The lines `2x + y -1=0`, `ax+ 3y-3=0` and `3x + 2y -2=0` are concurrent for -

Options

A. all a
B. a = 4 only
C. `- 1 le a le 3`
D. `a gt 0` only

Answer

To determine the value of \( a \) for which the lines \( 2x + y - 1 = 0 \), \( ax + 3y - 3 = 0 \), and \( 3x + 2y - 2 = 0 \) are concurrent, we need to set up the condition for concurrency. The lines are concurrent if the determinant formed by their coefficients is equal to zero.

### Step-by-Step Solution:

1. **Write the equations in standard form**:
- The first line is \( 2x + y - 1 = 0 \).
- The second line is \( ax + 3y - 3 = 0 \).
- The third line is \( 3x + 2y - 2 = 0 \).

2. **Identify coefficients**:
- From the first line: \( A_1 = 2, B_1 = 1, C_1 = -1 \)
- From the second line: \( A_2 = a, B_2 = 3, C_2 = -3 \)
- From the third line: \( A_3 = 3, B_3 = 2, C_3 = -2 \)

3. **Set up the determinant**:
The determinant for the coefficients is given by:
\[ \begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix} = 0 \]
Substituting the values:
\[ \begin{vmatrix} 2 & 1 & -1 \\ a & 3 & -3 \\ 3 & 2 & -2 \end{vmatrix} = 0 \]

4. **Calculate the determinant**:
Expanding the determinant:
\[ = 2 \begin{vmatrix} 3 & -3 \\ 2 & -2 \end{vmatrix} - 1 \begin{vmatrix} a & -3 \\ 3 & -2 \end{vmatrix} - 1 \begin{vmatrix} a & 3 \\ 3 & 2 \end{vmatrix} \]

Calculate each of the 2x2 determinants:
- First determinant:
\[ \begin{vmatrix} 3 & -3 \\ 2 & -2 \end{vmatrix} = (3)(-2) - (-3)(2) = -6 + 6 = 0 \]
- Second determinant:
\[ \begin{vmatrix} a & -3 \\ 3 & -2 \end{vmatrix} = (a)(-2) - (-3)(3) = -2a + 9 \]
- Third determinant:
\[ \begin{vmatrix} a & 3 \\ 3 & 2 \end{vmatrix} = (a)(2) - (3)(3) = 2a - 9 \]

5. **Substituting back into the determinant**:
Now substituting back:
\[ 2(0) - 1(-2a + 9) - 1(2a - 9) = 0 \]
Simplifying:
\[ 0 + 2a - 9 - 2a + 9 = 0 \]
Which simplifies to:
\[ 0 = 0 \]

6. **Conclusion**:
Since the determinant equals zero for all values of \( a \), the lines are concurrent for all values of \( a \).

### Final Answer:
The lines are concurrent for all values of \( a \).