The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of…
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:
-
$7: 16$
-
$7: 64$
-
$1: 4$
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$1: 32$
Solution
$
\begin{aligned}
& \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}
\end{aligned}
$
For independent term, $30-3 r=0 \Rightarrow r=10$
Hence the term independent of $x$,
$
\mathrm{T}_{11}={ }^{15} \mathrm{C}_{10} \times(2)^{10}
$
For term involve $x^{15}, 30-3 r=15 \Rightarrow r=5$
Hence coefficient of $x^{15}={ }^{15} \mathrm{C}_5 \times(2)^5$
Required ratio
$
\begin{aligned}
& =\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 \\
&
\end{aligned}
$
Asked in: JEE Main 2013 (09 Apr Online)
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