Statement I: `x^2-5x+6<0` if `2 < x < 3` Statement II: If `alpha` and `beta, (alpha < beta)` are the roots of the equation `ax^2+bx+c=0` and `alpha < x < beta` then `ax^2+bx+c` and `a` have opposite signs

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 True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze Statement I
We need to determine if the inequality \( x^2 - 5x + 6 < 0 \) holds true for the interval \( 2 < x < 3 \).

1. **Find the roots of the quadratic equation**:
\[ x^2 - 5x + 6 = 0 \]
We can factor this as:
\[ (x - 2)(x - 3) = 0 \]
Thus, the roots are \( x = 2 \) and \( x = 3 \).

2. **Determine the intervals**:
The roots divide the number line into intervals: \( (-\infty, 2) \), \( (2, 3) \), and \( (3, \infty) \).

3. **Test a point in the interval \( (2, 3) \)**:
Choose \( x = 2.5 \):
\[ f(2.5) = (2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25 \]
Since \( f(2.5) < 0 \), the inequality holds true in this interval.

4. **Conclusion for Statement I**:
Therefore, Statement I is **true**: \( x^2 - 5x + 6 < 0 \) for \( 2 < x < 3 \).

### Step 2: Analyze Statement II
We need to verify if the statement holds true: If \( \alpha \) and \( \beta \) are the roots of \( ax^2 + bx + c = 0 \) with \( \alpha < x < \beta \), then \( ax^2 + bx + c \) and \( a \) have opposite signs.

1. **Understanding the roots**:
The roots \( \alpha \) and \( \beta \) imply that the quadratic opens upwards if \( a > 0 \) and downwards if \( a < 0 \).

2. **Behavior of the quadratic between the roots**:
In the interval \( (\alpha, \beta) \):
- If \( a > 0 \), the quadratic \( ax^2 + bx + c \) will be negative between the roots \( \alpha \) and \( \beta \).
- If \( a < 0 \), the quadratic will be positive between the roots.

3. **Conclusion for Statement II**:
Thus, if \( \alpha < x < \beta \), \( ax^2 + bx + c \) and \( a \) will have opposite signs. Therefore, Statement II is also **true**.

### Final Conclusion
Both statements are true. The relationship between the two statements is that the reasoning in Statement II supports the conclusion of Statement I.