Question

The number of positive roots of the equation $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$ is

• A. 1
• B. 2
• C. 3
• D. 0

Answer 1

Answer: (B)

Given, $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$
$\Rightarrow x\left(x^2-12\right)-3(2x-14)+7(12-7x)=0$
$\Rightarrow x^3-12x-6x+42+84-49x=0$
$\Rightarrow x^3-67x+126=0$
$\Rightarrow (x-2)\left(x^2+2x-63\right)=0$
$\Rightarrow (x-2)\left(x^2+9x-7x-63\right)=0$
$\Rightarrow (x-2)\{x(x+9)-7(x+9)\}=0$
$\Rightarrow (x-2)(x+9)(x-7)=0$
$\Rightarrow x=2,7,-9$
Hence, number of positive roots is $2$ .