$A$ ray of light strikes a plane mirror $M$ at an angle of $45^{\circ}$ as shown in the figure. After reflection,the ray passes through a prism of refractive index $1.5$ whose apex angle is $4^{\circ}$. The total angle through which the ray is deviated is......$^{\circ}$

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.....$cm$

$A$ small fish,$0.4\,m$ below the surface of a lake,is viewed through a simple converging lens of focal length $3\,m$. The lens is kept at $0.2\,m$ above the water surface such that the fish lies on the optical axis of the lens. The image of the fish seen by the observer will be at $\left( \mu_{water} = \frac{4}{3} \right)$

$A$ diverging lens of focal length $10 \, cm$ is placed $10 \, cm$ in front of a plane mirror as shown in the figure. Light from a very far away source falls on the lens. The final image is at a distance

In a movie hall,the distance between the projector and the screen is increased by $1\%$. The illumination on the screen is:

An optical arrangement consists of two concave mirrors $M_1$ and $M_2$,and a convex lens $L$ with a common principal axis,as shown in the figure. The focal length of $L$ is $10 \text{ cm}$. The radii of curvature of $M_1$ and $M_2$ are $20 \text{ cm}$ and $24 \text{ cm}$,respectively. The distance between $L$ and $M_2$ is $20 \text{ cm}$. $A$ point object $S$ is placed at the mid-point between $L$ and $M_2$ on the axis. When the distance between $L$ and $M_1$ is $n/7 \text{ cm}$,one of the images coincides with $S$. The value of $n$ is. . . .