A concave mirror produces an image of an object such that the distance between the object and image is 20 cm . If the magnification of the image is ' -3 ', then the magnitude of the radius of curvature of the mirror is :

A concave mirror produces an image with object-image distance = 20 cm and magnification = $$-3$$.

We start by noting that for a concave mirror, using the standard convention: object distance $$u$$ is negative (object on the same side as incoming light), and for a real image, $$v$$ is also negative. The magnification is given by $$m = -\frac{v}{u} = -3$$, which leads to $$v = 3u$$, and since $$m = -3$$ (negative), the image is real and inverted.

Next, the distance between object and image is $$|v - u| = 20$$ cm. Substituting $$v = 3u$$ gives $$|3u - u| = |2u| = 20$$, so $$|u| = 10$$ cm. Therefore $$u = -10$$ cm (negative by convention) and $$v = 3(-10) = -30$$ cm.

Using the mirror formula, $$ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} = \frac{1}{-30} + \frac{1}{-10} = -\frac{1}{30} - \frac{1}{10} = -\frac{1}{30} - \frac{3}{30} = -\frac{4}{30} = -\frac{2}{15} $$. This gives $$ f = -\frac{15}{2} = -7.5 \text{ cm} $$.

The radius of curvature is $$R = 2f = 2 \times (-7.5) = -15$$ cm. The magnitude is $$|R| = 15$$ cm.

The correct answer is Option C: 15 cm.