Statement (1) : If a and b are integers and roots of `x^2 + ax + b = 0` are rational then they must be integers. Statement (2): If the coefficient of `x^2` in a quadratic equation is unity then its roots must be integers

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 given problem, we need to analyze both statements one by one.

### Step-by-Step Solution:

**Statement (1):** If \( a \) and \( b \) are integers and the roots of \( x^2 + ax + b = 0 \) are rational, then they must be integers.

1. **Understanding Rational Roots:**
- Let the roots of the quadratic equation \( x^2 + ax + b = 0 \) be \( \frac{p}{q} \) and \( \frac{r}{s} \), where \( p, q, r, s \) are integers and \( q, s \neq 0 \).

2. **Using Vieta's Formulas:**
- According to Vieta's formulas, the sum of the roots \( \left( \frac{p}{q} + \frac{r}{s} \right) = -a \) and the product of the roots \( \left( \frac{p}{q} \cdot \frac{r}{s} \right) = b \).

3. **Clearing Denominators:**
- Multiply the entire equation by \( qs \) (the product of the denominators) to eliminate fractions:
\[ ps + qr = -aqs \quad \text{(1)} \]
\[ pr = bqs \quad \text{(2)} \]

4. **Analyzing the Equations:**
- From equation (1), since \( ps + qr \) is an integer and \( -aqs \) is also an integer, we conclude that \( qs \) must divide \( ps + qr \).
- From equation (2), since \( pr \) is an integer, \( qs \) must divide \( bqs \).

5. **Divisibility Implications:**
- If \( qs \) divides \( pr \), and since \( q \) and \( s \) are integers, it follows that \( q \) must divide \( p \) and \( s \) must divide \( r \).
- Given that the highest common factor (HCF) of \( p \) and \( q \) is 1, it implies that \( q = 1 \) and \( s = 1 \).

6. **Conclusion for Statement (1):**
- Therefore, the roots \( \frac{p}{1} \) and \( \frac{r}{1} \) must be integers. Hence, Statement (1) is **True**.

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**Statement (2):** If the coefficient of \( x^2 \) in a quadratic equation is unity, then its roots must be integers.

1. **Consider a Quadratic Equation:**
- Let’s take an example of a quadratic equation where the coefficient of \( x^2 \) is 1, such as \( x^2 + x + 1 = 0 \).

2. **Finding the Roots:**
- The roots can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 1, c = 1 \):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm \sqrt{-3}}{2} \]

3. **Analyzing the Roots:**
- The roots are \( \frac{-1 \pm i\sqrt{3}}{2} \), which are complex numbers, not integers.

4. **Conclusion for Statement (2):**
- Therefore, it is not necessarily true that if the coefficient of \( x^2 \) is unity, the roots must be integers. Hence, Statement (2) is **False**.

### Final Answer:
- Statement (1) is **True**.
- Statement (2) is **False**.

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