L,C and R represent the physical quantities inductance, capacitance and resistance respectively. Which of the following combinations have dimensions of frequency?

Options

A. `RL^(-1)`
B. `R^(-1)L^(-1)`
C. `L^(-1//2)C^(-1/2)`
D. RCL

Explanation

To determine which combinations of inductance (L), capacitance (C), and resistance (R) have dimensions of frequency, we need to analyze the dimensions of each quantity and how they relate to frequency.

### Step-by-Step Solution:

1. **Understanding the Dimensions of L, C, and R**:
- Inductance (L) has the dimension of \( [L] = \text{ML}^2\text{T}^{-2}\text{A}^{-2} \).
- Capacitance (C) has the dimension of \( [C] = \text{M}^{-1}\text{L}^{-2}\text{T}^4\text{A}^2 \).
- Resistance (R) has the dimension of \( [R] = \text{ML}^2\text{T}^{-3}\text{A}^{-2} \).

2. **Finding the Dimension of Frequency**:
- Frequency (f) has the dimension of \( [f] = \text{T}^{-1} \).

3. **Analyzing the Combinations**:
- **Combination 1: \( \frac{R}{L} \)**:
- The dimension of \( R \) is \( \text{ML}^2\text{T}^{-3}\text{A}^{-2} \).
- The dimension of \( L \) is \( \text{ML}^2\text{T}^{-2}\text{A}^{-2} \).
- Therefore, \( \frac{R}{L} = \frac{\text{ML}^2\text{T}^{-3}\text{A}^{-2}}{\text{ML}^2\text{T}^{-2}\text{A}^{-2}} = \text{T}^{-1} \).
- This has the dimension of frequency.

- **Combination 2: \( \frac{1}{RC} \)**:
- The dimension of \( R \) is \( \text{ML}^2\text{T}^{-3}\text{A}^{-2} \).
- The dimension of \( C \) is \( \text{M}^{-1}\text{L}^{-2}\text{T}^4\text{A}^2 \).
- Therefore, \( RC = \text{ML}^2\text{T}^{-3}\text{A}^{-2} \times \text{M}^{-1}\text{L}^{-2}\text{T}^4\text{A}^2 = \text{T}^{-1} \).
- Thus, \( \frac{1}{RC} \) has the dimension of frequency.

- **Combination 3: \( \frac{1}{\sqrt{LC}} \)**:
- The dimension of \( LC \) is \( \text{ML}^2\text{T}^{-2}\text{A}^{-2} \times \text{M}^{-1}\text{L}^{-2}\text{T}^4\text{A}^2 = \text{T}^2 \).
- Therefore, \( \sqrt{LC} \) has the dimension of \( \text{T} \).
- Thus, \( \frac{1}{\sqrt{LC}} \) has the dimension of frequency.

4. **Conclusion**:
- All combinations \( \frac{R}{L} \), \( \frac{1}{RC} \), and \( \frac{1}{\sqrt{LC}} \) have dimensions of frequency.

### Final Answer:
The combinations that have dimensions of frequency are:
1. \( \frac{R}{L} \)
2. \( \frac{1}{RC} \)
3. \( \frac{1}{\sqrt{LC}} \)