If `f(x)=sin(sqrt([a])x)` (where [.] denotes the greatest integer function) has `pi` as its fundamental period, then

Options

A. a=1
B. `ain{1,2}`
C. `a=9`
D. `ain[4,5)`

Answer

To solve the problem, we need to determine the value of \( a \) such that the function \( f(x) = \sin(\sqrt{[a]} x) \) has a fundamental period of \( \pi \).

### Step-by-Step Solution:

1. **Understanding the Period of the Sine Function**:
The period of the function \( \sin(mx) \) is given by the formula:
\[ \text{Period} = \frac{2\pi}{m} \]
where \( m \) is a constant.

2. **Identifying \( m \)**:
In our case, we have:
\[ m = \sqrt{[a]} \]
Therefore, the period of \( f(x) \) can be expressed as:
\[ \text{Period} = \frac{2\pi}{\sqrt{[a]}} \]

3. **Setting the Period Equal to \( \pi \)**:
We know from the problem statement that the period of \( f(x) \) is \( \pi \). Thus, we set up the equation:
\[ \frac{2\pi}{\sqrt{[a]}} = \pi \]

4. **Solving for \( \sqrt{[a]} \)**:
To eliminate \( \pi \) from both sides, we divide:
\[ \frac{2}{\sqrt{[a]}} = 1 \]
Multiplying both sides by \( \sqrt{[a]} \) gives:
\[ 2 = \sqrt{[a]} \]

5. **Squaring Both Sides**:
To find \( [a] \), we square both sides:
\[ 4 = [a] \]

6. **Understanding the Greatest Integer Function**:
The greatest integer function \( [a] \) gives the largest integer less than or equal to \( a \). Therefore, if \( [a] = 4 \), it means:
\[ 4 \leq a < 5 \]

7. **Conclusion**:
The value of \( a \) must lie in the interval:
\[ a \in [4, 5) \]

### Final Answer:
Thus, the correct answer is:
\[ a \in [4, 5) \]