Question

# Assertion :Let $$a, b, c$$ and $$d$$ be distinct positive real numbers in H.P.$$a+d > b+c$$ Reason: $$\dfrac {1}{a}+\dfrac {1}{d}=\dfrac {1}{b}+\dfrac {1}{c}$$

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

Solution

## The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for AssertionStatement (1)As $$a, b, c, d$$ are in HP$$'b'$$ is the single H.M. between a and cAlso A.M. between $$a$$ and $$c$$ is $$\dfrac {a+c}{2}$$As, A.M. > H.M.$$\therefore \dfrac {a+c}{2} > b$$$$\therefore a+c > 2b$$ .....(i)$$'c'$$ is the single H.M. between b and dA.M. between b and d is $$\dfrac {b+d}{2}$$As, A.M. > H.M.$$\dfrac {b+d}{2} > c$$$$\therefore b+d > 2c .....(ii)$$Inequality (i) + (ii)$$a+c+b+d > 2b+2c$$$$a+d > b+c$$ so statement (1) is correctStatement (2) as $$a, b, c, d$$  are in H.P. $$\therefore \dfrac {1}{a}, \dfrac {1}{b}, \dfrac {1}{c}, \dfrac {1}{d}$$ will be in A.P.$$\therefore \dfrac {1}{b}-\dfrac {1}{a}=\dfrac {1}{d}-\dfrac {1}{c}$$ or, $$\therefore \dfrac {1}{a}+\dfrac {1}{d}=\dfrac {1}{b}+\dfrac {1}{c}$$So, statement (2) is correct and is not correct explanation for statement (1).Maths

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