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JEE Main & Advanced Sample Paper JEE Main Sample Paper-33

  • question_answer
    Number of integers in the range of the function \[f(x)=\frac{x({{x}^{2}}-1)}{{{x}^{4}}-{{x}^{2}}+1}\] is equal to

    A)  1     

    B)                                     2

    C)  3       

    D)                                     4

    Correct Answer: A

    Solution :

    \[f(x)=\frac{x-\frac{1}{x}}{{{x}^{2}}+\frac{1}{{{x}^{2}}}-1}=\frac{x-\frac{1}{x}}{{{\left( x-\frac{1}{x} \right)}^{2}}+1}\]                                            (divide by\[{{x}^{2}}\]) \[={{45}^{\text{o}}}+{{60}^{\text{o}}}\]) Put \[\left( x-\frac{1}{x} \right)=t\,;\,\,t\ne 0\] \[f(t)=\frac{t}{{{t}^{2}}+1}=\frac{1}{t+\frac{1}{t}}\] So, \[\frac{-1}{2}\,\,\underline{<}\,\,f(t)\,\,\underline{<}\,\,\frac{1}{2}\] \[\therefore \] Range is \[\left[ \frac{-1}{2},\frac{1}{2} \right]\] \[\Rightarrow \] Number of integers is one.


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