(B) The general term in the expansion of $(x + \frac{2}{x^2})^{15}$ is given by $T_{r+1} = ^{15}C_r (x)^{15-r} (\frac{2}{x^2})^r$. This simplifies to

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(B) The general term in the expansion of $(x + \frac{2}{x^2})^{15}$ is given by $T_{r+1} = ^{15}C_r (x)^{15-r} (\frac{2}{x^2})^r$. This simplifies to $T_{r+1} = ^{15}C_r \times 2^r \times x^{15-r} \times x^{-2r} = ^{15}C_r \times 2^r \times x^{15-3r}$. For the term to be independent of $x$,the exponent of $x$ must be $0$. So,$15 - 3r = 0$,which gives $r = 5$. Substituting $r = 5$ into the general term expression,we get the term independent of $x$ as $^{15}C_5 \times 2^5$.

Class 11 Mathematics Binomial Theorem General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

Easy

In the expansion of $(x + \frac{2}{x^2})^{15}$,the term independent of $x$ is

A
$^{15}C_6 \times 2^6$
B
$^{15}C_5 \times 2^5$
C
$^{15}C_4 \times 2^4$
D
$^{15}C_8 \times 2^8$

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Binomial Theorem

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General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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DifficultJEE MAIN 2024

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MediumWBJEE 2009

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MediumAIEEE 2003

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If for positive integers $r > 1$ and $n > 2$,the coefficients of the $(3r)^{th}$ and $(r + 2)^{th}$ powers of $x$ in the expansion of $(1 + x)^{2n}$ are equal,then:

DifficultAIEEE 2002

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The maximum value of the term independent of $t$ in the expansion of $\left( tx^{\frac{1}{5}} + \frac{(1-x)^{\frac{1}{10}}}{t} \right)^{10}$ where $x \in (0, 1)$ is

DifficultJEE MAIN 2021

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In the expansion of $(x + \frac{2}{x^2})^{15}$,the term independent of $x$ is

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