If constant term in expansion of left(frac{√[5]{3}}{x}+frac{2 x}{√[3]{5}}right)^{12}, x ≠ 0 ,

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7.Binomial Theorem

hard

If the constant term in the expansion of $\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25 \alpha$ is equal to :

$639$

$724$

$693$

$742$

(JEE MAIN-2024)

Solution

$ \mathrm{T}_{\mathrm{r}+1}={ }^{12} \mathrm{C}_{\mathrm{r}}\left(\frac{3^{1 / 5}}{\mathrm{x}}\right)^{12-\mathrm{r}}\left(\frac{2 \mathrm{x}}{5^{1 / 3}}\right)^{\mathrm{r}} $

$ \mathrm{T}_{\mathrm{r}+1}=\frac{{ }^{12} \mathrm{C}_{\mathrm{r}}(3)^{\frac{12-\mathrm{r}}{5}}(2)^{\mathrm{r}}(\mathrm{x})^{2 \mathrm{r}-12}}{(5)^{\mathrm{r} / 3}} $

$ \mathrm{r}=6 $

$ \mathrm{~T}_7=\frac{{ }^{12} \mathrm{C}_6(3)^{6 / 5}(2)^6}{5^2}=\left(\frac{9 \times 11 \times 7}{25}\right) 2^8 \cdot 3^{1 / 5} $

$ 25 \alpha=693$

Standard 11

Mathematics

The coefficient of $x^{18}$ in the expansion of $\left(x^4-\frac{1}{x^3}\right)^{15}$ is $………..$.

hard

(JEE MAIN-2023)

If $^n{C_{r – 2}} = 36$ , $^n{C_{r – 1}} = 84$ and $^n{C_r} = 126$ , then value of $^n{C_{2r}}$ is

normal

In ${\left( {\sqrt[3]{2} + \frac{1}{{\sqrt[3]{3}}}} \right)^n}$ if the ratio of ${7^{th}}$ term from the beginning to the ${7^{th}}$ term from the end is $\frac{1}{6}$, then $n = $

medium

The coefficient of $x^{18}$ in the product $(1+ x)(1- x)^{10} (1+ x + x^2 )^9$ is

hard

(JEE MAIN-2019)

If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n$ is $\sqrt{6}: 1$, then the third term from the beginning is:

hard

(JEE MAIN-2023)

If the constant term in the expansion of $\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25 \alpha$ is equal to :

Solution

Similar Questions

The coefficient of $x^{18}$ in the expansion of $\left(x^4-\frac{1}{x^3}\right)^{15}$ is $………..$.

If $^n{C_{r – 2}} = 36$ , $^n{C_{r – 1}} = 84$ and $^n{C_r} = 126$ , then value of $^n{C_{2r}}$ is

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