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Question

Let f(x)=2sinx, xmAsinx+B,m<x<mcosx, xm be a continuous function, where m is the principal solution of the equation sin3x+sinxcosx+cos3x=1. If m>0, then

A
A=1, B=1
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B
m=π
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C
m=π2
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D
A=1, B=1
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Solution

The correct options are
A A=1, B=1
C m=π2
sin3x+sinxcosx+cos3x=1
(sin3x+cos3x)(1sinxcosx)=0
(sinx+cosx)(1sinxcosx)(1sinxcosx)=0
(1sinxcosx)(sinx+cosx1)=0sinxcosx=1 or sinx+cosx=1sin2x=2 or sinx+cosx=1
sin2x=2 is not possible.

sinx+cosx=1
sinxsinπ4+cosxcosπ4=12
cos(xπ4)=cosπ4
x=0,π2,2π
But m is a principal solution with m>0. So, m=π2

Now, f(x)=⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪2sinx, xπ2Asinx+B,π2<x<π2cosx, xπ2

Continuity at x=π2
limxπ/2f(x)=limxπ/2+f(x)
2=A+B (1)

Continuity at x=π2
limxπ/2f(x)=limxπ/2+f(x)
A+B=0 (2)

Solving equation (1) and (2), we get
B=1, A=1

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