If `A = B +(C)/(D+E)` the dimensions of B and C are `[M^(0)L T^(-1)]` and `[M^(0)LT^(0)]` , respectively . Find the dimensions of A, D and E.

Options

A. `A=[M^(0)L^(0)T^(-1)]`, `D=[T]`, `E=[LT]`
B. `A=[MLT^(0)]`, `D=[T^(2)]`, `E=[T^(2)]`
C. `A=[M^(0)L^(0)T^(-1)]`, `D=[MT]`, `E=[MT]`
D. `A=[M^(0)LT^(-1)]`, `D=[T]`, `E=[T]`

Answer

To solve the problem, we need to find the dimensions of A, D, and E given the dimensions of B and C.

### Step-by-Step Solution:

1. **Identify the Given Dimensions**:
- The dimension of B is given as \( [M^0 L^1 T^{-1}] \).
- The dimension of C is given as \( [M^0 L^1 T^0] \).

2. **Understanding the Equation**:
- The equation is given as \( A = B + \frac{C}{D + E} \).
- Since we can only add quantities with the same dimensions, the dimensions of A must be the same as the dimensions of B.

3. **Determine the Dimension of A**:
- Therefore, the dimension of A is:
\[ [A] = [B] = [M^0 L^1 T^{-1}] \]

4. **Analyzing the Right Side of the Equation**:
- The right side of the equation can be expressed as:
\[ A = B + \frac{C}{D + E} \]
- Since \( D + E \) must have the same dimensions for the fraction \( \frac{C}{D + E} \) to be valid, we can denote the dimension of D and E as \( [D] = [E] \).

5. **Setting Up the Equation for Dimensions**:
- The dimension of \( \frac{C}{D + E} \) must also equal the dimension of A:
\[ [A] = \frac{[C]}{[D + E]} \]
- Since \( [D] = [E] \), we can write:
\[ [D + E] = [D] + [D] = 2[D] \]

6. **Substituting the Known Dimensions**:
- We know:
\[ [C] = [M^0 L^1 T^0] \]
- Thus, substituting into the equation gives:
\[ [A] = \frac{[C]}{[D]} = \frac{[M^0 L^1 T^0]}{[D]} \]

7. **Equating the Dimensions**:
- We already found that \( [A] = [M^0 L^1 T^{-1}] \):
\[ [M^0 L^1 T^{-1}] = \frac{[M^0 L^1 T^0]}{[D]} \]
- Rearranging gives:
\[ [D] = \frac{[M^0 L^1 T^0]}{[M^0 L^1 T^{-1}]} = [T^1] \]

8. **Conclusion**:
- Thus, the dimensions are:
- \( [A] = [M^0 L^1 T^{-1}] \)
- \( [D] = [T^1] \)
- Since \( [D] = [E] \), we also have \( [E] = [T^1] \).

### Final Dimensions:
- \( [A] = [M^0 L^1 T^{-1}] \)
- \( [D] = [T^1] \)
- \( [E] = [T^1] \)