1. Use magnification formula for mirrors

For a mirror, $m = -\frac{v}{u}$ Given magnification magnitude is $\left|m\right|=\frac14$

So, $-\frac{v}{u}=\frac14 \quad \text{or} \quad -\frac{v}{u}=-\frac14$ Depending on whether the image is real/inverted or virtual/erect.

Since this is an MCQ and distance between object and image is given, we test the physically valid case.

2. Case 1: Real diminished image

For a convex mirror, or for a concave mirror with object beyond center, diminished image has positive magnification in sign convention? Let us use the standard Cartesian sign convention carefully.

For a real object in front of mirror, $u<0$

If the image is real, $v<0$ Then $m=-\frac{v}{u}<0$ So for diminished real image, $m=-\frac14$

Thus, $-\frac{v}{u}=-\frac14 \Rightarrow \frac{v}{u}=\frac14 \Rightarrow v=\frac{u}{4}$

Since both $u$ and $v$ are negative, this is consistent.

Now object and image are on same side of mirror, so their separation is $|u-v|=40$ Substitute $v=\frac{u}{4}$: $\left|u-\frac{u}{4}\right|=40$ $\frac{3|u|}{4}=40$ $|u|=\frac{160}{3}$ So, $u=-\frac{160}{3}\text{ cm}, \qquad v=-\frac{40}{3}\text{ cm}$

Now apply mirror formula: $\frac{1}{f}=\frac{1}{v}+\frac{1}{u}$ $\frac{1}{f}=\frac{1}{-40/3}+\frac{1}{-160/3}$ $\frac{1}{f}=-\frac{3}{40}-\frac{3}{160}$ $\frac{1}{f}=-\frac{12+3}{160}=-\frac{15}{160}=-\frac{3}{32}$ $f=-\frac{32}{3}\text{ cm}$

Magnitude of focal length: $|f|=\frac{32}{3}\approx 10.7\text{ cm}$

3. Case 2: Virtual diminished image (convex mirror)

Then $m=+\frac14$ So, $-\frac{v}{u}=\frac14 \Rightarrow v=-\frac{u}{4}$ With $u<0$, this gives $v>0$, consistent.

Object and image are on opposite sides, so distance between them is $|u|+|v|=40$ $|u|+\frac{|u|}{4}=40$ $\frac{5|u|}{4}=40$ $|u|=32$ Thus, $u=-32,\quad v=+8$

Mirror formula: $\frac{1}{f}=\frac{1}{8}+\frac{1}{-32}=\frac{4-1}{32}=\frac{3}{32}$ $f=\frac{32}{3}\approx 10.7\text{ cm}$

Again, magnitude is $10.7\text{ cm}$

4. Final answer

The focal length (numerical value asked in options) is $\boxed{10.7\text{ cm}}$ So the correct option is C.