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Question

Assertion :For K>0, let f(x)=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪x21,x1Kπcos1xx+1,1<x<01x2,0x<11sin(x1)1tan(x1),x>1 then f is a discontinuous function Reason: f has a break point at x=1

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

The correct option is C Assertion is correct but Reason is incorrect
limx1f(x)=Klimx1πcos1xx+1=Klimyππy1+cosy(cos1x=y)
=Klimyπ12πycosy21p+y
=K22limyππ2y2sin(π2y2)×1π+y
=K2π0=f(1)

So f is discontinuous at x=1
limx1+f(x)=limx1+(1sin(x1)1tan(x1))=limu0(1sinu1tanu)
=limu01cosusinu=0=f(1)=limx1f(x)
Hence there is no break point at x=1 but there is a break point at x=1

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