Statement 1: The points `(k+1,k+2),(k,k+1),(k+1,k)` are collinear for any value of k.
Statement 2: If three points are collinear area of the triangle formed by them is zero.

Options

A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True and 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

Answer

To determine the validity of the two statements regarding collinearity of points, we will follow these steps:

### Step 1: Identify the Points
The points given are:
1. \( A(k+1, k+2) \)
2. \( B(k, k+1) \)
3. \( C(k+1, k) \)

### Step 2: Set Up the Area of Triangle Formula
To check if the points are collinear, we can use the formula for the area of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points:
- \( (x_1, y_1) = (k+1, k+2) \)
- \( (x_2, y_2) = (k, k+1) \)
- \( (x_3, y_3) = (k+1, k) \)

### Step 3: Substitute the Points into the Area Formula
Substituting the coordinates into the area formula, we have:
\[ \text{Area} = \frac{1}{2} \left| (k+1)((k+1) - k) + k(k - (k+2)) + (k+1)((k+2) - (k+1)) \right| \]

### Step 4: Simplify the Expression
Now, simplify the expression step by step:
1. Calculate each term:
- First term: \( (k+1)(1) = k + 1 \)
- Second term: \( k(k - (k + 2)) = k(-2) = -2k \)
- Third term: \( (k+1)(1) = k + 1 \)

2. Combine the terms:
\[ \text{Area} = \frac{1}{2} \left| (k + 1) - 2k + (k + 1) \right| = \frac{1}{2} \left| 2 - k \right| \]

### Step 5: Analyze the Result
The area is given by:
\[ \text{Area} = \frac{1}{2} \left| 2 - k \right| \]
This area will only be zero if \( k = 2 \). For any other value of \( k \), the area is not zero, indicating that the points are not collinear.

### Conclusion
- **Statement 1**: The points are collinear for any value of \( k \) is **False**.
- **Statement 2**: If three points are collinear, the area of the triangle formed by them is zero is **True**.

Thus, the final answer is:
- Statement 1 is false.
- Statement 2 is true.