Given that `y = a cos((t)/(P) - qx),` where t represents distance is metre. Which of the following statements is true ?

Options

A. unit of t is same as that of p
B. unit of t is same as that of q
C. unit of x is same as that of q
D. unit of x is same as that of P

Explanation

To analyze the given equation \( y = a \cos\left(\frac{t}{P} - qx\right) \), we need to determine the relationships between the variables \( t \), \( x \), \( P \), and \( q \).

### Step-by-Step Solution:

1. **Identify the Variables:**
- \( y \): Displacement (not specified in units, but typically in meters).
- \( a \): Amplitude (also typically in meters).
- \( t \): Represents distance in meters.
- \( P \): A constant that we need to analyze.
- \( q \): A constant that we need to analyze.
- \( x \): Another variable (distance) in meters.

2. **Analyze the Argument of the Cosine Function:**
- The argument of the cosine function is dimensionless. Therefore, the expression \( \frac{t}{P} - qx \) must also be dimensionless.
- This implies that both \( \frac{t}{P} \) and \( qx \) must have the same dimensions.

3. **Determine the Dimensions:**
- Since \( t \) is in meters, the dimension of \( t \) is [L].
- For \( \frac{t}{P} \) to be dimensionless, \( P \) must also have the dimension of length [L].
- Thus, we conclude that the dimensions of \( t \) and \( P \) are the same.

4. **Analyzing \( qx \):**
- Since \( qx \) must also be dimensionless, \( q \) must have dimensions that are the inverse of \( x \).
- Given that \( x \) is in meters, the dimension of \( q \) must be [L]⁻¹ (inverse length).

5. **Summarize the Relationships:**
- From the analysis, we have:
- The unit of \( t \) is the same as that of \( P \) (both are in meters).
- The unit of \( q \) is not the same as that of \( t \) or \( P \) (it is in [L]⁻¹).
- The unit of \( x \) is the same as that of \( t \) and \( P \) (both are in meters).

6. **Conclusion:**
- The correct statement is that the unit of \( t \) is the same as that of \( P \).

### Final Answer:
The true statement is: **The unit of \( t \) is the same as that of \( P \)**.