Binomial Theorem · JEE Main previous year questions

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x{)}^{26}$ and $(1-\alpha x{)}^{28}$, $\alpha \ne 0$, are equal, then the value of $\alpha$ is:

1. A.1
2. B.$\frac{14}{13}$
3. C.$\frac{27}{7}$
4. D.$\frac{7}{27}$

If the sum of the coefficients of $x^7$ and $x^{14}$ in the expansion of ${\left(\frac{1}{x^3}-x^4\right)}^n$, $x\ne 0$, is zero, then the value of $n$ is ____.

The coefficient of $x^2$ in the expansion of ${\left(2x^2+\frac{1}{x}\right)}^{10}$, $x\ne 0$, is :

1. A.$3240$
2. B.$3360$
3. C.$3480$
4. D.$3600$

Let the smallest value of $k\in \mathbb{N}$, for which the coefficient of $x^3$ in $(1+x{)}^3+(1+x{)}^4+(1+x{)}^5+...+(1+x{)}^{99}+(1+kx{)}^{100}$, $x\ne 0$, is $\left(43n+\frac{101}{4}\right)\left({}^{100}{\mathrm{C}}_3\right)$ for some $n\in \mathbb{N}$, be $p$. Then the value of $p+n$ is:

1. A.10
2. B.11
3. C.12
4. D.13

Pairs of elements with the same number of electrons in their respective 4f orbital are [Atomic number. Eu-63, Gd-64, Dy-66, Ho-67, Tm-69, Yb-70, Lu-71, Hf-72] A. (Eu and Gd) B. (Dy and Ho) C. ( Yb and Hf ) D. ( Lu and Tm) Choose the correct answer from the options given below :

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

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x{)}^{1000}+x(1+x{)}^{999}+x^2(1+x{)}^{998}+\cdots +x^{1000}$ is :

1. A.${}^{1002}{C}_{501}$
2. B.${}^{1001}{C}_{501}$
3. C.${}^{1000}{C}_{501}$
4. D.${}^{1002}{C}_{500}$

The coefficient of $x^{48}$ in $(1+x)+2(1+x{)}^2+3(1+x{)}^3+\dots +100(1+x{)}^{100}$ is equal to

1. A.$100\cdot {}^{100}{\mathrm{C}}_{49}-{}^{100}{\mathrm{C}}_{48}$
2. B.$100\cdot {}^{101}{\mathrm{C}}_{49}-{}^{101}{\mathrm{C}}_{50}$
3. C.${}^{100}{\mathrm{C}}_{50}+{}^{101}{\mathrm{C}}_{49}$
4. D.$100\cdot {}^{100}{\mathrm{C}}_{49}-{}^{100}{\mathrm{C}}_{50}$

If the coefficient of $x$ in the expansion of $(a{x}^2+bx+c)(1-2x{)}^{26}$ is -56 and the coefficients of $x^2$ and $x^3$ are both zero, then $a+b+c$ is equal to :

1. A.1483
2. B.1300
3. C.1500
4. D.1403

For an integer $n\ge 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y{)}^{2n-3}$ is $16$, then the distance of the point $\mathrm{P}\left(2n-1,n^2-4n\right)$ from the line $x+y=8$ is

1. A.$\sqrt{2}$
2. B.$2\sqrt{2}$
3. C.$5\sqrt{2}$
4. D.$3\sqrt{2}$

In the expansion of ${\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)}^n,n\in \mathbb{N}$, if the ratio of $15^{\mathrm{th}}$ term from the beginning to the $15^{\mathrm{th}}$ term from the end is $\frac{1}{6}$, then the value of ${}^n{\mathrm{C}}_3$ is

1. A.$4960$
2. B.$2300$
3. C.$1040$
4. D.$4060$

The sum of all rational terms in the expansion of $(2+\sqrt{3}{)}^8$ is :

1. A.16923
2. B.18817
3. C.3763
4. D.33845

The term independent of $x$ in the expansion of ${\left(\frac{(x+1)}{(x^{2/3}+1-x^{1/3})}-\frac{(x-1)}{(x-x^{1/2})}\right)}^{10},x>1$, is :

1. A.240
2. B.120
3. C.150
4. D.210

For some $n\ne 10$, let the coefficients of the $5\text{ th}$, $6\text{ th}$ and $7\text{ th}$ terms in the binomial expansion of $(1+x{)}^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+x{)}^{n+4}$ is:

1. A.$10$
2. B.$35$
3. C.$70$
4. D.$20$

Suppose $A$ and $B$ are the coefficients of $30^{\mathrm{th}}$ and $12^{\mathrm{th}}$ terms respectively in the binomial expansion of $(1+x{)}^{2n-1}$. If $2\mathrm{A}=5\mathrm{B}$, then $\mathrm{n}$ is equal to:

1. A.20
2. B.19
3. C.22
4. D.21

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

1. A.8
2. B.20
3. C.13
4. D.18

The coefficient of $x^{70}$ in $x^2(1+x{)}^{98}+x^3(1+x{)}^{97}+x^4(1+x{)}^{96}+\cdots +x^{54}(1+x{)}^{46}$ is ${}^{99}{\mathrm{C}}_p-{}^{46}{\mathrm{C}}_q$. Then a possible value of $p+q$ is :

1. A.61
2. B.83
3. C.55
4. D.68

The sum of the coefficients of $x^{2/3}$ and $x^{-2/5}$ in the binomial expansion of ${\left(x^{2/3}+\frac{1}{2}{x}^{-2/5}\right)}^9$ is

1. A.19/4
2. B.69/16
3. C.63/16
4. D.21/4

If the term independent of $x$ in the expansion of ${\left(\sqrt{a}{x}^2+\frac{1}{2x^3}\right)}^{10}$ is 105 , then $a^2$ is equal to :

1. A.6
2. B.4
3. C.2
4. D.9

Let $0\le r\le n$. If ${}^{n+1}{\mathrm{C}}_{r+1}:{}^n{\mathrm{C}}_r:{}^{n-1}{\mathrm{C}}_{r-1}=55:35:21$, then $2n+5r$ is equal to :

1. A.62
2. B.60
3. C.55
4. D.50

If $\mathrm{S}(x)=(1+x)+2(1+x{)}^2+3(1+x{)}^3+\cdots +60(1+x{)}^{60},x\ne 0$, and $(60{)}^2\mathrm{S}(60)=a(b{)}^b+b$, where $a,b\in N$, then $(a+b)$ equal to ____.

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

1. A.724
2. B.742
3. C.693
4. D.639

The sum of all rational terms in the expansion of ${\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)}^{15}$ is equal to :

1. A.633
2. B.6131
3. C.3133
4. D.931

If the coefficient of $x^{30}$ in the expansion of ${\left(1+\frac{1}{x}\right)}^6{\left(1+x^2\right)}^7{\left(1-x^3\right)}^8;x\ne 0$ is $\alpha$, then $|\alpha |$ equals ____.

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

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

In the expansion of $(1+x)\left(1-x^2\right){\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)}^5,x\ne 0$, the sum of the coefficients of $x^3$ and $x^{-13}$ is equal to ____.