Question

Assertion :The set of all $x$ satisfying the equation $x^{{\log }_5{x}^2+({\log }_{5}x{)}^2-12}=\frac{1}{x^4}.....\left(1\right)$ is $\{1,25,\frac{1}{125},\frac{1}{625}\}$ Reason: A polynomial equation of degree $n$ can have at most $n$ real roots.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

$x^{{\log }_5{x}^2+({\log }_{5}x{)}^2-12}=\frac{1}{x^4}$ ....(1)
Equation (1) is valid when $x>0$

Taking log to the base $5$ of both the sides of $\left(1\right)$, we obtain

$[{\log }_5{x}^2+{\left({\log }_{5}x\right)}^2-12]{\log }_{5}x=-4{\log }_{5}x[\because \log a^m=m\log a]$

$\Rightarrow {\log }_{5}x=0or{\left({\log }_{5}x\right)}^2+2{\log }_{5}x-8=0$

$\Rightarrow x=1or{\log }_{5}x=2,-4$

$\Rightarrow x=1,25,\frac{1}{625}$