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JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

  • question_answer
    The coefficient of \[{{x}^{-9}}\] in the expansion of \[{{\left( \frac{{{x}^{2}}}{2}-\frac{2}{x} \right)}^{9}}\] is   [Kerala (Engg.) 2001]

    A) 512

    B) - 512

    C) 521

    D) 251

    Correct Answer: B

    Solution :

    Here \[{{T}_{r+1}}={}^{\text{9}}{{C}_{r}}{{\left( \frac{{{x}^{2}}}{2} \right)}^{9-r}}{{\left( \frac{-2}{x} \right)}^{r}}\] = \[{}^{9}{{C}_{r}}\frac{{{x}^{18-3r}}{{(-2)}^{r}}}{{{2}^{9-r}}},\] this contains \[{{x}^{-9}}\] if 18 - 3r = - 9 i.e. if r = 9. Coefficient of \[{{x}^{-9}}\] = \[{}^{9}{{C}_{9}}\frac{{{(-2)}^{9}}}{{{2}^{0}}}=-{{2}^{9}}=-512\].


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