The $45^{\circ}-45^{\circ}-90^{\circ}$ prism of height $10 \,cm$ (see image) has a refractive index $\mu = 2$,with a silvered hypotenuse surface. $A$ convex lens of focal length $10 \,cm$ is placed $15 \,cm$ in front of a wall. The system produces a sharp image of point $P$ on the wall. The value of $h$ (in $cm$) is closest to:

$A$ $2\,m$ long scale with a least count of $0.2\,cm$ is used to measure the locations of objects on an optical bench. While measuring the focal length of a convex lens, the object pin and the convex lens are placed at $80\,cm$ mark and $1\,m$ mark, respectively. The image of the object pin on the other side of the lens coincides with the image pin that is kept at $180\,cm$ mark. The $\%$ error in the estimation of focal length is

An object is placed at a distance of $15\,cm$ from a convex lens of focal length $10\,cm$. On the other side of the lens,a convex mirror is placed at its focus such that the image formed by the combination coincides with the object itself. The focal length of the convex mirror is.....$cm$

The mirrors are perpendicular to each other as shown in the figure. $A$ light ray $AB$ is incident on the mirror $M_1$. The reflected ray then undergoes a reflection from the mirror $M_2$. The final ray after reflection from $M_2$ will be parallel to the incident ray, if

$Lux$ is equal to:

Two plane mirrors of length $L$ are separated by a distance $L$,and a man $M_2$ is standing at a distance $L$ from the connecting line of the mirrors,as shown in the figure. $A$ man $M_1$ is walking in a straight line at a distance $2L$ parallel to the mirrors at a speed $u$. Then,the man $M_2$ at $O$ will be able to see the image of $M_1$ for a total time of: