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

Options

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

Answer

To solve the problem, we need to analyze the given equation and the implications of the terms involved. The equation given is:

\[ y = a \cos\left(\frac{t}{P} - qx\right) \]

where \( t \) represents distance in meters. We need to determine which of the statements regarding the units of \( t \), \( P \), \( q \), and \( x \) is true.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The expression inside the cosine function, \( \frac{t}{P} - qx \), must be dimensionless because the cosine function can only accept dimensionless arguments.

2. **Analyzing the Terms**:
- \( t \) represents distance, so its unit is meters (m).
- \( P \) must also have units that allow \( \frac{t}{P} \) to be dimensionless. Therefore, \( P \) must also have the unit of meters (m).
- \( qx \) must also be dimensionless. Since \( x \) is a distance (in meters), \( q \) must have units that make \( qx \) dimensionless. Thus, \( q \) must have units of \( \text{m}^{-1} \) (inverse meters).

3. **Conclusion on Units**:
- Since \( t \) and \( P \) both have units of meters, we can conclude:
\[ \text{Unit of } t = \text{Unit of } P \]
- The unit of \( x \) is the same as the unit of \( t \) and \( P \) (both are in meters).
- The unit of \( q \) is \( \text{m}^{-1} \).

4. **Evaluating the Statements**:
- **Statement 1**: The unit of \( t \) is the same as that of \( q \) → **False** (units are different).
- **Statement 2**: The unit of \( x \) is the same as that of \( P \) → **True** (both are in meters).
- **Statement 3**: The unit of \( x \) is the same as that of \( q \) → **False** (units are different).
- **Statement 4**: The unit of \( t \) is the same as that of \( P \) → **True** (both are in meters).

5. **Final Answer**:
The correct statements are that the unit of \( t \) is the same as that of \( P \), and the unit of \( x \) is the same as that of \( P \). Therefore, the true statement is:

**The unit of \( t \) is the same as that of \( P \)**.