Question

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

(a) 4
(b) 5
(c) - 4
(d) 9

Answer 1

(c) We have, $\Delta =\left|\begin{array}{ccc}1& -2& 5\\ 2& a& -1\\ 0& 4& 2a\end{array}\right|=86$
$\Rightarrow 1(2a^2+4)-2(-4a-20)+0=86$ [expanding along first column]
$\Rightarrow 2a^2+4+8a+40=86\Rightarrow 2a^2+8a+44-86=0\Rightarrow a^2+4a-21=0\Rightarrow a^2+7a-3a-21=0\Rightarrow (a+7)(a-3)=0$
a = - 7 and 3
$\therefore$ Required sum = - 7 + 3 = -4