Distance between an object and three times magnified real image is 40 cm. The focal length of the minor used is ___ cm.

We need to find the focal length of a concave mirror that produces a 3x magnified real image with object-image distance of 40 cm.

Magnification: m = -3 (real image is inverted, so negative for a mirror)

Distance between object and image = 40 cm

Set up equations:

For a mirror: $$m = -\frac{v}{u}$$

$$-3 = -\frac{v}{u} \implies v = 3u$$

Using sign convention (both u and v are negative for concave mirror with real image):

Let u = -x (object distance), then v = -3x (image distance, real image).

Distance between object and image: |v - u| = |-3x - (-x)| = |-2x| = 2x = 40 cm

So x = 20 cm, u = -20 cm, v = -60 cm.

Find focal length using mirror formula:

$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u} = \frac{1}{-60} + \frac{1}{-20} = -\frac{1}{60} - \frac{1}{20} = -\frac{1+3}{60} = -\frac{4}{60} = -\frac{1}{15}$$

$$f = -15$$ cm

Therefore, the focal length is Option 2: -15 cm.