Step-by-step Solution

1. Principle of a Metre Bridge: A metre bridge works on the principle of a balanced Wheatstone bridge. When the galvanometer shows a null deflection, the ratio of resistances is equal to the ratio of the corresponding lengths of the bridge wire. The formula for a balanced metre bridge is: $\frac{R}{S} = \frac{l}{100 - l}$ where:

   • R is the unknown resistance.
   • S is the standard known resistance.
   • l is the balancing length from the end connected to R.

2. Identify Given Values: From the problem statement:

   • Standard resistance, $S = 90 \Omega$
   • Balancing length, $l = 40.0 \text{ cm}$
   • Least count of the scale, $\Delta l = 1 \text{ mm} = 0.1 \text{ cm}$. This represents the error in the length measurement.

3. Calculate the Unknown Resistance (R): Using the metre bridge formula, we can solve for R: $R = S \left( \frac{l}{100 - l} \right)$ Substitute the given values: $R = 90 \left( \frac{40.0}{100 - 40.0} \right)$ $R = 90 \left( \frac{40.0}{60.0} \right)$ $R = 90 \times \frac{2}{3}$ $R = 60 \Omega$

4. Error Analysis: To find the error in R ($\Delta R$), we use the formula for error propagation. The relation is $R = S \frac{l}{100 - l}$. Assuming the standard resistance S is error-free, the error in R depends on the error in measuring l.

   The relative error in R can be found by taking the logarithm and differentiating: $\ln(R) = \ln(S) + \ln(l) - \ln(100 - l)$ Differentiating to find the fractional error: $\frac{\Delta R}{R} = \frac{\Delta l}{l} + \frac{\Delta(100 - l)}{100 - l}$ Since $\Delta(100 - l)$ is the error in (100-l) which is the same as the error in l, i.e., $\Delta l$, we have: $\frac{\Delta R}{R} = \frac{\Delta l}{l} + \frac{\Delta l}{100 - l}$ $\frac{\Delta R}{R} = \Delta l \left( \frac{1}{l} + \frac{1}{100 - l} \right)$ $\frac{\Delta R}{R} = \Delta l \left( \frac{100 - l + l}{l(100 - l)} \right)$ $\frac{\Delta R}{R} = \frac{100 \Delta l}{l(100 - l)}$

5. Calculate the Absolute Error (\Delta R): Now, we can find $\Delta R$: $\Delta R = R \left( \frac{100 \Delta l}{l(100 - l)} \right)$ Substitute the known values:

   • $R = 60 \Omega$
   • $\Delta l = 0.1 \text{ cm}$
   • $l = 40.0 \text{ cm}$ $\Delta R = 60 \left( \frac{100 \times 0.1}{40.0 \times (100 - 40.0)} \right)$ $\Delta R = 60 \left( \frac{10}{40 \times 60} \right)$ $\Delta R = 60 \times \frac{10}{2400}$ $\Delta R = \frac{600}{2400} = \frac{1}{4}$ $\Delta R = 0.25 \Omega$

6. Final Result: The unknown resistance is expressed as $R \pm \Delta R$. Therefore, the unknown resistance is $60 \pm 0.25 \Omega$.

7. Conclusion: Comparing our result with the given options: A: 60 $\pm$ 0.15 $\Omega$ B: 135 $\pm$ 0.56 $\Omega$ C: 60 $\pm$ 0.25 $\Omega$ D: 135 $\pm$ 0.23 $\Omega$

   Our calculated value matches option C.