An object is 30 cm from a spherical mirror, along the central axis. The absolute value of lateral magnification is $\frac{1}{2}$.The image produced is inverted. The focal length of the mirror is

Given that an objects placed

 $u=-30 cm$ from a spherical mirror.

Absolute volume of lateral magnification $\left|m\right|=\frac{1}{2}\Rightarrow m=\frac{+1}{2}or \frac{-1}{2}$

Image produced is inverted. So we take $m=\frac{-1}{2}$

We know $m=\frac{-v}{u}$

$\frac{-1}{2}=\frac{-v}{u}$

$\Rightarrow v=\frac{+u}{2}=\frac{+(-30)}{2}=-15 cm$$\frac{1}{-30}-\frac{1}{15}=\frac{1}{f}$

$\frac{-2-1}{30}=\frac{1}{f}$

$f=-10 cm$