If \(\begin{vmatrix}1&-2&5\\2&a&-1\\0&4&2a\end{vmatrix} = 86\), then product of all values of \(a\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → −21
Expanding the determinant along the first column, which has a zero:
\(\displaystyle 1\begin{vmatrix}a&-1\\4&2a\end{vmatrix} - 2\begin{vmatrix}-2&5\\4&2a\end{vmatrix} = 86\)
\((2a^2 + 4) - 2(-4a - 20) = 86\)
\(2a^2 + 8a + 44 = 86\)
\(a^2 + 4a - 21 = 0\)
\((a + 7)(a - 3) = 0,\quad a = -7 \text{ or } 3\)
So the product of all values of a is
\((-7)(3) = -21\)