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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
    If the \[{{(r+1)}^{th}}\] term in the expansion of \[{{\left( \sqrt[3]{\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{a}}} \right)}^{21}}\] has the same power of a and b, then the value of r is

    A) 9

    B) 10

    C) 8

    D) 6

    Correct Answer: A

    Solution :

    We have \[{{T}_{r+1}}={{\,}^{21}}{{C}_{r}}{{\left( \sqrt[3]{\frac{a}{\sqrt{b}}} \right)}^{21-r}}{{\left( \sqrt{\frac{b}{\sqrt[3]{a}}} \right)}^{r}}\] \[={{\,}^{21}}{{C}_{r}}{{a}^{7-(r/2)}}{{b}^{(2/3)r-(7/2)}}\] Since the powers of a and b are the same, therefore \[7-\frac{r}{2}=\frac{2}{3}r-\frac{7}{2}\Rightarrow r=9\]


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