Question

Let $a,b\text{and}c$ be three real numbers satisfying
$\left[\begin{array}{ccc}a& b& c\end{array}\right]\left[\begin{array}{ccc}1& 9& 7\\ 8& 2& 7\\ 7& 3& 7\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$ . . . (i)

Let b = 6, with a and c satisfying Eq. (i). If $\alpha and\beta$ are the roots of the quadratic equation $a{x}^2$ + bx + c = 0, then $\alpha \beta$ is equal to

• A. 1
• B. 7
• C. -1
• D. -7

Answer 1

The correct option is D -7

On multiplying matrices, we have
$\left[\begin{array}{ccc}a+8b+7c& 9a+2b+3c& 7a+7b+7c\end{array}\right]=\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$
On solving the 3 equations, we get $b=6a\text{and}c=-7a$

If b = 6 $\Rightarrow$ a = 1 and c =-7
$\therefore a{x}^2+bx+c=0\Rightarrow x^2+6x-7=0$
$\Rightarrow$ (x+7) (x-1) = 0
$\therefore$ x = 1, - 7
$\alpha \beta =1\times -7=-7$