A lamp is 5.0 m from a wall. Find the focal length of a concave mirror which will form a four times magnified image of the lamp on the wall.

Given, 

Type of mirror = Concave Mirror (we know that concave mirror always forms magnified image when the image is real and inverted)

Object distance, u = -5m (all the distance to the left side of the mirror are to be measured in negative and positive to the right side of the mirror)

Magnification, m = -4 (distance below the principal axis are to be measured in negative and positive to the above side of the principal axis)

Image distance, v = ?

Focal length, f = ?

We know that magnification is equal to the ratio of image distance to the object distance. So, it can be given as-

$m=\frac{v}{u}$

Substituting the values in the formula we get-

$-4=\frac{v}{-5}$

$v=20m$

$v=-20m$ (the distance of the image is measured in negative, as the image is real and inverted)

Now to find the focal length, we will use the mirror formula, which is given as-

$\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

Substituting the given values in the formula we get-

$\frac{1}{f}=\frac{1}{-20}+\frac{1}{-5}$

$\frac{1}{f}=\frac{-1-4}{20}$

$\frac{1}{f}=\frac{-5}{20}$

$\frac{1}{f}=-\frac{1}{4}$

$f=-4m$ (by ceoss-multiplication)

Hence, the focal length of a concave mirror will be -4m, to form a four times magnified image of the lamp on the wall.