Binomial Theorem · JEE Main previous year questions

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x{)}^{10}$, then $\sum _{r=1}^{10}{r}^3{\left(\frac{a_r}{a_{r-1}}\right)}^2$ is equal to

1. A.$3025$
2. B.$4895$
3. C.$5445$
4. D.$1210$

The constant term in the expansion of ${\left(2x+\frac{1}{x^7}+3x^2\right)}^5$ is ____.

Let the sum of the coefficients of the first three terms in the expansion of ${\left(x-\frac{3}{x^2}\right)}^n,x\ne 0,n\in \mathbb{N}$, be 376. Then the coefficient of $x^4$ is _______.

Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ${\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)}^n$, in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}:1$. If the sixth term from the beginning is $\frac{\alpha }{\sqrt[4]{3}}$, then $\alpha$ is equal to ____.

Let the coefficients of the middle terms in the expansion of ${\left(\frac{1}{\sqrt{6}}+\beta x\right)}^4$, $(1-3\beta x{)}^2$ and ${\left(1-\frac{\beta }{2}x\right)}^6$, $\beta >0$, respectively form the first three terms of an A.P. If d is the common difference of this A.P. , then $50-\frac{2d}{{\beta }^2}$ is equal to _________.

Let for the $9^{\mathrm{th}}$ term in the binomial expansion of $(3+6x{)}^n$, in the increasing powers of $6x$, to be the greatest for $x=\frac{3}{2}$, the least value of $n$ is $n_0$. If $k$ is the ratio of the coefficient of $x^6$ to the coefficient of $x^3$, then $k+n_0$ is equal to :

If the coefficients of $x$ and $x^2$ in the expansion of $(1+x{)}^p(1-x{)}^q$, $p,q\le 15$, are $-3$ and $-5$ respectively, then the coefficient of $x^3$ is equal to ____ .

For two positive real numbers $a$ and $b$ such that $\frac{1}{a^2}+\frac{1}{b^3}=4$, then minimum value of the constant term in the expansion of ${\left(a{x}^{\frac{1}{8}}+b{x}^{-\frac{1}{12}}\right)}^{10}$ is :

1. A.$\frac{105}{2}$
2. B.$\frac{105}{4}$
3. C.$\frac{105}{8}$
4. D.$\frac{105}{16}$

If a random variable X follows the Binomial distribution B(5, p) such that $\mathrm{P}(X=0)=\mathrm{P}(X=1)$, then $\frac{P(X=2)}{P(X=3)}$ is equal to :

1. A.1
2. B.10
3. C.25
4. D.5

The term independent of x in the expansion of $(1-x^2+3x^3){\left(\frac{5}{2}{x}^3-\frac{1}{5x^2}\right)}^{11}$, $x\ne 0$ is :

1. A.$\frac{7}{40}$
2. B.$\frac{33}{200}$
3. C.$\frac{39}{200}$
4. D.$\frac{11}{50}$

If the coefficient of $x^{10}$ in the binomial expansion of ${\left(\frac{\sqrt{x}}{5^{\frac{1}{4}}}+\frac{\sqrt{5}}{x^{\frac{1}{3}}}\right)}^{60}$ is $5^k\cdot l$, where $l,k\in \mathrm{N}$ and $l$ is co-prime to 5, then $k$ is equal to ____.

If the sum of the coefficients of all the positive powers of $x$, in the Binomial expansion of ${\left(x^n+\frac{2}{x^5}\right)}^7$ is $939$, then the sum of all the possible integral values of $n$ is ________.

Let ${\mathrm{C}}_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x{)}^{10}$. If for $\alpha ,\beta \in \mathbb{R}$, $C_1+3\cdot 2C_2+5\cdot 3C_3+\dots \text{ upto }10\text{ terms}=\frac{\alpha \times 2^{11}}{2^{\beta }-1}\left(C_0+\frac{C_1}{2}+\frac{C_2}{3}+\dots \text{ upto }10\text{ terms}\right)$ then the value of $\alpha +\beta$ is equal to $\underline{}$.

The coefficient of $x^{101}$ in the expression $(5+x{)}^{500}+x(5+x{)}^{499}+x^2(5+x{)}^{498}+\cdots +x^{500}$, $x>0$, is

1. A.${}^{501}{\mathrm{C}}_{101}(5{)}^{399}$
2. B.${}^{501}{\mathrm{C}}_{101}(5{)}^{400}$
3. C.${}^{501}{\mathrm{C}}_{100}(5{)}^{400}$
4. D.${}^{500}{\mathrm{C}}_{101}(5{)}^{399}$

If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of ${\left(2x^3+\frac{3}{x}\right)}^{10}$ is $5^{10}-\beta \cdot 3^9$, then $\beta$ is equal to $\underline{}$.

If the sum of the coefficients in the expansion of $(x+y{)}^n$ is 4096, then the greatest coefficient in the expansion is ___________.

The ratio of the coefficient of the middle term in the expansion of $(1+x{)}^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x{)}^{19}$ is ____.

The term independent of 'x' in the expansion of ${\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)}^{10}$, where $x\ne 0,1$ is equal to ____.

The sum of all those terms which are rational numbers in the expansion of $(2^{1/3}+3^{1/4}{)}^{12}$ is :

1. A.89
2. B.27
3. C.35
4. D.43

If the greatest value of the term independent of 'x' in the expansion of ${\left(x\sin \alpha +a\frac{\cos \alpha }{x}\right)}^{10}$ is $\frac{10!}{(5!{)}^2}$, then the value of 'a' is equal to :

1. A.$-1$
2. B.1
3. C.$-2$
4. D.2

If ${}^n{P}_r={}^n{P}_{r+1}$ and ${}^n{C}_r={}^n{C}_{r-1}$, then the value of r is equal to :

1. A.1
2. B.4
3. C.2
4. D.3

If the co-efficient of $x^7$ and $x^8$ in the expansion of ${\left(2+\frac{x}{3}\right)}^n$ are equal, then the value of n is equal to ____.

If the constant term, in binomial expansion of ${\left(2x^r+\frac{1}{x^2}\right)}^{10}$ is 180, then r is equal to ________.

The coefficient of $x^{256}$ in the expansion of $(1-x{)}^{101}(x^2+x+1{)}^{100}$ is :

1. A.${}^{100}{C}_{16}$
2. B.${}^{100}{C}_{15}$
3. C.$-{}^{100}{C}_{16}$
4. D.$-{}^{100}{C}_{15}$

The number of rational terms in the binomial expansion of ${\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)}^{120}$ is ________.