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JEE Main & Advanced Mathematics Relations Question Bank Mock Test - Relations and Functions

  • question_answer
    A function F(x) satisfies the functional equation \[{{x}^{2}}F(x)+F(1-x)=2x-{{x}^{4}}\]for all real x. F(x) must be

    A) \[{{x}^{2}}\]              

    B) \[1-{{x}^{2}}\]

    C) \[1+{{x}^{2}}\]         

    D) \[{{x}^{2}}+x+1\]

    Correct Answer: C

    Solution :

    [c] \[{{x}^{2}}F(x)+F(1-x)=2x-{{x}^{4}}\]  ...(1) Replacing \[x\]by \[1-x,\]we get \[{{(1-x)}^{2}}F(1-x)+F(x)=2(1-x)-{{(1-x)}^{4}}\]                                                        ...(2) Eliminating \[F(1-x)\] from (1) and (2), we get \[F(x)=1-{{x}^{2}}.\]


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