STATEMENT-1: An elastic spring of force constant k is stretched by a small length x. The work done in extending the spring by a further length x is `2kx^(2)`.
because
STATEMENT-2: The work done in extending an elastic spring by a length x is proportional to `x^(2)`.

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 regarding the work done on an elastic spring and determine their validity.

### Step 1: Understanding the Work Done on a Spring
The work done \( W \) in stretching a spring can be calculated using the formula:
\[ W = \int F \, dx \]
where \( F \) is the force exerted by the spring, which follows Hooke's Law:
\[ F = kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the natural length.

### Step 2: Calculate Work Done in Stretching the Spring
1. **Initial Stretch**: When the spring is stretched from its natural length to a length \( x \):
\[ W_1 = \int_0^x kx \, dx = \left[ \frac{1}{2} kx^2 \right]_0^x = \frac{1}{2} kx^2 \]

2. **Further Stretching**: Now, if we stretch the spring further by an additional length \( x \) (total length becomes \( 2x \)):
\[ W_2 = \int_x^{2x} kx \, dx \]
To evaluate this integral, we need to express the force in terms of the new limits:
\[ W_2 = \int_x^{2x} kx \, dx = \left[ \frac{1}{2} kx^2 \right]_x^{2x} = \frac{1}{2} k(2x)^2 - \frac{1}{2} kx^2 = \frac{1}{2} k(4x^2) - \frac{1}{2} kx^2 = \frac{4kx^2}{2} - \frac{kx^2}{2} = \frac{3kx^2}{2} \]

### Step 3: Total Work Done
The total work done in stretching the spring from its natural length to \( 2x \) is:
\[ W_{\text{total}} = W_1 + W_2 = \frac{1}{2} kx^2 + \frac{3}{2} kx^2 = 2kx^2 \]

### Step 4: Evaluating the Statements
- **Statement 1**: "The work done in extending the spring by a further length \( x \) is \( 2kx^2 \)."
- This statement is **True** based on our calculation.

- **Statement 2**: "The work done in extending an elastic spring by a length \( x \) is proportional to \( x^2 \)."
- This statement is also **True** since the work done is given by \( W = \frac{1}{2} kx^2 \), which is proportional to \( x^2 \).

### Conclusion
Both statements are actually true, which contradicts the initial assessment in the video transcript. Therefore, the correct conclusion is that both statements are true.