STATEMENT - 1 : If `x^(2)+2x+3=0` and `7x^(2)+lx+k = 0` have a common roots, then `l+k = 35`. Given `l, k epsilon R`.
STATEMENT - 2 : If `a, b, c epsilon R` and roots of a quadratic equation `ax^(2)+bx+c=0` are imaginary then roots occurs in conjugate pair.

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 two statements provided and derive the necessary conclusions step by step.

### Step 1: Analyze the first quadratic equation
The first equation given is:
\[ x^2 + 2x + 3 = 0 \]
To find the roots, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = 2\), and \(c = 3\).

### Step 2: Calculate the discriminant
The discriminant \(D\) is given by:
\[ D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \]
Since the discriminant is negative, the roots are imaginary.

### Step 3: Roots of the first equation
The roots can be expressed as:
\[ x = \frac{-2 \pm \sqrt{-8}}{2 \cdot 1} = \frac{-2 \pm 2i\sqrt{2}}{2} = -1 \pm i\sqrt{2} \]
Let’s denote the roots as:
\[ \alpha = -1 + i\sqrt{2}, \quad \beta = -1 - i\sqrt{2} \]

### Step 4: Analyze the second quadratic equation
The second equation is:
\[ 7x^2 + lx + k = 0 \]
We need to find conditions for this equation to have a common root with the first equation.

### Step 5: Use Vieta's formulas
From Vieta's formulas for the second equation:
- The sum of the roots (\(\alpha + \beta\)) is given by:
\[ \alpha + \beta = -\frac{l}{7} \]
- The product of the roots (\(\alpha \beta\)) is given by:
\[ \alpha \beta = \frac{k}{7} \]

### Step 6: Set up equations using known values
From the first equation, we know:
\[ \alpha + \beta = -2 \quad \text{and} \quad \alpha \beta = 3 \]
Thus, we can set up the following equations:
1. \(-\frac{l}{7} = -2\) implies \(l = 14\)
2. \(\frac{k}{7} = 3\) implies \(k = 21\)

### Step 7: Calculate \(l + k\)
Now, we can find \(l + k\):
\[ l + k = 14 + 21 = 35 \]

### Conclusion for Statement 1
Thus, we have shown that if the two equations have a common root, then \(l + k = 35\) is indeed correct.

### Step 8: Analyze Statement 2
The second statement claims that if the roots of a quadratic equation \(ax^2 + bx + c = 0\) are imaginary, then they occur in conjugate pairs. This is a well-known property of polynomials with real coefficients. Therefore, this statement is also correct.

### Final Conclusion
Both statements are correct.