Question

The following system of equations \[ x + y + z = 1 \] \[ 2x + 3y - mz = 2 \] \[ 3x + 5y + 3z = 3 \] has no unique solution. Then the value of \( m \) is equal to

• A. \(3 \)
• B. \(5 \)
• C. \(2 \)
• D. \(-2 \)
• E. \(-3 \)

Hint:

For system of equations, no unique solution \(\Rightarrow\) determinant of coefficient matrix is zero.

Answer 1

The Correct Option is D

Concept: System has no unique solution when determinant of coefficient matrix is zero.

Step 1: Form determinant.
\[ \begin{vmatrix} 1 & 1 & 1 2 & 3 & -m 3 & 5 & 3 \end{vmatrix} = 0 \]

Step 2: Expand determinant.
\[ = 1 \begin{vmatrix} 3 & -m 5 & 3 \end{vmatrix} -1 \begin{vmatrix} 2 & -m 3 & 3 \end{vmatrix} +1 \begin{vmatrix} 2 & 3 3 & 5 \end{vmatrix} \]

Step 3: Compute minors.
\[ = 1(9 + 5m) - 1(6 + 3m) + 1(10 - 9) \] \[ = (9 + 5m) - (6 + 3m) + 1 \] \[ = 9 + 5m - 6 - 3m + 1 \] \[ = 2m + 4 \]

Step 4: Set equal to zero.
\[ 2m + 4 = 0 \Rightarrow m = -2 \]