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

  • question_answer
    If \[f(x+y)=f(x)+f(y)-xy-1\,\forall \,x,\,\,y\in R\] and\[f(1)=1,\] then the number of solutions of  \[f\left( n \right)=n,\]\[n\in N\], is

    A) 0                     

    B) 1

    C) 2                     

    D) more then 2

    Correct Answer: B

    Solution :

    [b] Given \[f(x+y)=f(x)+f(y)-xy-1\forall x,y\in R\] \[f(1)=1\] \[f(2)=f(1+1)=f(1)+f(1)-1-1=0\]
    \[f(3)=f(2+1)=f(2)+f(1)-2\times 1-1=-2\]
    \[f(n+1)=f(n)+f(1)-n-1=f(n)-n<f(n)\]
    \[Thus,\text{ }f(1)>f(2)>f(3)>....,\,\,and\text{ }f(1)=1.\]
    \[Therefore,\text{ }f(1)=1\,\,and\,\,f(n)<1,for\text{ }n>1\]
    \[Hence,\text{ }f(n)=n,n\in N,hasonlyonesolution\text{ }n=1.\]
     


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