Question

The number of real roots of the equation $\left|\begin{array}{ccc}{x}^2+1& 2x^3& x\\ 2x^3& x& x^2+1\\ x& x^2+1& 2x^3\end{array}\right|=0$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (B)

Cyclic determinant $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=0\Rightarrow a+b+c=0ora=b=c$
$\Rightarrow x^2+1+2x^3+x=0$ or $x^2+1=2x^3=x$ (reject)
$\Rightarrow 2x^3+x^2+x+1=0$
$f\left(x\right)=2x^3+x^2+x+1$
$f^{{}^{\prime }}\left(x\right)=6x^2+2x+1>0$ , $\forall x\in R$ , as $D<0$ so $f\left(x\right)$ is strictly increasing.
$\therefore f\left(x\right)=0$ has exactly one real root as $f\left(x\right)$ is odd degree polynomial.