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Question

If S denotes the set of all real values of x such that (x2010+1)(1+x2+x4++x2008)=2010x2009, then the number of elements in set S is

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Solution

Given ; (x2010+1)(1+x2+x4++x2008)=2010x2009
As x0, so dividing by x2009, we get
(x+1x2009)(1+x2+x4++x2008)=2010x+x3+x5++x2009+1x2009+1x2007++1x=2010x+1x+x3+1x3++x2009+1x2009=2010

We know that,
x+1x(,2][2,)x3+1x3(,2][2,)x2009+1x2009(,2][2,)
So,
x+1x+x3+1x3++x2009+1x20092×1005x+1x+x3+1x3++x2009+1x20092010
Therefore,
x+1x+x3+1x3++x2009+1x2009=2010
When
x+1x=2x=1

Hence, the number of elements in set S is 1.

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