Second, third and fourth terms in binomial expansion (x+a)^n are 240,720 and 1080, respectively. Fin

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7.Binomial Theorem

hard

The second, third and fourth terms in the binomial expansion $(x+a)^n$ are $240,720$ and $1080,$ respectively. Find $x, a$ and $n$

Option A

Option B

Option C

Option D

Solution

Given that second term $T_{2}=240$

We have ${T_2} = {\,^n}{C_1}{x^{n – 1}} \cdot a$

So ${\,^n}{C_1}{x^{n – 1}} \cdot a = 240$ ……….$(1)$

Similarly ${\,^n}{C_2}{x^{n – 2}}{a^2} = 720$ ………..$(2)$

and $^{n} C_{x} x^{n-3} a^{3}=1080$ ………….$(3)$

Dividing $(2)$ by $(1),$ we get

$\frac{{{\,^n}{C_2}{x^{n – 2}}{a^2}}}{{^n{C_1}{x^{n – 1}}a}} = \frac{{720}}{{240}}$ i.e., $\frac{(n-1) !}{(n-2) !} \cdot \frac{a}{x}=6$

or $\frac{a}{x}=\frac{6}{(n-1)}$ ………..$(4)$

Dividing $(3)$ by $(2),$ we have

$\frac{a}{x}=\frac{9}{2(n-2)}$ ………..$(5)$

From $(4)$ and $(5),$

$\frac{6}{n-1}=\frac{9}{2(n-2)}$ Thus, $n=5$

Hence, from $(1), 5 x^{4} a=240,$ and from $(4), \frac{a}{x}=\frac{3}{2}$

Solving these equations for $a$ and $x,$ we get $x=2$ and $a=3$

Standard 11

Mathematics

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell}}\right)^9$ is $-84$ and the Coefficient of $x^{-3 \ell}$ is $2^\alpha \beta$, where $\beta < 0$ is an odd number, Then $|\alpha \ell-\beta|$ is equal to

hard

(JEE MAIN-2023)

If the coefficient of the middle term in the expansion of ${(1 + x)^{2n + 2}}$ is $p$ and the coefficients of middle terms in the expansion of ${(1 + x)^{2n + 1}}$ are $q$ and $r$, then

medium

Let $\mathrm{m}$ and $\mathrm{n}$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3} \mathrm{x}^{\frac{1}{3}}+\frac{1}{2 \mathrm{x}^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{\mathrm{n}}{\mathrm{m}}\right)^{\frac{1}{3}}$ is :

hard

(JEE MAIN-2024)

The greatest coefficient in the expansion of ${(1 + x)^{2n + 1}}$ is

medium

If $\left(\frac{3^{6}}{4^{4}}\right) \mathrm{k}$ is the term, independent of $\mathrm{x}$, in the binomial expansion of $\left(\frac{\mathrm{x}}{4}-\frac{12}{\mathrm{x}^{2}}\right)^{12}$, then $\mathrm{k}$ is equal to …… .

medium

(JEE MAIN-2021)

The second, third and fourth terms in the binomial expansion $(x+a)^n$ are $240,720$ and $1080,$ respectively. Find $x, a$ and $n$

Solution

Similar Questions

If the constant term in the binomial expansion of $\left(\frac{x^{\frac{5}{2}}}{2}-\frac{4}{x^{\ell}}\right)^9$ is $-84$ and the Coefficient of $x^{-3 \ell}$ is $2^\alpha \beta$, where $\beta < 0$ is an odd number, Then $|\alpha \ell-\beta|$ is equal to

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