The pair of the quantities having same dimensions is

Options

A. Displacement, velocity
B. Time, frequency
C. Wavelength focal length
D. Force, acceleration

Answer

To solve the question regarding which pair of quantities have the same dimensions, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Identify the first option: Displacement and Velocity**
- **Displacement (L)**: Displacement is defined as the shortest distance from the initial to the final position of an object. Its dimension is length (L).
- **Velocity (V)**: Velocity is defined as the rate of change of displacement with respect to time. It can be expressed as:
\[ V = \frac{Displacement}{Time} = \frac{L}{T} \]
Thus, the dimension of velocity is \( LT^{-1} \).
- **Conclusion**: The dimensions of displacement (L) and velocity (LT^{-1}) are not the same.

2. **Identify the second option: Time and Frequency**
- **Time (T)**: The dimension of time is simply \( T \).
- **Frequency (f)**: Frequency is defined as the number of occurrences of a repeating event per unit time. Its dimension can be expressed as:
\[ f = \frac{1}{T} \]
Thus, the dimension of frequency is \( T^{-1} \).
- **Conclusion**: The dimensions of time (T) and frequency (T^{-1}) are not the same.

3. **Identify the third option: Wavelength and Focal Length**
- **Wavelength (λ)**: Wavelength is the distance between successive crests of a wave. Its dimension is length (L).
- **Focal Length (F)**: Focal length is the distance from the lens or mirror to the focus. It is also a measure of length, thus its dimension is \( L \).
- **Conclusion**: The dimensions of wavelength (L) and focal length (L) are the same.

4. **Identify the fourth option: Force**
- **Force (F)**: According to Newton's second law, force is defined as mass times acceleration. The dimension of force can be expressed as:
\[ F = m \cdot a \]
where mass (m) has dimension \( M \) and acceleration (a) has dimension \( LT^{-2} \). Thus, the dimension of force is:
\[ F = MLT^{-2} \]
- **Conclusion**: The dimension of force (MLT^{-2}) is not the same as any of the previous options.

### Final Conclusion:
The only pair of quantities that have the same dimensions are **wavelength and focal length**.