(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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Class 11

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Chapter

Binomial Theorem

Subtopic

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

If the $6^{th}$ term in the expansion of the binomial $\left[ \frac{1}{x^{8/3}} + x^2 \log_{10} x \right]^8$ is $5600$,then $x$ equals to

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If in the expansion of $(a-2b)^{n}$, the sum of the $5^{th}$ and $6^{th}$ term is zero, then the value of $\frac{a}{b}$ is

MediumWBJEE 2010

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In the expansion of $(1+x)^n$,the coefficients of the $p^{th}$ and $(p+1)^{th}$ terms are respectively $p$ and $q$. Then $p+q$ is equal to:

MediumTS EAMCET 2017

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Let $[t]$ denote the greatest integer $\leq t$. If the constant term in the expansion of $\left(3x^2 - \frac{1}{2x^5}\right)^7$ is $\alpha$,then $[\alpha]$ is equal to $............$.

DifficultJEE MAIN 2023

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In the expansion of $(1 + x)^{43}$,if the coefficients of the $(2r + 1)^{th}$ and the $(r + 2)^{th}$ terms are equal,the value of $r$ is:

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

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If the $6^{th}$ term in the expansion of the binomial $\left[ \frac{1}{x^{8/3}} + x^2 \log_{10} x \right]^8$ is $5600$,then $x$ equals to

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