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17. If x gt 1, y gt 1, z gt1 are in G.P,...

17. If `x gt 1, y gt 1, z gt1` are in G.P, then `1/(1+ ln x), 1/( 1+ ln y), 1/(1+ lnz)` are in

A

AP

B

HP

C

GP

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let the common ratio of the GP be r. Then,
`y = xr and z = xr^(2)`
`rArr ln y = ln x + ln r and ln z = ln x + 2 ln r`
Let `A = 1 + ln x, D = ln r`
Then, `(1)/(1 + ln x) = (1)/(A), (1)/(1 + ln y) = (1)/(1 + ln x + ln r) = (1)/(A +D)`
and `(1)/(1 + lnz) = (1)/(1 + ln x + 2ln r) = (1)/(A + 2D)`
Therefore, `(1)/(1 + ln x),(1)/(ln y), (1)/(1 + ln z)` are in HP.
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