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KVPY Sample Paper KVPY Stream-SX Model Paper-23

  • question_answer
    If \[f(x)\cdot f(y)=f(x)+f(y)+f(xy)-2\forall x,y\in R\] and if \[f(x)\] is not a constant function, then the value of \[f(1)\] is equal to

    A) 1

    B) 2      

    C) 0

    D) \[-\,1\]

    Correct Answer: B

    Solution :

    We have,
    \[f(x)\cdot f(y)=f(x)+f(y)+f(xy)-2\]
    Put \[x=y=1,\]we get
    \[{{(f(1))}^{2}}=3f(1)-2\] \[\Rightarrow \]\[f(1)=1\]or 2
    Let \[f(1)=1,\]then put \[y=1\]     
    \[\therefore \]\[f(x)\cdot f(1)=f(x)+f(1)+f(x)-2\] \[\Rightarrow \]\[f(x)=2f(x)+1-2\] \[\Rightarrow \]\[f(x)=1\] constant function, then \[f(1)=2,f(1)\ne 1\]


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