11th Class Mathematics Sample Paper Mathematics Olympiad - Sample Paper-3

  • question_answer
    The coefficient of \[{{\mathbf{x}}^{\mathbf{3}}}\]in \[{{\left( \sqrt{{{\mathbf{x}}^{\mathbf{5}}}}+\frac{3}{\sqrt{{{x}^{3}}}} \right)}^{6}}\]  is

    A)  120                             

    B)  540          

    C)  150                             

    D)  250

    Correct Answer: B

    Solution :

    [b] Let \[{{(r+1)}^{th}}\] term be the coefficient of x3 \[{{t}_{r+1}}{{=}^{6}}{{C}_{r}}{{\left( \sqrt{{{x}^{5}}} \right)}^{6-r}}{{\left( \frac{3}{\sqrt{{{x}^{3}}}} \right)}^{r}}{{=}^{6}}{{C}_{r.}}{{(x)}^{15-\frac{5r}{2}-\frac{3r}{2}}}.{{(3)}^{r}}\]\[{{=}^{6}}{{C}_{r}}.{{x}^{15-4r}}.{{(3)}^{r}}\] \[\because 15-4r=3\Rightarrow 4r=12\] \[\therefore r=3\] \[\therefore \]Coefficient of \[{{x}^{3}}{{=}^{6}}{{C}_{3}}.{{(3)}^{3}}\] \[=\frac{6}{3.3}\times 27=\frac{6\times 5\times 4}{3\times 2}\times 27=5\times 4\times 27=540\]. Hence, option [b] is correct.

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