The size of an image of an object formed by a mirror with a focal length of $20\, cm$ is observed to be reduced to $1/3^{rd}$ of its size. At what distance is the object placed from the mirror? What is the nature of the image and the mirror?

(D) Given: Focal length $f = -20\, cm$ (for a concave mirror),Magnification $m = -1/3$ (for a real and inverted image).
Using the magnification formula $m = -v/u$,we get $-1/3 = -v/u$,which implies $v = u/3$.
Using the mirror formula $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$,substitute $v = u/3$:
$\frac{1}{u/3} + \frac{1}{u} = \frac{1}{-20}$
$\frac{3}{u} + \frac{1}{u} = -\frac{1}{20}$
$\frac{4}{u} = -\frac{1}{20}$
$u = -80\, cm$.
The object is placed at a distance of $80\, cm$ from the mirror.
The image is real and inverted,and the mirror is concave.