A convex mirror has an object distance of 2 m and a magnification of 1/3. Calculate the focal length of the mirror.

The object is placed in front of a convex mirror.
Object distance, $u = 2$ m. According to the Cartesian sign convention, for a real object, $u$ is negative.
So, $u = -2$ m.
Magnification, $m = \frac{1}{3}$. For a convex mirror, the image is always virtual and erect, so the magnification is positive.
We need to find the focal length, $f$. For a convex mirror, the focal length $f$ is positive.

The magnification formula for mirrors is:

Substitute the known values:

Multiply both sides by 2 to solve for $v$:

The positive value of $v$ indicates that the image is virtual and formed behind the mirror, which is consistent with a convex mirror.

The mirror formula is:

Substitute the values of $u$ and $v$:

Combine the fractions:

The positive value of $f$ is consistent with the focal length of a convex mirror.

The focal length of the mirror is $\text{1 m}$.