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11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-5

  • question_answer
    Let \[f(x)=\frac{\alpha .x}{x+1},x\ne -1\], then for what value of a Is \[\left\{ \mathbf{f}\left\{ \mathbf{f}\left( \mathbf{x} \right) \right\} \right\}=\mathbf{x}?\]

    A) (a) 1                                         

    B) (b) \[-1\]            

    C) (c) \[\sqrt{2}\]                           

    D) (d) \[-\sqrt{2}\]

    Correct Answer: B

    Solution :

    [b] \[\therefore f(x)=\frac{\alpha .x}{x+1}\] \[x=\frac{\alpha .\frac{\alpha x}{x+1}}{\frac{\alpha x}{x+1}+1}=\frac{{{\alpha }^{2}}x}{\alpha x+x+1}x=\frac{{{\alpha }^{2}}x}{(\alpha +1)x+1}\] \[\Rightarrow (a+1){{x}^{2}}+(1-{{a}^{2}})x=0\] \[\{{{x}^{2}}+(1-\alpha )x\}=0\] If \[\alpha +1=0\] \[\therefore a=-1,\forall =x\] Hence, option [b]is correct.


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