Binomial Theorem · JEE Main previous year questions

Let the coefficient of $x^r$ in the expansion of $(x+3{)}^{n-1}+(x+3{)}^{n-2}(x+2)+(x+3{)}^{n-3}(x+2{)}^2+\dots \dots \dots .+(x+2{)}^{n-1}$ be ${\alpha }_r$. If ${\sum }_{r=0}^n{\alpha }_r={\beta }^n-{\gamma }^n,\beta ,\gamma \in \mathbb{N}$, then the value of ${\beta }^2+{\gamma }^2$ equals ____.

If A denotes the sum of all the coefficients in the expansion of ${\left(1-3x+10x^2\right)}^n$ and B denotes the sum of all the coefficients in the expansion of ${\left(1+x^2\right)}^n$, then :

1. A.$\mathrm{B}={\mathrm{A}}^3$
2. B.$3\mathrm{A}=\mathrm{B}$
3. C.$A=3B$
4. D.$A={\mathrm{B}}^3$

The coefficient of $x^{2012}$ in the expansion of $(1-x{)}^{2008}(1+x+x^2{)}^{2007}$ is equal to _______.

Let $\alpha$ be the constant term in the binomial expansion of ${\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)}^n$, $n\le 15$. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{-n}$ is $\lambda \alpha$, then $\lambda$ is equal to ___________.

The coefficient of $x^5$ in the expansion of ${\left(2x^3-\frac{1}{3x^2}\right)}^5$ is :

1. A.$\frac{26}{3}$
2. B.$\frac{80}{9}$
3. C.$9$
4. D.$8$

The sum, of the coefficients of the first 50 terms in the binomial expansion of $(1-x{)}^{100}$, is equal to

1. A.${}^{99}{\mathrm{C}}_{49}$
2. B.${}^{101}{\mathrm{C}}_{50}$
3. C.$-{}^{99}{\mathrm{C}}_{49}$
4. D.$-{}^{101}{\mathrm{C}}_{50}$

The mean of the coefficients of $x,x^2,\dots ,x^7$ in the binomial expansion of $(2+x{)}^9$ is _________.

The number of integral terms in the expansion of ${\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)}^{680}$ is equal to _________.

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x{)}^{n+2}$, which are in the ratio $1:3:5$, is equal to :

1. A.$63$
2. B.$92$
3. C.$25$
4. D.$41$

If the $1011^{\text{th}}$ term from the end in the binomial expansion of ${\left(\frac{4x}{5}-\frac{5}{2x}\right)}^{2022}$ is $1024$ times $1011^{\text{th}}$ term from the beginning, then $|x|$ is equal to

1. A.$\frac{5}{16}$
2. B.$8$
3. C.$12$
4. D.$15$

If the coefficient of $x^7$ in ${\left(ax-\frac{1}{b{x}^2}\right)}^{13}$ and the coefficient of $x^{-5}$ in ${\left(ax+\frac{1}{b{x}^2}\right)}^{13}$ are equal, then $a^4{b}^4$ is equal to :

1. A.22
2. B.33
3. C.44
4. D.11

The coefficient of $x^7$ in $(1-x+2x^3{)}^{10}$ is _________.

If the coefficients of $x$ and $x^2$ in $(1+x{)}^p(1-x{)}^q$ are 4 and $-5$ respectively, then $2p+3q$ is equal to :

1. A.66
2. B.60
3. C.69
4. D.63

If the coefficients of three consecutive terms in the expansion of $(1+x{)}^n$ are in the ratio $1:5:20$, then the coefficient of the fourth term is

1. A.$3654$
2. B.$1827$
3. C.$5481$
4. D.$2436$

If the ratio of the fifth term from the beginning 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 :

1. A.$30\sqrt{2}$
2. B.$60\sqrt{3}$
3. C.$60\sqrt{2}$
4. D.$30\sqrt{3}$

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

If the coefficient of $x^7$ in ${\left(a{x}^2+\frac{1}{2bx}\right)}^{11}$ and $x^{-7}$ in ${\left(ax-\frac{1}{3b{x}^2}\right)}^{11}$ are equal, then :

1. A.$243ab=64$
2. B.$32ab=729$
3. C.$64ab=243$
4. D.$729ab=32$

Let $\alpha >0$, be the smallest number such that the expansion of ${\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)}^{30}$ has a term $\beta x^{-\alpha },\beta \in \mathbb{N}$. Then $\alpha$ is equal to ___________.

The coefficient of $x^{-6}$, in the expansion of ${\left(\frac{4x}{5}+\frac{5}{2x^2}\right)}^9$, is

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

If the coefficient of $x^{15}$ in the expansion of ${\left(a{x}^3+\frac{1}{b{x}^{1/3}}\right)}^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of ${\left(a{x}^{1/3}-\frac{1}{b{x}^3}\right)}^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a,b)$ :

1. A.$a=3b$
2. B.$ab=1$
3. C.$ab=3$
4. D.$a=b$

The coefficient of $x^{301}$ in $(1+x{)}^{500}+x(1+x{)}^{499}+x^2(1+x{)}^{498}+\dots +x^{500}$ is :

1. A.${}^{500}{C}_{300}$
2. B.${}^{501}{C}_{200}$
3. C.${}^{500}{C}_{301}$
4. D.${}^{501}{C}_{302}$

Let the coefficients of three consecutive terms in the binomial expansion of $(1+2x{)}^n$ be in the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of those three terms, is ____.

Let K be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x{)}^{99}$. Let $a$ be the middle term in the expansion of ${\left(2+\frac{1}{\sqrt{2}}\right)}^{200}$. If $\frac{{}^{200}{C}_{99}K}{a}=\frac{2^{l}m}{n}$, where m and n are odd numbers, then the ordered pair $(l,\mathrm{n})$ is equal to

1. A.(50, 101)
2. B.(50, 51)
3. C.(51, 101)
4. D.(51, 99)

Let $y(x)=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$. Then $y^{\prime }-y^{\prime \prime }$ at $x=-1$ is equal to

1. A.$496$
2. B.$976$
3. C.$464$
4. D.$944$