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 the point that does not lie in the half-plane defined by the inequality \(2x + 3y - 12 \leq 0\), we will evaluate several points to see which one does not satisfy the inequality.

### Step-by-Step Solution:

1. **Understand the half-plane inequality**: The inequality \(2x + 3y - 12 \leq 0\) represents a half-plane in the Cartesian coordinate system. Points that satisfy this inequality lie on or below the line defined by the equation \(2x + 3y - 12 = 0\).

2. **Rearranging the inequality**: To find the boundary line, we set \(2x + 3y - 12 = 0\). Rearranging gives us:
\[ 3y = 12 - 2x \implies y = \frac{12 - 2x}{3} \]
This line divides the plane into two regions: one where the inequality holds (the half-plane) and one where it does not.

3. **Testing points**: We will test various points to see if they satisfy the inequality \(2x + 3y - 12 \leq 0\).

- **Point (1, 2)**:
\[ 2(1) + 3(2) - 12 = 2 + 6 - 12 = -4 \quad (\text{which is } \leq 0) \]
This point lies in the half-plane.

- **Point (2, 1)**:
\[ 2(2) + 3(1) - 12 = 4 + 3 - 12 = -5 \quad (\text{which is } \leq 0) \]
This point also lies in the half-plane.

- **Point (2, 3)**:
\[ 2(2) + 3(3) - 12 = 4 + 9 - 12 = 1 \quad (\text{which is } > 0) \]
This point does not lie in the half-plane.

- **Point (-3, 2)**:
\[ 2(-3) + 3(2) - 12 = -6 + 6 - 12 = -12 \quad (\text{which is } \leq 0) \]
This point lies in the half-plane.

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

### Final Answer:
The point which does not lie in the half-plane is **(2, 3)**.