An object is placed at a distance of $30~\mathrm{cm}$ from the pole of a concave mirror. If its real and inverted image is formed at $60~\mathrm{cm}$ in front of the mirror, the focal length of the mirror is:

By the New Cartesian Sign Convention, distances are measured from the pole, with distances in front of the mirror (to the left) taken as negative. A real, inverted image formed in front of a concave mirror also lies on the same (negative) side.

Given: $u=-30\ \mathrm{cm}$, $v=-60\ \mathrm{cm}$ (real image in front of the mirror).

Using the mirror formula $\dfrac{1}{v}+\dfrac{1}{u}=\dfrac{1}{f}$:

$$\frac{1}{f}=\frac{1}{-60}+\frac{1}{-30}=\frac{-1-2}{60}=\frac{-3}{60}=-\frac{1}{20}$$

$$f=-20\ \mathrm{cm}$$

The negative sign confirms it is a concave mirror, consistent with a real, inverted image being formed.

(B) is correct.

(A) $-15\ \mathrm{cm}$ results from a sign/arithmetic slip while combining fractions.

(C) $+20\ \mathrm{cm}$ has the correct magnitude but the wrong sign (would imply a convex mirror, which cannot form a real image).

(D) $+15\ \mathrm{cm}$ compounds both a sign error and an arithmetic error.