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KVPY Sample Paper KVPY Stream-SX Model Paper-1

  • question_answer
    If \[\left| z-4+3i \right|\le 1\] and \[\alpha \] and \[\beta \] are the least and greatest value of \[\left| z \right|\] and K be the least value of \[\frac{{{x}^{2}}+{{x}^{2}}+4}{x}\] interval \[(0,\infty ),\] then K is equal to

    A) \[\alpha \]                                 

    B) \[\beta \]

    C) \[\alpha \beta \]                         

    D) None of these

    Correct Answer: B

    Solution :

    [b]
    We have,
    \[\left| z-4+3i \right|\le 1\]
    \[\left| z-(4+3i) \right|\le 1\]
    \[-\,1+\left| 4-3i \right|\le 1+\left| 4-3i \right|\]
    \[-1+5\le \left| z \right|\le 1+5\]
    \[4\le \left| z \right|\le 6\]
    \[\because \]\[\alpha =4,\]\[\beta =6\]
    Let \[y=\frac{{{x}^{2}}+{{x}^{2}}+4}{x}\]
    \[\Rightarrow \]\[y={{x}^{3}}+x+\frac{4}{x}\]
    \[\Rightarrow \]\[y={{x}^{3}}+x+\frac{1}{x}+\frac{1}{x}+\frac{1}{x}+\frac{1}{x}\]
    Since, \[y\in (0,\infty ),\]therefore\[{{x}^{3}},\]\[x,\frac{1}{x}\] are positive.
    Sum will be least of
    \[\Rightarrow \]               \[x=1\]
    \[\therefore \]Least value of y is 6
    \[K=6\]\[\Rightarrow \]\[K=\beta \]


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