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Q.

Which of the following is/are correct?

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a

(tan⁡x)ln⁡(sin⁡x)>(cot⁡x)ln⁡(sin⁡x),∀x∈(0,π/4)

b

4ln⁡cosec⁡x<5ln⁡cosec⁡x,∀x∈(0,π/2)

c

(1/2)ln⁡(cos⁡x)<(1/3)ln⁡(cos⁡x),∀x∈(0,π/2)

d

2ln⁡(tan⁡x)>2ln⁡(sin⁡x),∀x∈(0,π/2)

answer is A.

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Detailed Solution

1 For x∈0,π4,tan⁡x(cot⁡x)ln⁡(sin⁡x)2    For x∈0,π2,cosec⁡x≥1      ⇒ ln⁡(cosec⁡x)≥0      ⇒ 4ln⁡(cosec⁡x)<5ln⁡(cosec⁡x)3  x∈0,n2     ⇒cos⁡x∈(0,1)     ⇒ ln⁡(cos⁡x)<0      also 12>13      ⇒−12ln⁡(cos⁡x)<13ln⁡(cos⁡x)4  For x∈0,π2       Since sinx < tanx, we get ln(sinx) < ln(tanx)        ⇒ 2ln⁡(sin⁡x)<2ln⁡(tan⁡x)
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