Question

There are two values of a which makes the determinant $∆=\left|\begin{array}{ccc}1& -2& 5\\ 2& a& -1\\ 0& 4& 2a\end{array}\right|$ equal to 86. The sum of these two values is

(a) 4
(b) 5
(c) −4
(d) 9

Answer 1

$$\begin{gathered} ∆=\left|\begin{array}{ccc}1& -2& 5\\ 2& a& -1\\ 0& 4& 2a\end{array}\right|=86 \\ \Rightarrow 1\left(2a^2+4\right)-2\left(-4a-20\right)=86 \\ \Rightarrow 2a^2+4+8a+40=86 \\ \Rightarrow 2a^2+8a-42=0 \\ \Rightarrow a^2+4a-21=0 \\ \Rightarrow a^2+7a-3a-21=0 \\ \Rightarrow a\left(a+7\right)-3\left(a+7\right)=0 \\ \Rightarrow \left(a+7\right)\left(a-3\right)=0 \\ \Rightarrow a=-7,3 \\ \mathrm{Sum}\mathrm{of}\mathrm{the}\mathrm{two}\mathrm{values}\mathrm{of}a=-7+3=-4. \end{gathered}$$

Hence, the correct option is (c).