Binomial Theorem · JEE Main previous year questions

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

Let ${}^n{C}_r$ denote the binomial coefficient of $x^r$ in the expansion of $(1+x{)}^n$. If ${\sum }_{k=0}^{10}(2^2+3k){}^{10}{C}_k=\alpha \cdot 3^{10}+\beta \cdot 2^{10},\alpha ,\beta \in R$, then $\alpha +\beta$ is equal to ____.

If n is the number of irrational terms in the expansion of ${\left(3^{1/4}+5^{1/8}\right)}^{60}$, then $(n-1)$ is divisible by :

1. A.30
2. B.8
3. C.7
4. D.26

The maximum value of the term independent of 't' in the expansion of ${\left(t{x}^{\frac{1}{5}}+\frac{(1-x{)}^{\frac{1}{10}}}{t}\right)}^{10}$ where $x\in (0,1)$ is :

1. A.$\frac{10!}{\sqrt{3}(5!{)}^2}$
2. B.$\frac{2\cdot 10!}{3\sqrt{3}(5!{)}^2}$
3. C.$\frac{10!}{3(5!{)}^2}$
4. D.$\frac{2\cdot 10!}{3(5!{)}^2}$

If the constant term in the binomial expansion of ${\left(\sqrt{x}-\frac{k}{x^2}\right)}^{10}$ is 405, then $|k|$ equals :

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

If the term independent of x in the expansion of ${\left(\frac{3}{2}{x}^2-\frac{1}{3x}\right)}^9$ is k, then 18 k is equal to :

1. A.5
2. B.9
3. C.7
4. D.11

Let $\alpha >0$, $\beta >0$ be such that ${\alpha }^3+{\beta }^2=4$. If the maximum value of the term independent of x in the binomial expansion of ${\left(\alpha x^{\frac{1}{9}}+\beta x^{-\frac{1}{6}}\right)}^{10}$ is 10k, then k is equal to :

1. A.176
2. B.336
3. C.352
4. D.84

The coefficient of $x^4$ is the expansion of $(1+x+x^2{)}^{10}$ is _______.

In the expansion of ${\left(\frac{x}{\cos \theta }+\frac{1}{x\sin \theta }\right)}^{16}$, if ${\ell }_1$ is the least value of the term independent of x when $\frac{\pi }{8}\le \theta \le \frac{\pi }{4}$ and ${\ell }_2$ is the least value of the term independent of x when $\frac{\pi }{16}\le \theta \le \frac{\pi }{8}$, then the ratio ${\ell }_2:{\ell }_1$ is equal to :

1. A.$8:1$
2. B.$16:1$
3. C.$1:8$
4. D.$1:16$

If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of ${\left(x+\sqrt{x^2-1}\right)}^6+{\left(x-\sqrt{x^2-1}\right)}^6$, then

1. A.$\alpha +\beta =60$
2. B.$\alpha -\beta =60$
3. C.$\alpha +\beta =-30$
4. D.$\alpha -\beta =-132$

The coefficient of $x^7$ in the expression $(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\dots +x^{10}$ is:

1. A.120
2. B.330
3. C.420
4. D.210

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

1. A.126
2. B.-84
3. C.-126
4. D.84

The term independent of x in the expansion of $\left(\frac{1}{60}-\frac{x^8}{81}\right){\left(2x^2-\frac{3}{x^2}\right)}^6$ is equal to :

1. A.36
2. B.-108
3. C.-36
4. D.-72

If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $(1+ax+b{x}^2)(1-3x{)}^{15}$ in powers of x, then the ordered pair (a,b) is equal to :

1. A.(28, 861)
2. B.(28, 315)
3. C.(-21, 714)
4. D.(-54, 315)

The smallest natural number n, such that the coefficient of x in the expansion of ${\left(x^2+\frac{1}{x^3}\right)}^n$ is ${}^n{\mathrm{C}}_{23}$, is :

1. A.23
2. B.58
3. C.38
4. D.35

If the fourth term in the binomial expansion of ${\left(\frac{2}{x}+x^{{\log }_{8}x}\right)}^6$ (x > 0) is $20\times 8^7$, then a value of x is :

1. A.$8^{-2}$
2. B.$8^2$
3. C.$8^3$
4. D.8

If some three consecutive coefficients in the binomial expansion of $(x+1{)}^n$ in powers of x are in the ratio $2:15:70$, then the average of these three coefficients is :-

1. A.625
2. B.227
3. C.964
4. D.232

If the fourth term in the binomial expansion of ${\left(\sqrt{x^{\left(\frac{1}{1+{\log }_{10}x}\right)}}+x^{\frac{1}{12}}\right)}^6$ is equal to 200, and x > 1, then the value of x is :

1. A.100
2. B.$10^3$
3. C.10
4. D.$10^4$

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of ${\left(2^{1/3}+\frac{1}{2(3{)}^{1/3}}\right)}^{10}$ is :

1. A.$1:2(6{)}^{1/3}$
2. B.$1:4(6{)}^{1/3}$
3. C.$2(36{)}^{1/3}:1$
4. D.$4(36{)}^{1/3}:1$

The total number of irrational terms in the binomial expansion of $(7^{1/5}-3^{1/10}{)}^{60}$ is :

1. A.54
2. B.55
3. C.49
4. D.48

The coefficient of $x^2$ in the expansion of the product $(2-x^2).((1+2x+3x^2{)}^6+(1-4x^2{)}^6)$ is :

1. A.107
2. B.106
3. C.108
4. D.155

If n is the degree of the polynomial, $${\left[\frac{2}{\sqrt{5x^3+1}-\sqrt{5x^3-1}}\right]}^8+{\left[\frac{2}{\sqrt{5x^3+1}+\sqrt{5x^3-1}}\right]}^8$$ and m is the coefficient of $x^n$ in it, then the ordered pair (n, m) is equal to :

1. A.$(24,(10{)}^8)$
2. B.$(8,5(10{)}^4)$
3. C.$(12,(20{)}^4)$
4. D.$(12,8(10{)}^4)$

The sum of the co-efficients of all odd degree terms in the expansion of ${\left(x+\sqrt{x^3-1}\right)}^5+{\left(x-\sqrt{x^3-1}\right)}^5,(x>1)$ is

1. A.$2$
2. B.$-1$
3. C.$0$
4. D.$1$

The coefficient of $x^{-5}$ in the binomial expansion of ${\left(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}}\right)}^{10}$, where $x\ne 0,1$, is :

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

If the number of terms in the expansion of ${\left(1-\frac{2}{x}+\frac{4}{x^2}\right)}^n$, $x\ne 0$, is 28, then the sum of the coefficients of all the terms in this expansion, is :

1. A.$243$
2. B.$729$
3. C.$64$
4. D.$2187$