An object is placed at a distance of $5 \, cm$ from the first principal focus of a convex lens of focal length $10 \, cm$. If a real image is formed,find its distance from the lens in $cm$.

If in a plano-convex lens,the radius of curvature of the convex surface is $10 \ cm$ and the focal length of the lens is $30 \ cm,$ then the refractive index of the material of the lens will be

Inside a vessel filled with liquid, a converging lens is placed as shown in the figure. The lens has a focal length of $15 \,cm$ when in air and has a refractive index of $\frac{3}{2}$. If the liquid has a refractive index of $\frac{9}{5}$, the focal length of the lens in the liquid is (in $\,cm$)

If the image formed by a thin convex lens of power $P$ has magnification $m$,then the image distance $v$ is:

The word $KVPY$ is written on a board and viewed through different lenses such that the board is at a distance beyond the focal length of the lens.
Ignoring magnification effects,consider the following statements.
$(I)$ First image has been viewed from the planar side of a plano-concave lens and second image from the planar side of a plano-convex lens.
$(II)$ First image has been viewed from the concave side of a plano-concave lens and second image from the convex side of a plano-convex lens.
$(III)$ First image has been viewed from the concave side of a plano-concave lens and second image from the planar side of a plano-convex lens.
$(IV)$ First image has been viewed from the planar side of a plano-concave lens and second image from the convex side of a plano-convex lens.
Which of the above statements are correct?

$A$ thin plano-convex lens of focal length $73.5 \ cm$ has a circular aperture of diameter $8.4 \ cm$. If the refractive index of the material of the lens is $\frac{5}{3}$, then the thickness of the lens is nearly