Integration to Solve Modified Sum of Binomial Coefficients

Q. The value of $2c_0+\frac{2^2}{2}{C}_1+\frac{2^3}{3}{C}_2+\frac{2^4}{4}{C}_3+....+\frac{2^{11}}{11}{C}_{10}is$

1. $\frac{3^{11}-1}{11}$
2. $\frac{2^{11}-1}{11}$
3. $\frac{{11}^3-1}{11}$
4. $\frac{3^2-1}{11}$

Q. The sum to (n+1) terms of the following series $\frac{C_0}{2}-\frac{C_1}{3}+\frac{C_2}{4}-\frac{C_3}{5}+\cdots$ is

1. $\frac{1}{n+1}$
2. $\frac{1}{n+2}$
3. $\frac{1}{n(n+1)}$
4. None of these

Q. $\sum _{r=0}^{r=n}\frac{\left(}{\begin{array}{c}n\\ r\end{array}}$

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

Q. $\int \sqrt{x^2+2x+5}dx$ is equal to
$($where $C$ is integration constant$)$

1. $\ln |x+\sqrt{x^2+2x+5}|+C$
2. $\frac{1}{2}(x+1)\sqrt{x^2+2x+5}+2\ln |x+1+\sqrt{x^2+2x+5}|+C$
3. $\frac{1}{2}{\tan }^{-1}\left[\frac{x+1}{2}\right]+C$
4. $\frac{1}{2}(x+1)\sqrt{x^2+2x+5}+2{\sin }^{-1}\left(\frac{x+1}{2}\right)+C$

Q. The value of
$100\underset{n\to 200}{lim}[C_1-(1+\frac{1}{2})C_2+(1+\frac{1}{2}+\frac{1}{3})C_3-\cdots +(-1{)}^{n-1}(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n})C_n]$ is (where $C_r={}^n{C}_r$)

Q.

The value of $\frac{C_1}{2}$ + $\frac{C_3}{4}$ + $\frac{C_5}{6}$ + ...... is equal to

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

2. n.$2^n$

3. $\frac{2^n}{n}$

4. $2^{n-1}$