In an experiment,a convex lens of focal length $15 \, cm$ is placed coaxially on an optical bench in front of a convex mirror at a distance of $5 \, cm$ from it. It is found that an object and its image coincide if the object is placed at a distance of $20 \, cm$ from the lens. The focal length of the convex mirror is.......$cm$.

$A$ plano-convex lens $(f = 20 \ cm)$ is silvered at the plane surface. Now,the new focal length $F$ will be........$cm$.

$A$ bi-convex lens of focal length $10\,cm$ is cut into two identical parts along a plane perpendicular to the principal axis. The power of each lens after the cut is $...........\,D$.

$A$ point-like object is placed at a distance of $1\, m$ in front of a convex lens of focal length $0.5\, m$. $A$ plane mirror is placed at a distance of $2\, m$ behind the lens. Find the position and nature of the final image formed by the system.

$A$ thin convex lens $L$ (refractive index $= 1.5$) is placed on a plane mirror $M$. When a pin is placed at $A$,such that $OA = 18\, cm$,its real inverted image is formed at $A$ itself,as shown in the figure. When a liquid of refractive index $\mu_l$ is put between the lens and the mirror,the pin has to be moved to $A'$,such that $OA' = 27\, cm$,to get its inverted real image at $A'$ itself. The value of $\mu_l$ will be

Two identical symmetric double convex lenses of focal length $f$ are cut into two equal parts $L_1, L_2$ by the $AB$ plane and $L_3, L_4$ by the $XY$ plane as shown in the figure respectively. The ratio of the focal lengths of lenses $L_1$ and $L_3$ is