Question

Assertion :If ${\log }_{0.2}\left(\frac{x+1}{x}\right)\ge 1$, then $x\in [-1.25,-1)$. Reason: If $0<a<1$, ${\log }_{a}x\ge {\log }_{a}y\iff y\ge x>0$.

• 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.

${\log }_{0.2}\left(\frac{x+1}{x}\right)$ is valid when $\frac{x+1}{x}>0$
$\Rightarrow x\in (-\infty ,-1)\cup (0,\infty )$ ....(1)
${\log }_{0.2}\left(\frac{x+1}{x}\right)\ge 1$
$\Rightarrow \frac{x+1}{x}\le \frac{1}{5}$
$\Rightarrow \frac{4x+5}{x}\le 0$
$\Rightarrow x\in [-\frac{5}{4},0)$ .....(2)
From (1) & (2), we get
$x\in [-1.25,-1)$

Ans: A