The point which does not lie in the half plane
`2x+3y-12 le 0` is

Options

A. (1,2)
B. (2,1)
C. (2,3)
D. (-3,2)

Explanation

To determine which point does not lie in the half-plane defined by the inequality \(2x + 3y - 12 \leq 0\), we will evaluate each option by substituting the coordinates of the points into the inequality.

### Step-by-step Solution:

1. **Understand the Inequality**: The inequality \(2x + 3y - 12 \leq 0\) defines a half-plane. We need to find a point that does not satisfy this inequality.

2. **Evaluate the Points**: We will check each point one by one to see if it satisfies the inequality.

- **Point (1, 2)**:
\[ 2(1) + 3(2) - 12 \leq 0 \\ 2 + 6 - 12 \leq 0 \\ 8 - 12 \leq 0 \\ -4 \leq 0 \quad \text{(True)} \]
This point satisfies the inequality.

- **Point (2, 1)**:
\[ 2(2) + 3(1) - 12 \leq 0 \\ 4 + 3 - 12 \leq 0 \\ 7 - 12 \leq 0 \\ -5 \leq 0 \quad \text{(True)} \]
This point satisfies the inequality.

- **Point (2, 3)**:
\[ 2(2) + 3(3) - 12 \leq 0 \\ 4 + 9 - 12 \leq 0 \\ 13 - 12 \leq 0 \\ 1 \leq 0 \quad \text{(False)} \]
This point does not satisfy the inequality.

- **Point (-3, 2)**:
\[ 2(-3) + 3(2) - 12 \leq 0 \\ -6 + 6 - 12 \leq 0 \\ 0 - 12 \leq 0 \\ -12 \leq 0 \quad \text{(True)} \]
This point satisfies the inequality.

3. **Conclusion**: The point that does not lie in the half-plane \(2x + 3y - 12 \leq 0\) is \((2, 3)\).

### Summary of Steps:
- Substitute the coordinates of each point into the inequality.
- Check if the resulting expression is less than or equal to zero.
- Identify the point that does not satisfy the inequality.