Consider three planes :
`2x+py+6z=8, x+2y+qz=5 and x+y+3z=4`
Q. Three planes intersect at a point if :

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`

Answer

To determine the conditions under which three planes intersect at a point, we need to analyze the coefficients of the planes given by the equations:

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

### Step 1: Write the coefficient matrix
The coefficient matrix \(A\) for the three planes is given by:

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

### Step 2: Calculate the determinant of the coefficient matrix
To find the conditions for the planes to intersect at a single point, we need to ensure that the determinant of this matrix is not equal to zero. We calculate the determinant \(D\) as follows:

\[ D = \begin{vmatrix} 2 & p & 6 \\ 1 & 2 & q \\ 1 & 1 & 3 \end{vmatrix} \]

### Step 3: Expand the determinant
Using the rule of Sarrus or cofactor expansion, we can calculate the determinant:

\[ D = 2 \begin{vmatrix} 2 & q \\ 1 & 3 \end{vmatrix} - p \begin{vmatrix} 1 & q \\ 1 & 3 \end{vmatrix} + 6 \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \(\begin{vmatrix} 2 & q \\ 1 & 3 \end{vmatrix} = (2 \cdot 3) - (q \cdot 1) = 6 - q\)

2. \(\begin{vmatrix} 1 & q \\ 1 & 3 \end{vmatrix} = (1 \cdot 3) - (q \cdot 1) = 3 - q\)

3. \(\begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} = (1 \cdot 1) - (2 \cdot 1) = 1 - 2 = -1\)

Now substituting back into the determinant \(D\):

\[ D = 2(6 - q) - p(3 - q) + 6(-1) \]

### Step 4: Simplify the expression
Expanding this gives:

\[ D = 12 - 2q - 3p + pq - 6 \]

Combining like terms results in:

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

### Step 5: Set the determinant not equal to zero
For the three planes to intersect at a single point, we require:

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

### Conclusion
Thus, the conditions under which the three planes intersect at a point are that the expression \(pq - 3p - 2q + 6\) should not equal zero.