Question

The system of linear equations 3x - 2y - kz = 10, 2x - 4y - 2z = 6, x + 2y - z = 5m is inconsistent if :

• A. $k = 3, m = 4/5$
• B. $k \neq 3, m = 4/5$
• C. $k = 3, m \neq 4/5$
• D. $k \neq 3, m$ any real

Hint:

If $\Delta = 0$ and any of $\Delta_i \neq 0$, the system has no solution. If all $\Delta_i = 0$, the system may have infinite solutions.

Answer 1

The Correct Option is C

Step 1: For inconsistency, the determinant of coefficients \( \Delta = 0 \). \[ \Delta = \begin{vmatrix} 3 & -2 & -k \\ 2 & -4 & -2 \\ 1 & 2 & -1 \end{vmatrix} = 3(4 + 4) + 2(-2 + 2) - k(4 + 4) = 24 - 8k = 0 \Rightarrow k = 3. \]
Step 2: Check \( \Delta_z \) (or others). For inconsistency, at least one of \( \Delta_x, \Delta_y, \Delta_z \neq 0 \). \[ \Delta_z = \begin{vmatrix} 3 & -2 & 10 \\ 2 & -4 & 6 \\ 1 & 2 & 5m \end{vmatrix} = 3(-20m - 12) + 2(10m - 6) + 10(4 + 4) = 32 - 40m. \]
Step 3: For inconsistency, \[ 32 - 40m \neq 0 \Rightarrow m \neq \frac{4}{5}. \]