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Question

The values of a and b so that the functionf(x)= ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪x+a2sinx,0x<π42xcotx+b,π4xπ2acos2xbsinx,π2<xπ is continuous at x=π4,π2 are

A
a=π6
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B
a=π6
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C
b=π12
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D
b=π
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Solution

The correct option is C b=π12
f(x) is continuous in the interval 0x<π4,π4<x<π2,π2<xπ
We need to make the function continuous at x=π4,π2
For continuity at x=π4,limxπ4f(x)=limxπ4+f(x)=f(π4)
limx(π4)(x+a2sinx)=limx(π4)+(2xcotx+b)=f(π4)
π4+a2sin(π4)=2.π4.cot(π4)+b=2π4.cot(π4)+b
π4+a=π2+bab=π4 ...(1)
For continuity at x=π2,limxπ2f(x)=limxπ2+f(x)=f(π2)
limxπ2(2xcotx+b)=limx(π2)+(acos2xbsinx)=acosπbsinπ2
0+b=aba+2b=0 ...(2)
From equation (1) and (2)
a=π6,b=π12

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