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If `f(x)+2f((1)/(x))=3x, x ne 0 and ` `S={x in R: f(x) = f(-x)}, " then " S`A. is an empty setB. contains exactly one elementsC. contains exactly two elementsD. contains more than two elements |
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Answer» Correct Answer - C We have, `f(x)+2f((1)/(x))=3x, x ne 0 " …(i)" ` On replacing x by `(1)/(x)` in the above equation, we get `f((1)/(x))+2f(x)=(3)/(x)` `rArr 2f(x)+f((1)/(x))=(3)/(x) " ….(ii)" ` On multiplying Eq. (ii) by 2 and subtracting Eq. (i) from Eq. (ii), we get `4f(x)+2f((1)/(x))=(6)/(x)` `(underset (-)(f)(x)underset(-)(+)2f((1)/(x))underset(-)(=)3x)/(3f(x)=(6)/(x)-3x)` `rArr f(x)=(2)/(x) -x` Now, consider ` f(x)=f(-x)` `rArr (2)/(x)-x=-(2)/(x)+x rArr (4)/(x)=2x` `rArr 2x^(2)=4 rArr x^(2) =2` `rArr x = pm sqrt(2)` Hence, S contains exactly two elements. |
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