For what values of p and q the system of equations `2x+py+6z=8, x+2y+qz=5, x+y+3z=4` has no solution. (a).`p=2, q ne 3`. (b).`p ne 2, q ne 3` (c). `p ne 2, q = 3` (d). `p =2, q=3`

Options

A. `p=2, q ne 3`.
B. `p ne 2, q ne 3`
C. `p ne 2, q = 3`
D. `p =2, q=3`

Explanation

To determine the values of \( p \) and \( q \) for which the system of equations has no solution, we will analyze the given equations:

1. \( 2x + py + 6z = 8 \) (Equation 1)
2. \( x + 2y + qz = 5 \) (Equation 2)
3. \( x + y + 3z = 4 \) (Equation 3)

For a system of linear equations to have no solution, the determinant of the coefficients must be equal to zero. We will set up the determinant using the coefficients of \( x \), \( y \), and \( z \) from the equations.

The coefficient matrix is:

\[ \begin{bmatrix} 2 & p & 6 \\ 1 & 2 & q \\ 1 & 1 & 3 \end{bmatrix} \]

The determinant of this matrix can be calculated using the formula for the determinant of a 3x3 matrix:

\[ \text{Det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where:
- \( a = 2, b = p, c = 6 \)
- \( d = 1, e = 2, f = q \)
- \( g = 1, h = 1, i = 3 \)

Calculating the determinant step by step:

1. **Calculate \( ei - fh \)**:
\[ ei = 2 \cdot 3 = 6 \]
\[ fh = q \cdot 1 = q \]
\[ ei - fh = 6 - q \]

2. **Calculate \( di - fg \)**:
\[ di = 1 \cdot 3 = 3 \]
\[ fg = q \cdot 1 = q \]
\[ di - fg = 3 - q \]

3. **Calculate \( dh - eg \)**:
\[ dh = 1 \cdot 1 = 1 \]
\[ eg = 2 \cdot 1 = 2 \]
\[ dh - eg = 1 - 2 = -1 \]

Now substituting these into the determinant formula:

\[ \text{Det} = 2(6 - q) - p(3 - q) + 6(-1) \]

Expanding this gives:

\[ \text{Det} = 12 - 2q - 3p + pq - 6 \]
\[ = pq - 3p - 2q + 6 \]

Setting the determinant to zero for the system to have no solution:

\[ pq - 3p - 2q + 6 = 0 \]

Rearranging this, we can factor it:

\[ pq - 3p - 2q + 6 = (q - 3)(2 - p) = 0 \]

This gives us two equations:

1. \( q - 3 = 0 \) → \( q = 3 \)
2. \( 2 - p = 0 \) → \( p = 2 \)

Thus, the values of \( p \) and \( q \) for which the system has no solution are \( p = 2 \) and \( q = 3 \).

### Final Answer:
The correct option is (d) \( p = 2, q = 3 \).