The resistance of a conductor is `5Omega` at `50^(@)`C and `6Omega` at `100^(@)C`. The resistance at `0^(@)C` is

Options

A. `2Omega`
B. `4 Omega`
C. `2.5Omega`
D. `4.5 Omega`

Explanation

To find the resistance of a conductor at 0°C given its resistance at two different temperatures, we can use the formula that relates resistance and temperature:

\[ R_t = R_0 (1 + \alpha \Delta T) \]

Where:
- \( R_t \) is the resistance at temperature \( t \)
- \( R_0 \) is the resistance at 0°C
- \( \alpha \) is the temperature coefficient of resistance
- \( \Delta T \) is the change in temperature from 0°C

### Step 1: Set up the equations

From the problem, we have:
- \( R_{50} = 5 \, \Omega \) at \( 50°C \)
- \( R_{100} = 6 \, \Omega \) at \( 100°C \)

We can set up two equations based on the formula:

1. For \( 50°C \):
\[ 5 = R_0 (1 + \alpha (50 - 0)) \quad \text{(Equation 1)} \]

2. For \( 100°C \):
\[ 6 = R_0 (1 + \alpha (100 - 0)) \quad \text{(Equation 2)} \]

### Step 2: Rearranging the equations

From Equation 1:
\[ 5 = R_0 (1 + 50\alpha) \]
\[ 5 = R_0 + 50 R_0 \alpha \quad \text{(Rearranging)} \]

From Equation 2:
\[ 6 = R_0 (1 + 100\alpha) \]
\[ 6 = R_0 + 100 R_0 \alpha \quad \text{(Rearranging)} \]

### Step 3: Express \( R_0 \alpha \)

Now, we can express \( R_0 \alpha \) from both equations:

From Equation 1:
\[ 50 R_0 \alpha = 5 - R_0 \quad \Rightarrow \quad R_0 \alpha = \frac{5 - R_0}{50} \quad \text{(Equation 3)} \]

From Equation 2:
\[ 100 R_0 \alpha = 6 - R_0 \quad \Rightarrow \quad R_0 \alpha = \frac{6 - R_0}{100} \quad \text{(Equation 4)} \]

### Step 4: Equate the two expressions for \( R_0 \alpha \)

Set Equation 3 equal to Equation 4:
\[ \frac{5 - R_0}{50} = \frac{6 - R_0}{100} \]

### Step 5: Cross-multiply and solve for \( R_0 \)

Cross-multiplying gives:
\[ 100(5 - R_0) = 50(6 - R_0) \]
\[ 500 - 100 R_0 = 300 - 50 R_0 \]

Rearranging:
\[ 500 - 300 = 100 R_0 - 50 R_0 \]
\[ 200 = 50 R_0 \]
\[ R_0 = \frac{200}{50} = 4 \, \Omega \]

### Conclusion

The resistance at \( 0°C \) is \( R_0 = 4 \, \Omega \).

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