There are two values of $a$ which makes the determinant $\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|$ equal to 86 . The sum of these two values is
(a) 4
(b) 5
(c) $-4$
(d) 9
$\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a\end{array}\right|=86$
$\Rightarrow 1\left(2 a^{2}+4\right)-2(-4 a-20)=86$
$\Rightarrow 2 a^{2}+4+8 a+40=86$
$\Rightarrow 2 a^{2}+8 a-42=0$
$\Rightarrow a^{2}+4 a-21=0$
$\Rightarrow a^{2}+7 a-3 a-21=0$
$\Rightarrow a(a+7)-3(a+7)=0$
$\Rightarrow(a+7)(a-3)=0$
$\Rightarrow a=-7,3$
Sum of the two values of $a=-7+3=-4$.
Hence, the correct option is (c).
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.