1. Identify the type of image and mirror

A virtual, magnified image by a curved mirror is produced only by a concave mirror when the object is placed between the pole and the focus.

So, for a concave mirror:

• focal length $f<0$
• object distance $u<0$
• virtual image distance $v>0$

2. Use magnification condition

The image is two times magnified, so $m=+2$

For mirrors, $m=-\frac{v}{u}$

Hence, $-\frac{v}{u}=2 \Rightarrow v=-2u$

3. Use the given separation between object and image

The object is in front of the mirror and the virtual image is behind the mirror, so the distance between them is $|u|+|v|=15$

Since $u<0$ and $v>0$, $-u+v=15$

Using $v=-2u$, $-u+(-2u)=15$ $-3u=15$ $u=-5\text{ cm}$

Therefore, $v=10\text{ cm}$

4. Apply mirror formula

Mirror formula: $\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$

Substitute $u=-5$ cm and $v=10$ cm: $\frac{1}{f}=\frac{1}{10}+\frac{1}{-5}$ $\frac{1}{f}=\frac{1}{10}-\frac{2}{10}=-\frac{1}{10}$

So, $f=-10\text{ cm}$

5. Check options

• A: $-10$ cm ✅
• B: $-12$ cm ❌
• C: $15$ cm ❌
• D: $10/3$ cm ❌

Thus, the correct answer is Option A.