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Question

f(x) satisfies the relation f(x)λπ/20sinxcostf(t) dt=sinx.

If λ>2, then f(x) decreases in which of the following interval?


A
(0,π)
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B
(π2,3π2)
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C
(π2,π2)
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D
none of these 
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Solution

The correct option is C (π2,π2)
f(x)λπ/20sinxcost f(t) dt=sinxf(x)λsinxπ/20cost f(t) dt=sinx
f(x)Asinx=sinx
where, A=λπ/20cost f(t) dt

f(x)=(A+1)sinx
f(t)=(A+1)sint

A=λπ/20(A+1)costsint dtA=λ(A+1)2π/20sin2t dtA=λ(A+1)2[cos2t2]π/20A=λ(A+1)2A=λ2λ
So,
f(x)=(λ2λ+1)sinxf(x)=(22λ)sinx

Now, λ>2
f(x)=ksinx,  (k>0)
Hence, f(x) is decreasing when sinx is increasing.
From the given options, sinx is increasing in (π2,π2).

Therefore, f(x) is decreasing in (π2,π2).
 

Mathematics

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