Sum of Product of Binomial Coefficients

Q. The sum of the series $2C_0+\frac{C_1}{2}\cdot 2^2+\frac{C_2}{3}\cdot 2^3+\frac{C_3}{4}\cdot 2^4+\cdots +\frac{C_n}{n+1}\cdot 2^{n+1}$ is equal to , where$\left(C_r{=}^n{C}_r\right)$

1. $\frac{3^{n+1}-1}{n+1}$
2. $\frac{3^{n+1}+1}{n+1}$
3. $\frac{3^{n+1}-1}{n-1}$
4. $\frac{3^{n-1}-1}{n+1}$

Q.

The term independent of x in the expansion of $(1+x{)}^n(1+\frac{1}{x}{)}^n$ is :

1. 

2. 

3. 

4. 

Q.

If the coefficients of $x^{7}and{x}^8$ in the expansion of ${(2+\frac{x}{3})}^n$ are equal then n =

1. 55

2. 35

3. 27

4. 45

Q.

$\left({}^{30}{C}_0\right)\left({}^{30}{C}_{10}\right)-\left({}^{30}{C}_1\right)\left({}^{30}{C}_{11}\right)+.....\left({}^{30}{C}_{20}\right)\left({}^{30}{C}_{30}\right)$ is equal to

1. 

2. 

3. 

4. 

Q. The value of
$(\frac{30}{0})(\frac{30}{10})-(\frac{30}{1})(\frac{30}{11})+(\frac{30}{2})(\frac{30}{12})...+(\frac{30}{20})(\frac{30}{30})$ is; where $(\frac{n}{r}){=}^n{C}_r$

1. ${(}_{10}^{30})$
2. ${(}_{15}^{30})$
3. ${(}_{30}^{60})$
4. ${(}_{10}^{31})$

Q. If $(1+x{)}^n=\sum _{r=0}^n{}^n{C}_r{x}^n,$ then $\frac{C_0}{1\cdot 2}{2}^2+\frac{C_1}{2\cdot 3}{2}^3+\frac{C_2}{3\cdot 4}{2}^4+\cdots +\frac{C_n}{(n+1)(n+2)}{2}^{n+2}$ is equal to

1. $\frac{3^{n+2}-2n-5}{(n+1)(n+2)}$
2. $\frac{3^{n+2}+2n-5}{(n+1)(n+2)}$
3. $\frac{3^{n+2}-2n+5}{(n+1)(n+2)}$
4. $\frac{3^{n+2}+2n+5}{(n+1)(n+2)}$

Q. If ${}^n{C}_0,{}^n{C}_1,{}^n{C}_2,\cdots ,{}^n{C}_n$ denote the binomial coefficients in the expansion of $(1+x{)}^n$ and $p+q=1$ , then $\sum _{r=0}^n{r}^2{}^n{C}_r{p}^r{q}^{n-r}$ is

1. $np$
2. None of these
3. $npq$
4. $n^2{p}^2+npq$

Q. The coefficient of $x^n{y}^n$ in the expansion of $\left\{\right(1+x\left)\right(1+y\left)\right(x+y){\}}^n$ is

1. $\sum _{r=0}^n{C_r}^3$
2. $\sum _{r+s=0}^n{}^n{C}_r\cdot {{}^n{C}_s}^2$
3. None of these
4. $\sum _{r=0}^n{C_r}^2$

Q.

$\int \frac{x^2}{x^4-x^2-12}dx$

Q. The term independent of $x$ in the expansion of ${(2x+\frac{1}{x})}^{10}$ is

1. $6520$
2. $8464$
3. $6720$
4. $8064$

Q. Term independent of $x$ in the expansion of ${(x-\frac{1}{x^2})}^{3n}$ (where $n$ is even natural number)

1. $\frac{3n!}{n!\times 2n!}$
2. $\frac{3n!}{n!}$
3. $\frac{3n!}{2n!}$
4. $\frac{3n!}{(n!{)}^2}$

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

1. 64
2. 2187
3. 729
4. 243

Q. For n >3 , ${1.2}^n{C}_r-{2.3}^n{C}_{r-1}+.....+(-1{)}^r(r+1\left)\right(r+2)$ is

1. 
2. 
3. 
4. 

Q. If $f\left(x\right)$ is continuous for all real values of $x$, then $\sum _{r=1}^n\underset{0}{\overset{1}{\int }}f(r-1+x)dx$ is equal to, where n is a natural number

1. $\underset{0}{\overset{n}{\int }}f\left(x\right)dx$
2. $\underset{0}{\overset{1}{\int }}f\left(x\right)dx$
3. $n\underset{0}{\overset{1}{\int }}f\left(x\right)dx$
4. $(n-1)\underset{0}{\overset{1}{\int }}f\left(x\right)dx$

Q. If the coefficient of $x^{7}and{x}^{8}in{(2+\frac{x}{3})}^n$ are equal, then n is equal to

1. 45
2. 15
3. 56
4. 55

Q. If the sum of the series ${}^{12}{C}_5+7{.}^{12}{C}_6+21{.}^{12}{C}_7+35{.}^{12}{C}_8+35{.}^{12}{C}_9+21{.}^{12}{C}_{10}+{7.}^{12}{C}_{11}{+}^{12}{C}_{12}$ is equal to ${}^{2n+1}{C}_{12}$ then $n$ is

Q. If ${}^n{C}_0,{}^n{C}_1,{}^n{C}_2,\cdots ,{}^n{C}_n$ denote the binomial coeffients in the expansion of $(1+x{)}^n$ and $p+q=1,$ then $\sum _{r=0}^{n}r{}^n{C}_r{p}^r{q}^{n-r}$ is

1. $np$
2. None of these
3. $n{p}^2$
4. $npq$

Q. The sum of the coefficients of all odd degree terms in the expansion of ${(x+\sqrt{x^3-1})}^5+{(x-\sqrt{x^3-1})}^5$ $,(x>1)$ is

1. $1$
2. $0$
3. $-1$
4. $2$

Q. For integers $n$ and $r,$ let $\left(\begin{array}{c}n\\ r\end{array}\right)=\{\begin{array}{cc}{}^n{C}_r,& \text{if}n\ge r\ge 0\\ 0,& \text{otherwise}\end{array}$

The maximum value of $k$ for which the sum $\sum _{i=0}^k\left(\begin{array}{c}10\\ i\end{array}\right)\left(\begin{array}{c}15\\ k-i\end{array}\right)+\sum _{i=0}^{k+1}\left(\begin{array}{c}12\\ i\end{array}\right)\left(\begin{array}{c}13\\ k+1-i\end{array}\right)$ exists, is equal to

Q. $\underset{x\to 0}{lim}\frac{a^x+a^{-x}-2}{x^2}$ is equal to

1. $\log a$
2. $(\log a{)}^2$
3. $0$
4. None of these

Q. Find the sum of n terms of the series $\frac{5}{1.2}\cdot \frac{1}{3}+\frac{7}{2.3}\cdot \frac{1}{3^2}+\frac{9}{3.4}\cdot \frac{1}{3^3}+\frac{11}{4.5}\cdot \frac{1}{3^4}+....$

Q. If $(1+x{)}^n=\sum _{r=0}^n{}^n{C}_r{x}^r$ and $\sum _{r=0}^n\frac{1}{{}^n{C}_r}=a,$ then the value of $\sum _{0\le i\le n}\sum _{0\le j\le n}(\frac{i}{{}^n{C}_i}+\frac{j}{{}^n{C}_j})$ is equal to

1. $n^{2}a$
2. $\frac{n^2}{2a}$
3. $\frac{na}{2}$
4. $na(n+1)$

Q. If $(1+x{)}^n=\sum _{r=0}^n{}^n{C}_r{x}^n,$ then $\frac{C_0}{1\cdot 2}{2}^2+\frac{C_1}{2\cdot 3}{2}^3+\frac{C_2}{3\cdot 4}{2}^4+\cdots +\frac{C_n}{(n+1)(n+2)}{2}^{n+2}$ is equal to

1. $\frac{3^{n+2}+2n-5}{(n+1)(n+2)}$
2. $\frac{3^{n+2}-2n+5}{(n+1)(n+2)}$
3. $\frac{3^{n+2}-2n-5}{(n+1)(n+2)}$
4. $\frac{3^{n+2}+2n+5}{(n+1)(n+2)}$

Q.

The sum of the first $\mathrm{n}$ terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+......$ is equal to

1. $2^{2\mathrm{n}}-\mathrm{n}-1$

2. $1-2^{-\mathrm{n}}$

3. $\mathrm{n}+2^{-\mathrm{n}}-1$

4. $2^{\mathrm{n}}+1$

Q. ${\sum }_{k=1}^{10}\frac{(-1{)}^{k-1}}{K}.\left({10}_{C_K}\right)=$

1. 
2. 
3. 
4. 

Q. The value of ${\sum }_{r=0}^n(-1{)}^r\left(\frac{{}^n{C}_r}{{}^{r+3}{C}_r}\right)$ is

1. $n$
2. $\frac{3!}{2(n+3)}$
3. $3!\left(n^2\right)$
4. 0

Q. $\text{Statement 1}:\text{The sum of the series}1+(1+2+4)+(4+6+9)+....+(361+380+400)=8000$

$\text{Statement 2}$ :
$\sum _{k=1}^n(k^3-(k-1{)}^3)=n^3$, for any natural number, $n$

1. statement $1$ is false, statement $2$ is true
2. statement $1$ is true , statement $2$ is true ; statement $2$ is a correct explanation for statement $1$
3. statement $1$ is true , statement $2$ is true ; statement $2$ is not a correct explanation for statement $1$
4. statement $1$ is true , statement $2$ is false

Q. Find: $a^3+\frac{1}{a^3}$, if $a+\frac{1}{a}=5$

Q. The value of $r$ for which ${}^{20}{C}_r{}^{20}{C}_0+{}^{20}{C}_{r-1}{}^{20}{C}_1+{}^{20}{C}_{r-2}{}^{20}{C}_2+\dots +{}^{20}{C}_0{}^{20}{C}_r$ is maximum, is :

1. $15$
2. $10$
3. $20$
4. $11$

Q.

Find the sum of the series ${}^n{C}_0{}^n{C}_2+{}^n{C}_1{}^n{C}_3+{}^n{C}_2{}^n{C}_4.....{}^n{C}_{n-2}{}^n{C}_n$

1. ${}^{2n}{C}_{n-1}$

2. ${}^n{C}_{n-2}$

3. ${}^{2n}{C}_{n-2}$

4. $\frac{{}^{2n}{C}_{n-2}}{2}$