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KVPY Sample Paper KVPY Stream-SX Model Paper-16

  • question_answer
    If S be the sum of coefficients in the expansion of \[{{(px+qy-rz)}^{n}}\] (where p, q, r > 0) then the value of \[\underset{n\to \infty }{\mathop{\lim }}\,\,\,\,\frac{S}{{{({{S}^{1/n}}+1)}^{n}}},\] is:

    A) \[\frac{pq}{r}\]

    B) \[{{e}^{\frac{pq}{r}}}\]

    C) 0

    D) 1

    Correct Answer: C

    Solution :

    \[S={{(p+q+r)}^{n}}\]\[\{Putting\,x=y=z=1\}\]
    \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{S}{{{({{S}^{1/n}}+1)}^{n}}}=\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{(p+q-r)}^{n}}}{{{\{(p+q-r)+1\}}^{n}}}\]
    \[\underset{n\to \infty }{\mathop{\lim }}\,{{\left( \frac{p+q-r}{(p+q-r+1)} \right)}^{n}}=0\]
    \[\left\{ as\left( \frac{p+q-r}{(p+q-r+1)} \right)<1 \right\}\]


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