Two objects `A` and `B` are of lengths `5cm` and `7cm` determined with errors `0.1cm` and `0.2cm` respecitively. The error in determining (a) the total length and (b) the difference in their lengths are

Options

A. `(12 +- 0.3), (2 +- 0.3)`
B. `(7 +- 0.3), (2 +- 0.3)`
C. `(12 +- 0.3), (12 +- 0.3)`
D. `(12 +- 0.3), (2 +- 0.6)`

Explanation

To solve the problem, we need to determine the errors in two scenarios: (a) the total length of objects A and B, and (b) the difference in their lengths.

### Step-by-Step Solution:

1. **Identify the lengths and their respective errors:**
- Length of object A, \( L_A = 5 \, \text{cm} \) with an error \( \Delta L_A = 0.1 \, \text{cm} \)
- Length of object B, \( L_B = 7 \, \text{cm} \) with an error \( \Delta L_B = 0.2 \, \text{cm} \)

2. **Calculate the total length:**
- The total length \( L_{total} \) is given by:
\[ L_{total} = L_A + L_B = 5 \, \text{cm} + 7 \, \text{cm} = 12 \, \text{cm} \]

3. **Calculate the error in the total length:**
- The error in the total length \( \Delta L_{total} \) is the sum of the individual errors:
\[ \Delta L_{total} = \Delta L_A + \Delta L_B = 0.1 \, \text{cm} + 0.2 \, \text{cm} = 0.3 \, \text{cm} \]

4. **Calculate the difference in lengths:**
- The difference in lengths \( L_{difference} \) is given by:
\[ L_{difference} = L_A - L_B = 5 \, \text{cm} - 7 \, \text{cm} = -2 \, \text{cm} \]

5. **Calculate the error in the difference of lengths:**
- The error in the difference \( \Delta L_{difference} \) is also the sum of the individual errors:
\[ \Delta L_{difference} = \Delta L_A + \Delta L_B = 0.1 \, \text{cm} + 0.2 \, \text{cm} = 0.3 \, \text{cm} \]

### Final Results:
- (a) The total length is \( 12 \, \text{cm} \) with an error of \( 0.3 \, \text{cm} \).
- (b) The difference in lengths is \( -2 \, \text{cm} \) with an error of \( 0.3 \, \text{cm} \).

### Summary:
- Total Length: \( 12 \, \text{cm} \pm 0.3 \, \text{cm} \)
- Difference in Length: \( -2 \, \text{cm} \pm 0.3 \, \text{cm} \)