Two conductors have the same resistance at `0^@C` but their temperature coefficient of resistanc are `alpha_1 and alpha_2`. The respective temperature coefficients of their series and parallel combinations are nearly

Options

A. `(alpha_(1)+alpha_(2))/(2),alpha_(1)+alpha_(2)`
B. `alpha_(1)+alpha_(2),(alpha_(1)+alpha_(2))/(2)`
C. `alpha_(1)+alpha_(2),(alpha_(1)+alpha_(2))/(alpha_(1)+alpha_(2))`
D. `(alpha_(1)+alpha_(2))/(2),(alpha_(1)+alpha_(2))/(2)`

Answer

To solve the problem, we need to find the temperature coefficients of resistance for two conductors when they are connected in series and parallel. The conductors have the same resistance at 0°C but different temperature coefficients of resistance, denoted as α₁ and α₂.

### Step-by-Step Solution:

1. **Understanding Resistance and Temperature Coefficient**:
The resistance \( R \) of a conductor at a temperature \( T \) is given by the formula:
\[ R(T) = R_0 (1 + \alpha T) \]
where \( R_0 \) is the resistance at 0°C, and \( \alpha \) is the temperature coefficient of resistance.

2. **Resistance in Series**:
For two resistors \( R_1 \) and \( R_2 \) connected in series, the total resistance \( R_s \) at temperature \( T \) is:
\[ R_s(T) = R_1(T) + R_2(T) \]
Substituting the resistance formula:
\[ R_s(T) = R_0(1 + \alpha_1 T) + R_0(1 + \alpha_2 T) \]
Simplifying this:
\[ R_s(T) = R_0(2 + (\alpha_1 + \alpha_2) T) \]
The temperature coefficient of resistance for the series combination \( \alpha_s \) is given by:
\[ \alpha_s = \frac{\alpha_1 + \alpha_2}{2} \]

3. **Resistance in Parallel**:
For two resistors \( R_1 \) and \( R_2 \) connected in parallel, the equivalent resistance \( R_p \) at temperature \( T \) is:
\[ \frac{1}{R_p(T)} = \frac{1}{R_1(T)} + \frac{1}{R_2(T)} \]
Substituting the resistance formula:
\[ \frac{1}{R_p(T)} = \frac{1}{R_0(1 + \alpha_1 T)} + \frac{1}{R_0(1 + \alpha_2 T)} \]
Simplifying this:
\[ \frac{1}{R_p(T)} = \frac{(1 + \alpha_2 T) + (1 + \alpha_1 T)}{R_0(1 + \alpha_1 T)(1 + \alpha_2 T)} \]
\[ = \frac{2 + (\alpha_1 + \alpha_2) T}{R_0(1 + \alpha_1 T)(1 + \alpha_2 T)} \]
For small \( \alpha_1 \) and \( \alpha_2 \), we can approximate:
\[ R_p(T) \approx \frac{R_0}{2} (1 + \frac{\alpha_1 + \alpha_2}{2} T) \]
The temperature coefficient of resistance for the parallel combination \( \alpha_p \) is:
\[ \alpha_p = \frac{\alpha_1 + \alpha_2}{2} \]

4. **Final Results**:
- The temperature coefficient of resistance for the series combination is:
\[ \alpha_s = \frac{\alpha_1 + \alpha_2}{2} \]
- The temperature coefficient of resistance for the parallel combination is:
\[ \alpha_p = \frac{\alpha_1 + \alpha_2}{2} \]

### Conclusion:
Thus, the respective temperature coefficients of resistance for the series and parallel combinations are both approximately \( \frac{\alpha_1 + \alpha_2}{2} \).