The value of K, if \(\begin{vmatrix} 1 & \mathrm{K} & 3\\ 3 & \mathrm{K} & -2\\ 2 & 3 & -1\end{vmatrix} = 33\), is :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 0
Expanding along the first row, term by term:
\( 1 \times \begin{vmatrix} \mathrm{K} & -2\\ 3 & -1\end{vmatrix} = -\mathrm{K} + 6 \)
\( -\mathrm{K} \times \begin{vmatrix} 3 & -2\\ 2 & -1\end{vmatrix} = -\mathrm{K}(-3 + 4) = -\mathrm{K} \)
\( 3 \times \begin{vmatrix} 3 & \mathrm{K}\\ 2 & 3\end{vmatrix} = 3(9 - 2\mathrm{K}) = 27 - 6\mathrm{K} \)
Adding the three terms:
\(-\mathrm{K} + 6 - \mathrm{K} + 27 - 6\mathrm{K} = 33 - 8\mathrm{K}\)
Equating to 33:
\(33 - 8\mathrm{K} = 33\)
\(\mathrm{K} = 0\)