The set of real values of k for which the lines `x + 3y +1=0, kx + 2y-2=0` and `2x-y + 3 = 0` form a triangle is

Options

A. `R-{-4,(2)/(3)}`
B. `R-{-4,(-6)/(5),(2)/(3)}`
C. `R-{(-2)/(3),4}`
D. R

Explanation

To determine the set of real values of \( k \) for which the lines \( x + 3y + 1 = 0 \), \( kx + 2y - 2 = 0 \), and \( 2x - y + 3 = 0 \) form a triangle, we need to ensure that the lines are not parallel and not concurrent.

### Step 1: Find the slopes of the lines
1. **First line**: \( x + 3y + 1 = 0 \)
- Rearranging gives \( 3y = -x - 1 \) or \( y = -\frac{1}{3}x - \frac{1}{3} \)
- **Slope**: \( m_1 = -\frac{1}{3} \)

2. **Second line**: \( kx + 2y - 2 = 0 \)
- Rearranging gives \( 2y = -kx + 2 \) or \( y = -\frac{k}{2}x + 1 \)
- **Slope**: \( m_2 = -\frac{k}{2} \)

3. **Third line**: \( 2x - y + 3 = 0 \)
- Rearranging gives \( y = 2x + 3 \)
- **Slope**: \( m_3 = 2 \)

### Step 2: Ensure the lines are not parallel
For the lines to form a triangle, they must not be parallel. This means their slopes must be different.

1. **Condition 1**: \( m_1 \) and \( m_2 \) should not be equal:
\[ -\frac{k}{2} \neq -\frac{1}{3} \]
This simplifies to:
\[ k \neq \frac{2}{3} \]

2. **Condition 2**: \( m_2 \) and \( m_3 \) should not be equal:
\[ -\frac{k}{2} \neq 2 \]
This simplifies to:
\[ k \neq -4 \]

### Step 3: Ensure the lines are not concurrent
The lines are concurrent if the determinant of the coefficients of \( x \), \( y \), and the constant terms is zero. We form the matrix:

\[ \begin{vmatrix} 1 & 3 & -1 \\ k & 2 & 2 \\ 2 & -1 & -3 \end{vmatrix} \]

Calculating the determinant:
\[ = 1 \cdot (2 \cdot (-3) - 2 \cdot (-1)) - 3 \cdot (k \cdot (-3) - 2 \cdot 2) - 1 \cdot (k \cdot (-1) - 2 \cdot 2) \]
\[ = 1 \cdot (-6 + 2) - 3 \cdot (-3k - 4) - 1 \cdot (-k - 4) \]
\[ = -4 + 9k + 12 + k + 4 \]
\[ = 10k + 12 \]

Setting the determinant not equal to zero for non-concurrency:
\[ 10k + 12 \neq 0 \]
\[ 10k \neq -12 \quad \Rightarrow \quad k \neq -\frac{6}{5} \]

### Step 4: Combine conditions
The values of \( k \) for which the lines form a triangle are:
1. \( k \neq \frac{2}{3} \)
2. \( k \neq -4 \)
3. \( k \neq -\frac{6}{5} \)

### Final Answer
The set of real values of \( k \) for which the lines form a triangle is:
\[ k \in \mathbb{R} \setminus \left\{ \frac{2}{3}, -4, -\frac{6}{5} \right\} \]