Question

Assertion :Let $a,b,c$ and $d$ be distinct positive real numbers in H.P.

$a+d>b+c$

Reason: $\frac{1}{a}+\frac{1}{d}=\frac{1}{b}+\frac{1}{c}$

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Statement (1)
As $a,b,c,d$ are in HP
${}^{\prime }{b}^{\prime }$ is the single H.M. between a and c
Also A.M. between $a$ and $c$ is $\frac{a+c}{2}$
As, A.M. > H.M.
$\therefore \frac{a+c}{2}>b$
$\therefore a+c>2b$ .....(i)
${}^{\prime }{c}^{\prime }$ is the single H.M. between b and d
A.M. between b and d is $\frac{b+d}{2}$
As, A.M. > H.M.
$\frac{b+d}{2}>c$
$\therefore b+d>2c.....\left(ii\right)$
Inequality (i) + (ii)
$a+c+b+d>2b+2c$
$a+d>b+c$ so statement (1) is correct
Statement (2) as $a,b,c,d$ are in H.P. $\therefore \frac{1}{a},\frac{1}{b},\frac{1}{c},\frac{1}{d}$ will be in A.P.
$\therefore \frac{1}{b}-\frac{1}{a}=\frac{1}{d}-\frac{1}{c}$ or, $\therefore \frac{1}{a}+\frac{1}{d}=\frac{1}{b}+\frac{1}{c}$
So, statement (2) is correct and is not correct explanation for statement (1).