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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
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect but Reason is correct
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Solution

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
$$'b'$$ is the single H.M. between a and c
Also 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 d
A.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 correct
Statement (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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