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Question

Consider the following statements
S1: If f(x)=x25|x|+6, then f(52)=0
S2: The slope of tangent at x=1 on the curve y=sin1(cosπx) is π
S3: If x=f(t),y=g(t), then d2xdy2=f(t)g(t)
Which of the following is/are correct about the truth value of above statements

A
S1 is true
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B
S2 is false
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C
all are true
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D
S3 is false
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Solution

The correct option is D S3 is false
S1:f(x)=|x2+5x+6|, for x<0
f(x)=(x2+5x+6), if 3x2
f(x)=2x5
f(52)=0

S2:y=sin1(cosπx)
=π2cos1(cosπx)=⎪ ⎪⎪ ⎪π2πx,0x1πx3π2,1<x2
Function is not differentiable at x=1.
Hence, no tangent can be drawn at this point.

S3:dxdt=f(t),dydt=g(t)
dxdy=dxdtdtdy=f(t)g(t)
d2xdy2=ddy(dxdy)=ddt(dxdy)dtdy
d2xdy2=f(t)g′′(t)f′′(t)g(t)(f(t))3

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