The distance between an object and its image (magnified by $-\frac{1}{3}$) is $30 \ cm$. The focal length of the mirror used is $\left(\frac{x}{4}\right) \ cm$,where the magnitude of the value of $x$ is . . . . . . .

$A$ driving mirror consists of a cylindrical mirror of radius of curvature $10 \,cm$ and the length over the curved surface is $10 \,cm$. If the eye of the driver is assumed to be at a great distance from the mirror,then the field of view in radians is: (in $.0$)

An object of length $f/3$ is placed on the principal axis of a concave mirror of focal length $f$. The image of the end of the object farthest from the mirror is formed at the same position. Find the length of the image.

Use the mirror equation to deduce that:
$(a)$ an object placed between $f$ and $2f$ of a concave mirror produces a real image beyond $2f$.
$(b)$ a convex mirror always produces a virtual image independent of the location of the object.
$(c)$ the virtual image produced by a convex mirror is always diminished in size and is located between the focus and the pole.
$(d)$ an object placed between the pole and focus of a concave mirror produces a virtual and enlarged image.

$A$ small linear object of length $b$ is placed on the axis of a concave mirror of focal length $f$ at a distance $u$ from its pole. What is the approximate length of the image?

$A$ candle of height $3 \ cm$ is placed at $300 \ cm$ from a wall. $A$ concave mirror is kept at a distance $x$ from the wall in such a way that the image of the flame on the wall is $9 \ cm$. Then $x$ is $.... \ cm$.