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

Answer: (D)

$m^{2} + n^{2} - mn = 39$

Answer 2

To find the values of m and n such that the system has infinitely many solutions, we start by setting the determinant of the coefficient matrix to zero:

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

Expanding the determinant:

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

Setting $D = 0$:

$3m - 6 = 0 \implies m = 2.$

Next, consider the augmented matrix determinant $D_3$:

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

Expanding $D_3$ and setting it to zero gives:

$n = 7.$

Substitute $m = 2$ and $n = 7$ into the given equation:

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

Therefore, the correct answer is Option (4).