Statement-1: The equation `x^(3)+y^(3)+3xy=1` represents the combined equation of a straight line and a circle.
Statement-2: The equation of the straight line contained in `x^(3)+y^(3)+3xy=1` is `x+y=1`

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 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 given statements about the equation \( x^3 + y^3 + 3xy = 1 \).

### Step 1: Rewrite the Equation
We start with the equation:
\[ x^3 + y^3 + 3xy = 1 \]
Using the identity for the sum of cubes, we can rewrite \( x^3 + y^3 \) as:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
Thus, we can express the equation as:
\[ (x + y)(x^2 - xy + y^2) + 3xy = 1 \]

### Step 2: Rearranging the Equation
We can rearrange the equation to isolate the term on the right:
\[ (x + y)(x^2 - xy + y^2) + 3xy - 1 = 0 \]

### Step 3: Factor the Equation
We can factor the equation as follows:
\[ (x + y - 1)(x^2 + y^2 + xy + x + y + 1) = 0 \]
This gives us two factors:
1. \( x + y - 1 = 0 \) (which represents a straight line)
2. \( x^2 + y^2 + xy + x + y + 1 = 0 \) (which we need to analyze further)

### Step 4: Analyze the Second Factor
The second factor \( x^2 + y^2 + xy + x + y + 1 = 0 \) needs to be checked to see if it represents a circle. A circle in the general form is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
To determine if the equation represents a circle, we check if it can be transformed into this form. The presence of the \( xy \) term indicates that this is not a standard circle equation.

### Conclusion
- **Statement 1**: The equation \( x^3 + y^3 + 3xy = 1 \) represents the combined equation of a straight line and a circle. This statement is **false** because the second factor does not represent a circle.
- **Statement 2**: The equation of the straight line contained in \( x^3 + y^3 + 3xy = 1 \) is \( x + y = 1 \). This statement is **true** as we derived it directly from the factorization.

### Final Answer
Thus, the correct conclusion is that Statement 1 is false and Statement 2 is true.