If there are two values of $a$ which makes determinant, $\Delta = \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86$, then the sum of these number is

$-4$

The correct answer is Option (4) → $-4$ ##

We have,

$\Delta = \begin{vmatrix} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2a \end{vmatrix} = 86$

$\Rightarrow 1(2a^2 + 4) - 2(-4a - 20) + 0 = 86 \quad \text{[expanding along first column i.e., } C_1]$

$\Rightarrow 2a^2 + 4 + 8a + 40 = 86$

$\Rightarrow 2a^2 + 8a + 44 - 86 = 0$

$\Rightarrow 2a^2 + 8a - 42 = 0$

$\Rightarrow a^2 + 4a - 21 = 0$

$\Rightarrow a^2 + 7a - 3a - 21 = 0$

$\Rightarrow (a+7)(a-3) = 0$

$\Rightarrow a = -7 \text{ and } 3$

$∴\text{Required sum} = -7 + 3 = -4$