens is $..... \ cm$.