Statement-1 : If `f : [-3, 3] rarr R` is defined as `f(x) = [(x^(2))/(a)]`, then f(x) = 0, `AA x in D_(f)`, iff `a in (9, oo)`. ([x] denotes the greatest integer function)
and
Statement - 2: [x] = 0, `AA 0 le x lt 1`.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1.
C. Statement -1 is False, Statement -2 is False
D. Statement -1 is False, Statement -2 is True

Answer

To solve the problem, we need to analyze both statements provided in the question.

### Step 1: Analyze Statement 1
We are given a function \( f : [-3, 3] \to \mathbb{R} \) defined as:
\[ f(x) = \left\lfloor \frac{x^2}{a} \right\rfloor \]
where \( \lfloor x \rfloor \) denotes the greatest integer function. We need to determine when \( f(x) = 0 \) for all \( x \) in the domain \( D_f \).

**Domain of \( x \)**:
The domain is \( [-3, 3] \). Therefore, \( x^2 \) will range from \( 0 \) (when \( x = 0 \)) to \( 9 \) (when \( x = -3 \) or \( x = 3 \)). Thus, \( x^2 \in [0, 9] \).

**Condition for \( f(x) = 0 \)**:
For \( f(x) = 0 \), we need:
\[ \left\lfloor \frac{x^2}{a} \right\rfloor = 0 \]
This means:
\[ 0 \leq \frac{x^2}{a} < 1 \]
From this inequality, we can derive:
\[ 0 \leq x^2 < a \]
Since \( x^2 \) can take values from \( 0 \) to \( 9 \), we need:
\[ 9 < a \]
Thus, \( a \) must be greater than \( 9 \) for \( f(x) = 0 \) for all \( x \in D_f \).

### Step 2: Analyze Statement 2
Statement 2 states:
\[ \lfloor x \rfloor = 0 \quad \text{for all } 0 \leq x < 1 \]
This is indeed correct because the greatest integer function \( \lfloor x \rfloor \) returns \( 0 \) for any \( x \) in the interval \( [0, 1) \).

### Conclusion
- **Statement 1** is correct because \( f(x) = 0 \) holds for all \( x \in [-3, 3] \) if \( a > 9 \).
- **Statement 2** is also correct as it accurately describes the behavior of the greatest integer function in the specified interval.

Thus, both statements are true.

### Final Answer
Both Statement 1 and Statement 2 are correct.