If `f(x)=sin (sqrt([lambda])x)` is a function with period `pi` , [ ] where denotes the greatest integer less than or equal to `lambda" is "pi`, then

Options

A. `lambda in [4,5)`
B. `lambda in [4,5]`
C. `lambda=4,5`
D. None of these

Answer

To solve the problem, we need to determine the value of \(\lambda\) such that the function \(f(x) = \sin(\sqrt{[\lambda]} x)\) has a period of \(\pi\). Here, \([\lambda]\) denotes the greatest integer less than or equal to \(\lambda\).

### Step-by-Step Solution:

1. **Understand the Period of the Sine Function**:
The sine function \(\sin(kx)\) has a period given by:
\[ \text{Period} = \frac{2\pi}{k} \]
where \(k\) is the coefficient of \(x\).

2. **Set the Period Equal to \(\pi\)**:
In our case, we have:
\[ k = \sqrt{[\lambda]} \]
We want the period to be \(\pi\), so we set up the equation:
\[ \frac{2\pi}{\sqrt{[\lambda]}} = \pi \]

3. **Solve for \(\sqrt{[\lambda]}\)**:
To solve for \(\sqrt{[\lambda]}\), we can simplify the equation:
\[ 2 = \sqrt{[\lambda]} \]

4. **Square Both Sides**:
Squaring both sides gives:
\[ 4 = [\lambda] \]

5. **Determine the Range of \(\lambda\)**:
The greatest integer function \([\lambda]\) indicates that \(\lambda\) must be in the range:
\[ 4 \leq \lambda < 5 \]
This means \(\lambda\) can take any value starting from 4 up to, but not including, 5.

6. **Final Answer**:
Therefore, the value of \(\lambda\) is:
\[ \lambda \in [4, 5) \]