Statement-1: `If f(x) = 1 + x + (x^(2))/(2!) + (x^(3))/(3!) + (x^(4))/(4!)`, then the equation f(x) = 0 has two pairs of repeated roots.
Statement-2 Polynomial equation P(x) = 0 has repeated root `alpha`, if `P(alpha) = 0 and P'(alpha) = f0`

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, we need to analyze the function \( f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} \) and determine if the equation \( f(x) = 0 \) has two pairs of repeated roots.

### Step 1: Understanding the function \( f(x) \)

The function \( f(x) \) is a polynomial of degree 4. The terms are:

- Constant term: \( 1 \)
- Linear term: \( x \)
- Quadratic term: \( \frac{x^2}{2!} \)
- Cubic term: \( \frac{x^3}{3!} \)
- Quartic term: \( \frac{x^4}{4!} \)

### Step 2: Finding the derivative \( f'(x) \)

To check for repeated roots, we need to find the first derivative \( f'(x) \):

\[ f'(x) = 0 + 1 + \frac{2x}{2!} + \frac{3x^2}{3!} + \frac{4x^3}{4!} \]
\[ = 1 + \frac{x}{1} + \frac{x^2}{2} + \frac{x^3}{6} \]

### Step 3: Setting up the conditions for repeated roots

For \( f(x) \) to have repeated roots, we need to satisfy two conditions:

1. \( f(\alpha) = 0 \)
2. \( f'(\alpha) = 0 \)

### Step 4: Evaluating \( f(0) \) and \( f'(0) \)

Let's evaluate \( f(0) \):

\[ f(0) = 1 + 0 + 0 + 0 + 0 = 1 \quad (\text{not a root}) \]

Now, evaluate \( f'(0) \):

\[ f'(0) = 1 + 0 + 0 + 0 = 1 \quad (\text{not a repeated root}) \]

### Step 5: Analyzing the roots of \( f(x) \)

Since \( f(0) \neq 0 \) and \( f'(0) \neq 0 \), we conclude that \( x = 0 \) is not a root of \( f(x) \). Thus, we need to check if there are any other roots.

### Step 6: Finding the roots of \( f(x) \)

To find the roots of \( f(x) \), we can analyze the polynomial further or use numerical methods, but for our analysis, we can also note that \( f(x) \) is a sum of positive terms for \( x \geq 0 \), indicating that it does not cross the x-axis in that region.

### Conclusion

Since \( f(x) \) does not have \( x = 0 \) as a root and does not appear to have any repeated roots, we conclude that:

- **Statement 1 is false**: \( f(x) = 0 \) does not have two pairs of repeated roots.
- **Statement 2 is true**: The condition for repeated roots is correctly stated.