Dimensional formulae are used
A) to convert one system of units into another
(B) to find proportionality constants
C) to cheak the correctness of an equation

Options

A. Only a & b are true
B. Only C is true
C. A & C are true
D. All the true

Answer

To solve the question regarding the uses of dimensional formulae, we will analyze each option provided:

### Step-by-Step Solution:

1. **Understanding Dimensional Formulae**:
- Dimensional formulae represent the physical dimensions of a quantity in terms of basic dimensions (mass, length, time, etc.).
- They help in analyzing relationships between different physical quantities.

2. **Option A: To Convert One System of Units into Another**:
- Dimensional analysis can indeed be used to convert units. For example, if we know the dimensional formula of acceleration is \( [L T^{-2}] \), we can relate different units (like meters to centimeters).
- For instance, \( 1 \text{ m} = 100 \text{ cm} \) allows us to convert \( \text{m/s}^2 \) to \( \text{cm/s}^2 \).
- **Conclusion**: This option is **true**.

3. **Option B: To Find Proportionality Constants**:
- While dimensional analysis can help in finding the relationship between quantities (like \( t \) in terms of \( l \) and \( g \)), it does not provide the exact value of proportionality constants (like \( 2\pi \) in the equation \( t = 2\pi\sqrt{\frac{l}{g}} \)).
- Dimensional analysis can give the dimensions of the quantities involved, but not the numerical constants.
- **Conclusion**: This option is **false**.

4. **Option C: To Check the Correctness of an Equation**:
- Dimensional analysis can be used to verify if an equation is dimensionally consistent. For example, in the equation \( F = ma \), we can check if both sides have the same dimensions.
- The dimension of force \( F \) is \( [M L T^{-2}] \), and the dimension of \( ma \) (mass times acceleration) is also \( [M L T^{-2}] \). Since both sides match, the equation is dimensionally correct.
- **Conclusion**: This option is **true**.

5. **Final Conclusion**:
- Based on the analysis:
- Option A is true.
- Option B is false.
- Option C is true.
- Therefore, the correct answers are A and C.