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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 term independent of x in the expansion of  \[{{\left( \sqrt{\frac{x}{3}}+\frac{3}{2{{x}^{2}}} \right)}^{10}}\] will be  [IIT 1965; BIT Ranchi 1993; KCET 2000; UPSEAT 2001]

    A) 3/2

    B) 5/4

    C) 5/2

    D) None of these

    Correct Answer: B

    Solution :

    \[(10-r)\left( \frac{1}{2} \right)+r(-2)=0\Rightarrow 5-\frac{r}{2}-2r=0\Rightarrow r=2\] So the term independent of x \[{{=}^{10}}{{C}_{2}}\times {{\left( \frac{1}{3} \right)}^{4}}{{\left( \frac{3}{2} \right)}^{2}}=\frac{10\times 9}{2\times 1}\times \frac{1}{3\times 3\times 2\times 2}=\frac{5}{4}\]


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