In equation `y=x^(2)cos^(2)2pi(betagamma)/alpha`, the units of `x,alpha,beta` are m, `s^(-1)and(ms^(-1))^(-1)` respectively. The units of y and `gamma` are

Options

A. `m^(2),ms^(-2)`
B. `m,ms^(-1)`
C. `m^(2),m`
D. `m,ms^(-2)`

Answer

To solve the problem, we need to determine the units of \( y \) and \( \gamma \) from the given equation:

\[ y = x^2 \cos^2\left(2\pi \frac{\beta \gamma}{\alpha}\right) \]

### Step 1: Analyze the units of \( x \)
The problem states that the unit of \( x \) is meters (m).

### Step 2: Determine the units of \( y \)
Since \( y \) is equal to \( x^2 \cos^2\left(2\pi \frac{\beta \gamma}{\alpha}\right) \), we can analyze the units:

- The term \( \cos^2\left(2\pi \frac{\beta \gamma}{\alpha}\right) \) is dimensionless because cosine is a trigonometric function, which means it does not have any units.
- Therefore, the units of \( y \) will be the same as the units of \( x^2 \).

Calculating the units of \( x^2 \):

\[ \text{Units of } y = \text{Units of } x^2 = (\text{m})^2 = \text{m}^2 \]

### Step 3: Analyze the term inside the cosine function
Next, we need to analyze the term \( 2\pi \frac{\beta \gamma}{\alpha} \). For this term to be dimensionless, the units of \( \beta \gamma \) must be the same as the units of \( \alpha \).

### Step 4: Determine the units of \( \alpha \)
The problem states that the unit of \( \alpha \) is \( s^{-1} \) (per second).

### Step 5: Determine the units of \( \beta \)
The unit of \( \beta \) is given as \( (ms^{-1})^{-1} \), which simplifies to:

\[ \text{Units of } \beta = \frac{1}{\text{m/s}} = \frac{s}{\text{m}} \]

### Step 6: Set up the equation for \( \gamma \)
Using the dimensional analysis, we have:

\[ \text{Units of } \beta \cdot \text{Units of } \gamma = \text{Units of } \alpha \]

Substituting the known units:

\[ \left(\frac{s}{\text{m}}\right) \cdot \text{Units of } \gamma = s^{-1} \]

### Step 7: Solve for the units of \( \gamma \)
Rearranging the equation gives us:

\[ \text{Units of } \gamma = s^{-1} \cdot \frac{\text{m}}{s} = \frac{\text{m}}{s^2} \]

### Conclusion
Thus, we have determined the units of \( y \) and \( \gamma \):

- The units of \( y \) are \( \text{m}^2 \).
- The units of \( \gamma \) are \( \frac{\text{m}}{s^2} \).

### Final Result
- Units of \( y = \text{m}^2 \)
- Units of \( \gamma = \frac{\text{m}}{s^2} \)