General Term of Binomial Expansion

Q.

If the number of integral terms in the expansion of ${\left(3^{\frac{1}{2}}+5^{\frac{1}{8}}\right)}^n$ is exactly $33$, then the least value of n is

1. $128$

2. $248$

3. $256$

4. $264$

Q.

The coefficient of $x^n$ in expansion of ${\left(1+2x+3x^2+....\right)}^{\frac{1}{2}}$ is

1. $-1$

2. $0$

3. $2$

4. $1$

Q.

The sum of coefficients of even powers of $x$ in ${\left(x+\sqrt{x^3-1}\right)}^6+{\left(x-\sqrt{x^3-1}\right)}^6$

1. $23$

2. $24$

3. $18$

4. $21$

Q. The value of $x$, for which the $6th$ term in the expansion ${\{2^{{\log }_2\sqrt{(9^{x-1}+7)}}+\frac{1}{2^{\frac{1}{5}{\log }_2(3^{x-1}+1)}}\}}^7\text{is}84$, is equal to

1. $4$
2. $3$
3. $-2$
4. $1$

Q.

The 2nd, 3rd and 4th terms in the expansion of $(x+y{)}^n$ are 240, 720 and 1080 respectively. Find x, y, n.

1. n = 4, x = 2, y = 3

2. n = 5, x = 2, y = 3

3. n = 4, x = 3, y = 3

4. n = 6, x = 2, y = 3

Q.

$r^{th}$ term in the expansion of $(a+2x{)}^n$ is

1. $\frac{n(n+1).....(n-r+1)}{r!}{a}^{n-r+1}(2x{)}^r$

2. $\frac{n(n-1).....(n-r+2)}{(r-1)!}{a}^{n-r+1}(2x{)}^{r-1}$

3. $\frac{n(n+1).....(n-r)}{(r+1)!}{a}^{n-r}\left(x\right)r$

4. None of these

Q. Find the $6^{\text{th}}$ term of the expansion $(y^{1/2}+x^{1/3}{)}^n,$ if the value of coefficient of $3^{\text{rd}}$ term from the end is $45$

1. $232y^{5/2}{x}^{5/3}$
2. $252y^{5/2}{x}^{5/3}$
3. $232y^{5/3}{x}^{5/2}$
4. $252y^{5/3}{x}^{5/2}$

Q. For the expansion of $(\sqrt{2}+3^{\frac{1}{5}}{)}^{10},$ which of the following is (are) TRUE?

1. Sum of rational terms is $41$
2. There are two rational terms.
3. There are four rational terms.
4. Sum of rational terms is $61$

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

1. $48$
2. $49$
3. $54$
4. $55$

Q. The coefficient of $x^4$ in the expansion of $(x^2-x-2{)}^6$ is

1. $60$
2. $240$
3. $-280$
4. $-180$

Q. If $S=\frac{5}{1\times 3\times 7}+\frac{7}{3\times 5\times 9}+\frac{9}{5\times 7\times 11}+...,$ then the value of $45S$ is

Q. Find the $6^{\text{th}}$ term of the expansion $(y^{1/2}+x^{1/3}{)}^n,$ if the value of coefficient of $3^{\text{rd}}$ term from the end is $45$

1. $232y^{5/2}{x}^{5/3}$
2. $252y^{5/2}{x}^{5/3}$
3. $232y^{5/3}{x}^{5/2}$
4. $252y^{5/3}{x}^{5/2}$

Q.

If n is positive integer and three consecutive coefficients in the expansion of $(1+x{)}^n$ are in the ratio 6 : 33 : 110, then n =

1. 4

2. 6

3. 12

4. 16

Q.

If coefficient of $(2r+3{)}^{th}$ and $(r-1{)}^{th}$ terms in the expansion of $(1+x{)}^{15}$ are equal, then value of r is

1. 5

2. 6

3. 4

4. 3

Q. For the series $S=1+\frac{1}{(1+3)}(1+2{)}^2+\frac{1}{(1+3+5)}(1+2+3{)}^2+\frac{1}{(1+3+5+7)}(1+2+3+4{)}^2+\cdots$

1. $9^{\text{th}}$ term is $25$
2. $7^{\text{th}}$ term is $16$
3. Sum of first ten terms is $\frac{505}{4}$
4. Sum of first ten terms is $505$

Q. Let $X=({(}^{10}{C}_1{)}^2+2{(}^{10}{C}_2{)}^2+3{(}^{10}{C}_3{)}^2)+...+10{(}^{10}{C}_{10}{)}^2),$ where ${}^{10}{C}_r,r\in \{1,2,...,10\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}X$ is

Q. The sixth term in the expansion of ${[\sqrt{\left\{2^{\log (10-3^x)}\right\}}+\sqrt[5]{5\left\{2^{(x-2)\log 3}\right\}}]}^m$ is equal to $21$. If it is known that the binomial coefficient of the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion represents respectively the first, third and fifth terms of an A.P. (the symbol log stands for logarithm to the base $10$), then minimum value of expression is

1. $64$
2. $32$
3. $128$
4. None of these

Q.

If the $(r+1{)}^{th}$ term in the expansion of ${(\sqrt[3]{3\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{3a}}})}^{21}$ has the same power of a and b, then value of r is

1. 9

2. 10

3. 8

4. 6

Q. If the sixth term from the beginning is equal to the sixth term from the end in the expansion of ${(2^{1/5}+3^{-1/5})}^n$, then the sum of the rational terms in the expansion is

1. $\frac{1349}{9}$
2. $\frac{1549}{9}$
3. $\frac{1759}{9}$
4. $\frac{1119}{9}$

Q. The number of rational terms in the expansion of ${(9^{1/4}+4^{1/6})}^{1000}$

1. $167$
2. $153$
3. $247$
4. $183$

Q. $Therthtermintheexpansionof(a+b{)}^{n}is(\begin{array}{c}n\\ r\end{array})a^r{b}^{n-r}$

1. False
2. True

Q. $\underset{n\to \infty }{\text{lim}}\frac{1\cdot n^2+2{(n-1)}^2+3{(n-2)}^2+\cdots +n\cdot 1^2}{1^3+2^3+\cdots +n^3}$ is equal to

1. $\frac{4}{3}$
2. $\frac{8}{3}$
3. $\frac{1}{3}$
4. $\frac{2}{3}$

Q. The sixth term in the expansion of ${[\sqrt{\left\{2^{\log (10-3^x)}\right\}}+\sqrt[5]{5\left\{2^{(x-2)\log 3}\right\}}]}^m$ is equal to $21$. If it is known that the binomial coefficient of the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion represents respectively the first, third and fifth terms of an A.P. (the symbol log stands for logarithm to the base $10$), then sum of possible values of $x$ is

1. $1$
2. $3$
3. $4$
4. $2$

Q.

If the ratio of the coefficient of third and fourth term in the expansion of $(x-\frac{1}{2x}{)}^n$ is 1:2, then the value of n will be

1. 18

2. 16

3. 12

4. -10

Q.

If the coefficients of $p^{th}$, $(p+1{)}^{th}$ and $(p+2{)}^{th}$ terms in the expansion of $(1+x{)}^n$ are in A.P., then

1. $n^2-2np+4p^2$ = 0

2. $n^2-n(4p+1)+4p^2-2$ = 0

3. $n^2-n(4p+1)+4p^2$ = 0

4. None of these

Q.

In $(\sqrt[3]{32}+\frac{1}{\sqrt[3]{33}}{)}^n$ if the ratio of $7^{th}$ term from the beginning to the $7^{th}$ term from the end is $\frac{1}{6}$, then n =

1. 7

2. 8

3. 9

4. None of these

Q. If the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion of $(x+a{)}^n$ are $240,720$ and $1080$ respectively, then the sum of odd numbered terms in the expansion is

1. $1664$
2. $2376$
3. $1562$
4. $1486$

Q.

If the coefficient of $x^7$ in $(a{x}^2+\frac{1}{bx}{)}^{11}$ is equal to the coefficient of $x^{-7}$ in $(ax-\frac{1}{b{x}^2}{)}^{11}$, then ab =

1. 1

2. $\frac{1}{2}$

3. 2

4. 3

Q. If first three terms of $(1+ax{)}^n$ are $1,6x,16x^2$ respectively, then the value of $(18+n{)}^a$ is

Q. In the expansion of ${(x^3-\frac{1}{x^2})}^n,n\in N$, if the sum of the coefficients of $x^5$ and $x^{10}$ is $0$, then $n$ is

1. $25$
2. $20$
3. $15$
4. $10$