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Question

Let S denotes the set of all real values of x such that (x2016+1)(1+x2+x4++x2014)=2016x2015. Then

A
Number of elements in S is 2
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B
Number of elements in S is 1
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C
Point (x,2), xS lies inside the parabola Y=X22X+2, XR
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D
Image of the point (x,2) about the line y=x lies on the line x+y=4
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Solution

The correct options are
B Number of elements in S is 1
C Point (x,2), xS lies inside the parabola Y=X22X+2, XR
(x2016+1)(1+x2+x4++x2014)=2016x2015(x+1x2015)(1+x2+x4++x2014)=2016x+1x2015+x3+1x2013++x2015+1x=2016(x+1x)+(x3+1x3)++(x2015+1x2015)=2016

When x>0,x+1x2
When x<0,x+1x2
So, L.H.S. = R.H.S. only if x=1

Putting the point (1,2) in the parabola,
2>12+2
So, the point (1,2) lies inside the parabola.

Image of the point (1,2) about the line y=x is (2,1) which does not lie on the line x+y=4

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