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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 first 3 terms in the expansion of \[{{(1+ax)}^{n}}\] \[(n\ne 0)\] are 1, 6x and 16x2. Then the value of a and n are respectively [Kerala (Engg.) 2002]

    A) 2 and 9

    B) 3 and 2

    C) 2/3 and 9

    D) 3/2 and \[6\]

    Correct Answer: C

    Solution :

    \[{{T}_{1}}={}^{n}{{C}_{0}}=1\]                  ?..(i) \[{{T}_{2}}={}^{n}{{C}_{1}}ax=6x\]          ?..(ii) \[{{T}_{3}}={}^{n}{{C}_{2}}{{(ax)}^{2}}=16{{x}^{2}}\]                ?..(iii) From (ii), \[\frac{n!}{\left( n-1 \right)\,!}a=6\]  Þ \[n\,a\,=\,6\]    ?..(iv) From (iii), \[\frac{n(n-1)}{2}\,{{a}^{2}}=16\]      ?..(v) Only  is satisfying equation (iv) and (v).


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