Let `a, b, c, p, q` be the real numbers. Suppose `alpha,beta` are the roots of the equation `x^2+2px+ q=0`. and `alpha,1/beta` are the roots of the equation `ax^2+2 bx+ c=0`, where `beta !in {-1,0,1}`. Statement I `(p^2-q) (b^2-ac)>=0` Statement 11 `b !in pa` or `c !in qa`.

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.

Explanation

To solve the problem step by step, we will analyze the given equations and statements.

### Step 1: Identify the roots of the first equation
The first equation is given as:
\[ x^2 + 2px + q = 0 \]
Let the roots be \( \alpha \) and \( \beta \).

Using Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta = -2p \)
- The product of the roots \( \alpha \beta = q \)

### Step 2: Identify the roots of the second equation
The second equation is given as:
\[ ax^2 + 2bx + c = 0 \]
The roots of this equation are \( \alpha \) and \( \frac{1}{\beta} \).

Again, using Vieta's formulas, we have:
- The sum of the roots \( \alpha + \frac{1}{\beta} = -\frac{2b}{a} \)
- The product of the roots \( \alpha \cdot \frac{1}{\beta} = \frac{c}{a} \)

### Step 3: Express \( \beta \) in terms of \( \alpha \)
From the first equation:
\[ \beta = -2p - \alpha \]
Substituting this into the product of the roots gives:
\[ \alpha \left(-\frac{1}{2p + \alpha}\right) = \frac{c}{a} \]

### Step 4: Analyze the statements
**Statement I:**
\[ (p^2 - q)(b^2 - ac) \geq 0 \]
We know:
- \( p^2 = \left(\frac{\alpha + \beta}{-2}\right)^2 \)
- \( q = \alpha \beta \)

Substituting these into the expression, we can analyze whether it holds true.

**Statement II:**
\[ b \notin pa \quad \text{or} \quad c \notin qa \]
This statement suggests that \( b \) and \( c \) cannot be expressed in terms of \( p \) and \( q \) respectively.

### Step 5: Verify Statement I
We need to show that:
\[ (p^2 - q)(b^2 - ac) \geq 0 \]
Using the derived expressions for \( p^2 \) and \( q \), we can substitute and simplify to check if the inequality holds.

### Step 6: Verify Statement II
We need to check if \( b \) can be expressed as \( pa \) or if \( c \) can be expressed as \( qa \). Given that \( \beta \) is not equal to -1, 0, or 1, we can conclude that these expressions do not hold true.

### Conclusion
After analyzing both statements, we conclude that:
- **Statement I** is true.
- **Statement II** is also true but does not provide a correct explanation for Statement I.