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JEE Main & Advanced Sample Paper JEE Main Sample Paper-9

  • question_answer
    Passage (Q. - 69) Consider the rectangular hyperbola \[xy=15!,\] the number of points \[(\alpha ,\beta )\] lying the curve is \[\alpha ,\beta \in {{l}^{+}}\And HCF(\alpha ,\beta )=1\]is

    A)  64                         

    B)  625    

    C)  8064                     

    D)  49

    Correct Answer: A

    Solution :

    \[xy=15!={{2}^{11}}{{.3}^{6}}{{5}^{3}}{{.7}^{2}}.11.13\] No. of positive integral solutions = No. of ways of fixing x = Number of factors \[=15!=(11+1)(6+1)(3+1)\]\[(2+1)\]\[{{(1+1)}^{2}}=4032\] \[\therefore \] Total number of integral solutions \[=24032=8064\] (As solution can be positive or negative) As \[HCF(\alpha ,\beta )=1\] So, \[\alpha \And \beta \] will not have common factor other than 1, so identical prime number should not be separated. So, the number of solutions \[={{2}^{6}}.\]          (1)


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