Question

Assertion :If a, b, c are three distinct positive number in H.P., then
$\left(\frac{a+b}{2a-b}\right)+\left(\frac{c+b}{2c-b}\right)>4$ Reason: Sum of any number and it's reciprocal is always greater than or equal to 2.

• 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 C Assertion is correct but Reason is incorrect

Assertion: $\frac{\frac{1}{b}+\frac{1}{a}}{\frac{2}{b}-\frac{1}{a}}+\frac{\frac{1}{b}+\frac{1}{c}}{\frac{2}{b}-\frac{1}{c}}=\frac{\frac{1}{b}+\frac{1}{a}}{\frac{1}{c}}+\frac{\frac{1}{b}+\frac{1}{c}}{\frac{1}{a}}$

$=\frac{c}{b}+\frac{c}{a}+\frac{a}{b}+\frac{a}{b}=\frac{a+c}{b}+\left(\frac{c}{a}+\frac{a}{c}\right)$

$=\frac{(a+c{)}^2}{2ac}+\left(\frac{c}{a}+\frac{a}{c}\right)=\frac{1}{2}\left(\frac{a}{c}+\frac{c}{a}+2\right)+\left(\frac{c}{a}+\frac{a}{c}\right)>4$

$[\therefore x+\frac{1}{x}>2$ when $x>\ne \frac{1}{x}]$

The reason is False as for this numbers should be positive.

Assertion is correct but Reason is incorrect.