If the system of equations
\(x + y + z = 6\)
\(2x + ky + 3z = 14\)
\(x + 2y + 2z = 9\)
does not have a unique solution, then the value of k is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 3
A system AX = B has a unique solution exactly when \(|A| \ne 0\). So here the coefficient determinant must be zero:
\( \displaystyle |A| = \begin{vmatrix}1 & 1 & 1\\ 2 & k & 3\\ 1 & 2 & 2\end{vmatrix} \)
Expanding along the first row:
\( |A| = 1(2k - 6) - 1(4 - 3) + 1(4 - k) \)
\( = 2k - 6 - 1 + 4 - k \)
\( = k - 3 \)
Setting \(|A| = 0\):
\(k - 3 = 0\)
\(k = 3\)