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KVPY Sample Paper KVPY Stream-SX Model Paper-4

  • question_answer
    If \[\alpha =\underset{n\to \infty }{\mathop{\lim }}\,\left( \frac{1}{{{n}^{3}}+1}+\frac{4}{{{n}^{3}}+1}+......+\frac{{{n}^{2}}}{{{n}^{3}}+1} \right)\] and \[\beta =\underset{n\to \infty }{\mathop{\lim }}\,\,\frac{\sin 2x}{\sin 8x}\] then the quadratic equation whose roots are \[\alpha ,\]\[\beta \] is

    A) \[12{{x}^{2}}-7x+1=0\]

    B) \[{{x}^{2}}+19x-120=0\]

    C) \[{{x}^{2}}17x+60=0\]                        

    D) \[{{x}^{2}}-7x+12=0\]

    Correct Answer: A

    Solution :

    [A]
    \[\alpha =\underset{n\to \infty }{\mathop{\lim }}\,\frac{1+4+9+....+{{n}^{2}}}{{{n}^{3}}+1}\]\[=\underset{n\to \infty }{\mathop{\lim }}\,\frac{n\,\,(n+1)\,\,(2n+1)}{6\,\,({{n}^{3}}+1)}\]
    \[\beta =\underset{n\to \infty }{\mathop{\lim }}\,\left( \frac{\sin 2x}{2x} \right)\,\,\left( \frac{8x}{\sin 8x} \right)\,\,\times \,\,\left[ \frac{2x}{8x} \right]\]
    \[\beta =\frac{1}{4}\]
    \[\alpha \beta =\frac{1}{12}\]
    \[\alpha +\beta =\frac{7}{12}\]
    \[12{{x}^{2}}-7x+1=0\]


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