Question

The values of $m, n$, for which the system of equations
$x + y + z = 4,$
$2x + 5y + 5z = 17,$
$x + 2y + mz = n$
has infinitely many solutions, satisfy the equation :

• A. $m^{2} + n^{2} - m - n = 46$
• B. $m^{2} + n^{2} + m + n = 64$
• C. $m^{2} + n^{2} + mn = 68$
• D. $m^{2} + n^{2} - mn = 39$

Answer 1

The Correct Option is D

To achieve an infinite number of solutions for the system, the determinant of the coefficient matrix must be zero:

\[ D = \begin{vmatrix} 1 & 1 & 1 \\ 2 & 5 & 5 \\ 1 & 2 & m \end{vmatrix} = 0. \]

Calculating the determinant yields:

\[ D = 1 \cdot (5m - 10) - 1 \cdot (2m - 5) + 1 \cdot (4 - 5) = 3m - 6. \]

Setting \(D = 0\) to find \(m\):

\[ 3m - 6 = 0 \implies m = 2. \]

Next, the determinant of the augmented matrix \(D_3\) must also be zero:

\[ D_3 = \begin{vmatrix} 1 & 1 & 4 \\ 2 & 5 & 17 \\ 1 & 2 & n \end{vmatrix} = 0. \]

Expanding \(D_3\) and setting it to zero results in:

\[ n = 7. \]

Substituting \(m = 2\) and \(n = 7\) into the expression \(m^2 + n^2 - mn\):

\[ m^2 + n^2 - mn = 2^2 + 7^2 - (2 \times 7) = 4 + 49 - 14 = 39. \]

The correct option is (4).