The value of p for which the lines `2x+y-3=0, 3x-y-2=0` and `x-py+5=0` may intersect at a point is

Options

A. 2
B. 3
C. 5
D. 6

Answer

To find the value of \( p \) for which the lines \( 2x + y - 3 = 0 \), \( 3x - y - 2 = 0 \), and \( x - py + 5 = 0 \) intersect at a single point, we need to set up a determinant condition for the three lines.

### Step-by-Step Solution:

1. **Identify the coefficients of the lines:**
- For the line \( 2x + y - 3 = 0 \), the coefficients are \( (2, 1, -3) \).
- For the line \( 3x - y - 2 = 0 \), the coefficients are \( (3, -1, -2) \).
- For the line \( x - py + 5 = 0 \), the coefficients are \( (1, -p, 5) \).

2. **Set up the determinant:**
The lines will intersect at a point if the determinant of the coefficients is zero. The determinant \( D \) can be expressed as:
\[ D = \begin{vmatrix} 2 & 1 & -3 \\ 3 & -1 & -2 \\ 1 & -p & 5 \end{vmatrix} \]

3. **Calculate the determinant:**
We can calculate the determinant using the formula for a \( 3 \times 3 \) matrix:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the first row, \( d, e, f \) are the second row, and \( g, h, i \) are the third row.

Substituting the values:
\[ D = 2 \begin{vmatrix} -1 & -2 \\ -p & 5 \end{vmatrix} - 1 \begin{vmatrix} 3 & -2 \\ 1 & 5 \end{vmatrix} - 3 \begin{vmatrix} 3 & -1 \\ 1 & -p \end{vmatrix} \]

Now, calculating each of the \( 2 \times 2 \) determinants:
- For \( \begin{vmatrix} -1 & -2 \\ -p & 5 \end{vmatrix} = (-1)(5) - (-2)(-p) = -5 - 2p = -5 - 2p \)
- For \( \begin{vmatrix} 3 & -2 \\ 1 & 5 \end{vmatrix} = (3)(5) - (-2)(1) = 15 + 2 = 17 \)
- For \( \begin{vmatrix} 3 & -1 \\ 1 & -p \end{vmatrix} = (3)(-p) - (-1)(1) = -3p + 1 \)

Now substituting back into the determinant:
\[ D = 2(-5 - 2p) - 1(17) - 3(-3p + 1) \]
\[ D = -10 - 4p - 17 + 9p - 3 \]
\[ D = -30 + 5p \]

4. **Set the determinant to zero:**
For the lines to be concurrent, we set \( D = 0 \):
\[ -30 + 5p = 0 \]
\[ 5p = 30 \]
\[ p = \frac{30}{5} = 6 \]

5. **Final Answer:**
The value of \( p \) for which the lines intersect at a point is \( p = 6 \).