1.Relation and Function
hard

જો $f\left( x \right) + 2f\left( {\frac{1}{x}} \right) = 3x,x \ne 0$ અને $S = \left\{ {x \in R:f\left( x \right) = f\left( { - x} \right)} \right\}$;તો $S :$

A

બે કરતાં વધારે ઘટકો ધરાવે.

B

માત્ર બે જ ઘટકો ધરાવે.

C

ખાલી ગણ છે.

D

માત્ર બે જ ઘટક ધરાવે.

(JEE MAIN-2016)

Solution

$f(x)+2 f(1 / x)=3 x$        ……..$(1)$

$\mathrm{x} \rightarrow \frac{1}{\mathrm{x}} \Rightarrow \mathrm{f}(1 / \mathrm{x})+2 \mathrm{f}(\mathrm{x})=3 / \mathrm{x}$       ……..$(2)$

$\quad \mathrm{f}(\mathrm{x})+2\left(\frac{3}{\mathrm{x}}-2 \mathrm{f}(\mathrm{x})\right)=3 \mathrm{x}$

$\Rightarrow 3 \mathrm{f}(\mathrm{x})=\frac{6}{\mathrm{x}}-3 \mathrm{x}$

$\Rightarrow \mathrm{f}(\mathrm{x})=\frac{2}{\mathrm{x}}-\mathrm{x}$

For $\mathrm{s} \quad \mathrm{f}(\mathrm{x})=\mathrm{f}(-\mathrm{x}) \Rightarrow \frac{2}{\mathrm{x}}-\mathrm{x}=0$

$\Rightarrow \mathrm{x}=\pm \sqrt{2}$

Standard 12
Mathematics

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