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The ratio of the coefficient of  x^{15}  to the term independent of x  in the expansion of  \left ( x^{2}+\frac{2}{x} \right )^{15}is :

Option: 1

7:16


Option: 2

7:64


Option: 3

1:4


Option: 4

1:32


Answers (1)

best_answer

\\\mathrm{T}_{r+1}={ }^{15} \mathrm{C}_{r}\left(x^{2}\right)^{15-r} \cdot\left(2 x^{-1}\right)^{r} \\ ={ }^{15} \mathrm{C}_{r} \times(2)^{r} \times x^{30-3 r}

Term that is independent of x when 30 - 3r = 0

r = 10

The term independent of x is

\mathrm{T}_{11}={ }^{15} \mathrm{C}_{10} \times(2)^{10}

\\\text{For term involve }x^{15}, 30-3 r=15 \\\Rightarrow r=5\\\text{Hence coefficient of }x^{15}={ }^{15} \mathrm{C}_{5} \times(2)^{5} \\\text{Required ratio}

\\=\frac{{ }^{15} \mathrm{C}_{5} \times(2)^{5}}{{ }^{15} \mathrm{C}_{10} \times(2)^{10}}=\frac{\frac{15 !}{10 ! 5 !}}{\frac{15 !}{5 ! 10 !} \times(2)^{5}} \\ =1: 32

Posted by

Suraj Bhandari

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