Question

Assertion :If $1+{\log }_5(x^2+1)\ge {\log }_5({cx}^2+4x+c)$ for all $x\in R$, then $c\in (2,3)$ Reason: If $a{x}^2+bx+c>0$ for all $x\in R$, then $a>0$ and $b^2-4ac<0$

• 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 A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$1+{\log }_5(x^2+1)\ge {\log }_5(c{x}^2+4x+c)\Rightarrow {\log }_{5}5+{\log }_5(x^2+1)\ge {\log }_5(c{x}^2+4x+c)[\because {\log }_{a}a=1]\Rightarrow {\log }_5(5x^2+5)\ge {\log }_5(c{x}^2+4x+c)[\because \log a+\log b=\log (ab\left)\right]\Rightarrow 5x^2+5\ge c{x}^2+4x+c(5-c)x^2-4x+(5-c)\ge 0\text{and}c{x}^2+4x+c>0(5-c)x^2-4x+(5-c)\ge 0\forall xi.e5-c>0andD<0\text{simplifying gives}c\le 3c{x}^2+4x+c>0forallxi.ec>0andD<0givesc>2$
Hence$c\in (2,3]$
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion