$A$ beam of light converges at a point $P$. Now a lens is placed in the path of the convergent beam $12 \, cm$ from $P$. At what point does the beam converge if the lens is a convex lens of focal length $20 \, cm$ (in $, cm$)?

$A$ 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:

$A$ lens behaves as a converging lens in air and a diverging lens in water. The refractive index of the lens is

If a lens is moved towards the object from a distance of $40 \,cm$ to $30 \,cm$,the magnification of the image remains the same (numerically). The focal length of the lens is ......... $cm$.

The distance between an object and a screen is $100\, cm$. $A$ lens can produce a real image of the object on the screen for two different positions between the screen and the object. The distance between these two positions is $40\, cm$. If the power of the lens is close to $\left(\frac{N}{100}\right) D$ where $N$ is an integer,the value of $N$ is.......

$A$ convex lens is dipped in a liquid whose refractive index is equal to the refractive index of the lens material. Then its focal length will