Question

The solution set of the equation ${\log }_{\sqrt{3}+|x^2-1|}(3-x^2-\frac{1}{x^2})={\log }_{\sqrt{3}+|x^2-1|}(5+x^2-x)$ is

• A. $\varphi$
• B. $(1,3)$
• C. $(-\infty ,3)$
• D. $(0,1)$

Answer 1

The correct option is A $\varphi$

Given ${\log }_{\sqrt{3}+|x^2-1|}(3-x^2-\frac{1}{x^2})={\log }_{\sqrt{3}+|x^2-1|}(5+x^2-x)$
$\Rightarrow 3-x^2-\frac{1}{x^2}=5+x^2-x$
$\Rightarrow 2x^4-x^3+2x^2+1=0$
$\Rightarrow x^2(2x^2-x+2)=-1$ - (i)
Now, for $2x^2-x+2$, the discriminant $=1-4\times 2\times 2=-15$
Since the discriminant is negative, the equation has non real roots.
Hence, the RHS of (i) is always positive for real values of $x$, while the LHS is negative.
Hence, there is no real solution .
Hence, option A is correct.