Question

The number of positive roots of the equation $ \begin{vmatrix}x&3&7\\ 2&x&2\\ 7&6&x\end{vmatrix}=0 $ is

• A. 1
• B. 2
• C. 3
• D. 0

Answer 1

Answer: (B)

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