Sum of Coefficients of All Terms

Q. The value of $C_0+2C_1+3C_2+4C_3+\dots +(n+1)C_n$ is
<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> ( where $C_r={}^n{C}_r$)

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

Q. If $(1+x+x^2{)}^{20}=a_0+a_{1}x+a_2{x}^2+\cdots +a_{40}{x}^{40}$, then the value of $a_0+3a_1+5a_2+\cdots +81a_{40}$ is:

1. $161\times 3^{20}$
2. $41\times 3^{40}$
3. $41\times 3^{20}$
4. $161\times 3^{40}$

Q.

Number of greatest binomial coefficients in the expansion of $(1+x{)}^{2n}$ and $(1+x{)}^{2n+1}$ are p and q respectively (where n is a positive integer). Find the value of p+2q

___

Q. If $C_r$ denotes coefficient of $x^r$ in $(1+x{)}^{99}$, then the value of $C_0-2C_1+3C_2-4C_3+\cdots -100C_{99}$ is

Q.

$\frac{{}^{11}{C}_0}{1}$ + $\frac{{}^{11}{C}_1}{2}$ + $\frac{{}^{11}{C}_2}{3}$+.............. $\frac{{}^{11}{C}_{10}}{11}$ =

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

2. $\frac{2^{11}-1}{6}$

3. $\frac{3^{11}-1}{6}$

4. $\frac{3^{11}-1}{6}$

Q. The value of ${}^{15}{C}_0+{}^{15}{C}_1+{}^{15}{C}_2+\dots +{}^{15}{C}_7$ is

1. $2^{14}$
2. $2^{15}$
3. $2^{13}$
4. $2^7$

Q.

Find the term independent of x in the expansion of $(x^2+2x^4)(x+\frac{1}{x}{)}^{32}$

1. 

2. 

3. 

4. 

Q. If the sum of the coefficients in the expansion of ${(a^2{x}^2-2ax+1)}^{51}$ vanishes, then the value of $a$ is

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

Q. The value of ${(}^{21}{C}_1{-}^{10}{C}_1)+{(}^{21}{C}_2{-}^{10}{C}_2)+{(}^{21}{C}_3{-}^{10}{C}_3)+{(}^{21}{C}_4{-}^{10}{C}_4)+...+{(}^{21}{C}_{10}{-}^{10}{C}_{10})$ is

1. $2^{21}-2^{11}$
2. $2^{21}-2^{10}$
3. $2^{20}-2^9$
4. $2^{20}-2^{10}$

Q. If $\sum _{r=0}^n{\left(\frac{{}^n{C}_{r-1}}{{}^n{C}_r+{}^n{C}_{r-1}}\right)}^3=\frac{25}{24},$ then the value of $n$ is

1. $10$
2. $5$
3. $20$
4. $15$

Q. Consider the expansion of $(1+x{)}^{2n+1}$
If the coefficients of $x^r$ and $x^{r+1}$ are equal in the expansion, then $r$ is equal to

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

Q. Any complex number in the polar form can be expressed in Euler's form as $\cos \theta +i\sin \theta =e^{i\theta }$. This form of the complex number is useful in finding the sum of series $\sum _{r=0}^n{}^n{C}_r(\cos \theta +i\sin \theta {)}^r$.
$\sum _{r=0}^n{}^n{C}_r(\cos r\theta +i\sin r\theta )=\sum _{r=0}^n{}^n{C}_r{e}^{ir\theta }=\sum _{r=0}^n{}^n{C}_r(e^{i\theta }{)}^r=(1+e^{i\theta }{)}^n$
Also, we know that the sum of binomial series does not change if $r$ is replaced by $n-r$. Using these facts, answer the following questions.

If $f\left(x\right)=\frac{\sum _{r=0}^{50}{}^{50}{C}_r\sin 2rx}{\sum _{r=0}^{50}{}^{50}{C}_r\cos 2rx}$, then the value of $f\left(\frac{\pi }{8}\right)$ is equal to

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

Q. The value of ${}^{10}{C}_1+{}^{10}{C}_3+{}^{10}{C}_5+{}^{10}{C}_7+{}^{10}{C}_9$ is

1. $2^7$
2. $2^8$
3. $2^9$
4. $2^{10}$

Q. If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_n{x}^n,$ then $\underset{0\le r<s\le n}{\sum \sum }(r+s)(C_r+C_s+C_r{C}_s)$ is equal to

1. $n^2\cdot 2^n-\frac{n}{2}[2^{2n}-{}^{2n}{C}_n]$
2. $n^2\cdot 2^n+\frac{n}{2}[2^{2n}-{}^{2n}{C}_n]$
3. $n^2\cdot 2^n+\frac{n}{2}[2^{2n}+{}^{2n}{C}_n]$
4. None of these

Q. The coefficient of $x^{50}$ in $(1+x{)}^{1000}+x(1+x{)}^{999}+x^2(1+x{)}^{998}+...+x^{1000}$ is

1. ${}^{1001}{C}_{50}$
2. ${}^{1000}{C}_{50}$
3. ${}^{1001}{C}_{51}$
4. ${}^{1000}{C}_{51}$

Q. The value of $\sum _{r=1}^{19}\frac{{}^{20}{C}_{r+1}\cdot (-1{)}^r}{2^{2r+1}}$ is

1. $2({\left(\frac{3}{4}\right)}^{20}+4)$
2. $-2({\left(\frac{3}{4}\right)}^{20}+4)$
3. $2({\left(\frac{3}{4}\right)}^{20}-4)$
4. $-2({\left(\frac{3}{4}\right)}^{20}-4)$

Q. In the expansion of ${(5^{\frac{1}{2}}+7^{\frac{1}{8}})}^{1024}$, the number of integral terms is .

1. 128
2. 129
3. 130
4. 131

Q. Any complex number in the polar form can be expressed in Euler's form as $\cos \theta +i\sin \theta =e^{i\theta }$. This form of the complex number is useful in finding the sum of series $\sum _{r=0}^n{}^n{C}_r(\cos \theta +i\sin \theta {)}^r$.
$\sum _{r=0}^n{}^n{C}_r(\cos r\theta +i\sin r\theta )=\sum _{r=0}^n{}^n{C}_r{e}^{ir\theta }=\sum _{r=0}^n{}^n{C}_r(e^{i\theta }{)}^r=(1+e^{i\theta }{)}^n$
Also, we know that the sum of binomial series does not change if $r$ is replaced by $n-r$. Using these facts, answer the following questions.

The value of $\sum _{r=0}^{100}{}^{100}{C}_r\left(\sin rx\right)$ is equal to

1. $2^{100}{\cos }^{100}\left(\frac{x}{2}\right)\sin 50x$
2. $2^{100}\sin \left(50x\right)\cos \left(\frac{x}{2}\right)$
3. $2^{101}{\cos }^{100}\left(50x\right)\sin \left(\frac{x}{2}\right)$
4. $2^{101}{\sin }^{100}\left(50x\right)\cos \left(50x\right)$

Q. The coefficient of $x^n$ in the expansion of $(1-x)(1-x{)}^n$ is :

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

Q. In the following equation given below, find the value of ${}^{\prime }{a}^{\prime }$ :

1. $a=11$
2. $a=19$
3. $a=17$
4. $a=12$

Q. The expression $\frac{2^2+1}{2^2-1}+\frac{3^2+1}{3^2-1}+\frac{4^2+1}{4^2-1}+..........+\frac{(2011{)}^2+1}{(2011{)}^2-1}$ lies in the interval

Q. For $r=0,\cdots ,10,If{A}_r,B_r$ and $C_r$ denote respectively the coefficient of $x^r$ in the expansions of $(1+x{)}^{10},(1+x{)}^{20},and(1+x{)}^{30}$. Then the value of ${\sum }_{r=1}^{10}{A}_r(B_{10}{B}_r-C_{10}{A}_r)$ is

1. $B_{10}-C_{10}$
2. $A_{10}(B_{10}^2-c_{10}{A}_{10})$
3. \N
4. $C_{10}-B_{10}$

Q. $\sum _{r=0}^n{r}^2.{(}^n{C}_r{)}^2$ is equal to

1. $n^2.{}^{2n-2}{C}_{n-1}$
2. $n^2.{}^{2n}{C}_n$
3. $\frac{n^2.2^{n-1}\left(\prod _{n-1}^{r=1}(2r-1)\right)}{(n-1)!}$
4. $\frac{2^n\left(\prod _{r=1}^n(2r-1)\right)}{n!}$

Q. The value of $\sum _{r=0}^n(-2{)}^r\left(\frac{{}^n{C}_r}{{}^{r+2}{C}_r}\right)$ is

1. $\frac{1}{n+1},$ when $n$ is odd
2. $\frac{1}{n+2},$ when $n$ is odd
3. $\frac{1}{n+1},$ when $n$ is even
4. $\frac{1}{n+2},$ when $n$ is even