Sum of Coefficients of All Terms

Q.

Let $A$ be a$3\times 3$matrix with $\text{det}\left(A\right)=4$. Let $R_i$ denote the $i^{th}$ row of $A$.

If a matrix $B$ is obtained by performing the operation $R_2\to 2R_2+5R_3$ on $2A$, then $\text{det}\left(B\right)$ is equal to:

1. $64$

2. $16$

3. $80$

4. $128$

Q. The value of ${2000}_{C_2}+{2000}_{C_5}+{2000}_{C_8}+...+{2000}_{C_{2000}}=?$

1. $\frac{2^{1999}-1}{3}$
2. $\frac{2^{1999}+1}{3}$
3. $\frac{2^{2000}+1}{3}$
4. $\frac{2^{2000}-1}{3}$

Q.

$\sum _{\mathrm{n}=0}^{\mathrm{m}}{}^{\mathrm{n}+\mathrm{r}}\mathrm{C}_{\mathrm{n}}$is equal to?

1. ${}^{\mathrm{n}+\mathrm{m}+1}\mathrm{C}_{\mathrm{n}+1}$

2. ${}^{\mathrm{n}+\mathrm{m}+2}\mathrm{C}_{\mathrm{n}}$

3. ${}^{\mathrm{n}+\mathrm{m}+3}\mathrm{C}_{\mathrm{n}-1}$

4. None of these

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 $a_0^2-a_1^2+a_2^2-\cdots -a_{19}^2$ is

1. $\frac{1}{2}{a}_{20}(1-a_{20})$
2. $\frac{1}{2}{a}_{20}(1+a_{20})$
3. $\frac{1}{2}{a}_{20}^2$
4. \N

Q. The value of ${}^{20}{C}_0+{}^{20}{C}_1+\cdots +{}^{20}{C}_8$ is

1. $2^{19}-\frac{({}^{20}{C}_{10}+{}^{20}{C}_9)}{2}$
2. $2^{19}-\frac{({}^{20}{C}_{10}+2\times {}^{20}{C}_9)}{2}$
3. $2^{19}-\frac{{}^{20}{C}_{10}}{2}$
4. $2^{19}-\frac{(2\times {}^{20}{C}_{10}+{}^{20}{C}_9)}{2}$

Q. If $A_r,B_r,C_r$ denotes the coefficients of $x^r$ in the expansion of $(1+x{)}^{10},(x+1{)}^{20},(1+x{)}^{30}$ respectively, 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. $0$
4. $C_{10}-B_{10}$

Q. If ${}^{2n+1}{C}_0+{}^{2n+1}{C}_1+{}^{2n+1}{C}_2+\cdots +{}^{2n+1}{C}_n=4^{11}$ then which of the following is/are correct ?

1. ${}^n{C}_0+\frac{{}^n{C}_1}{2}+\frac{{}^n{C}_2}{2^2}+\cdots \frac{{}^n{C}_n}{2^n}={\left(\frac{3}{2}\right)}^{11}$
2. ${}^n{C}_0+\frac{{}^n{C}_1}{2}+\frac{{}^n{C}_2}{2^2}+\cdots \frac{{}^n{C}_n}{2^n}={\left(\frac{2}{3}\right)}^{11}$
3. ${}^n{C}_0-\frac{{}^n{C}_1}{2}+\frac{{}^n{C}_2}{2^2}-\cdots +(-1{)}^n\frac{{}^n{C}_n}{2^n}={\left(\frac{1}{2}\right)}^{11}$
4. ${}^n{C}_0-\frac{{}^n{C}_1}{2}+\frac{{}^n{C}_2}{2^2}-\cdots +(-1{)}^n\frac{{}^n{C}_n}{2^n}={\left(\frac{2}{3}\right)}^{11}$

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. If the value of ${}^{12}{C}_2+{}^{13}{C}_3+{}^{14}{C}_4+...+{}^{999}{C}_{989}$ is ${}^{1000}{C}_a+b$, then $|a+b|=$
(where ($a+b)$ is a single digit integer)

Q. If the expansion of $\frac{1}{(1-ax)(1-bx)}=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n+\cdots$, then $a_n$ is (where $a\ne b,|ax|,|bx|<1$)

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

Q. Let $m,n\in N$ and $\text{gcd}(2,n)=1.$ If $30\left(\begin{array}{c}30\\ 0\end{array}\right)+29\left(\begin{array}{c}30\\ 1\end{array}\right)+\cdots +2\left(\begin{array}{c}30\\ 28\end{array}\right)+1\left(\begin{array}{c}30\\ 29\end{array}\right)=n\cdot 2^m,$ then $n+m$ is equal to
$(\text{Here}\left(\begin{array}{c}n\\ k\end{array}\right)={}^n{C}_k)$

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

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

Q. The sum of last three digits of $7^{100}-3^{100}$ is

Q. If $(1+x{)}^n=C_0+C_{1}x+\cdots +C_n{x}^n,$ then the value of $\sum _{r=0}^n\sum _{s=0}^n{C}_r{C}_s$ is equal to

1. $2^n$
2. $2^{2n}$
3. $2^{n+1}$
4. $2^{3n}$

Q. If ${(x^{2006}+\frac{1}{x^{2006}}+2)}^{2010}=a_0+a_{1}x+a_2{x}^2+....a_n{x}^n$, then the value of $2a_0-a_1-a_2+2a_3-a_4-a_5+2a_6-...$ is

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

Q. If $k$ and $n$ be the two positive integers such that $S_t=1^t+2^t+\cdots +n^t$, then $\sum _{r=1}^m{(}^{m+1}{C}_r{S}_r)$ is equal to:

1. $(n+1{)}^{m+1}-(n+1)$
2. $(n+1{)}^{m+1}+(n+1)$
3. $(n-1{)}^{m+1}-(n-1)$
4. None of these

Q.

What is $\frac{\mathrm{\pi }}{2}$radians in degree?

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. $\sum _{r=0}^{300}{a}_r{x}^r=(1+x+x^2+x^3{)}^{100}$. If $a=\sum _{r=0}^{300}{a}_r$, then $\sum _{r=0}^{300}r{a}_r$ is equal to

1. $300a$
2. $100a$
3. $150a$
4. $75a$

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

1. $n[2^{2n-1}-{}^{2n-1}{C}_{n-1}]$
2. $n[2^{2n-1}+{}^{2n-1}{C}_{n-1}]$
3. $2n[2^{2n-1}-{}^{2n-1}{C}_{n-1}]$
4. $2n[2^{2n-1}+{}^{2n-1}{C}_{n-1}]$

Q. Sum of the binomial coefficients in the expansion of ${\left(a+b+c+d+e\right)}^n$ is

1. $n^5$
2. $n+1$
3. $5^n$
4. $2^n$

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. If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+\cdots +C_n{x}^n,$ then the value of $\underset{0\le r<s\le n}{\sum \sum }(r\cdot s)C_r{C}_s$ is

1. $n^2[2^{2n-3}+\frac{1}{2}{}^{2n-2}{C}_{n-1}]$
2. $[2^{2n-3}-\frac{1}{2}{}^{2n-2}{C}_{n-1}]$
3. $n^2[2^{2n-3}-\frac{1}{2}{}^{2n-2}{C}_{n-1}]$
4. $[2^{2n-3}+\frac{1}{2}{}^{2n-2}{C}_{n-1}]$

Q. Consider $(1+x{)}^{2n}+{(1+2x+x^2)}^n=\sum _{r=0}^{2n}{a}_r{x}^r,n\in N.$ If $\sum _{r=0}^{2n}{a}_r=f\left(n\right),$ then

1. $\sum _{n=1}^{\infty }\frac{1}{f\left(n\right)}=\frac{1}{6}$
2. $\sum _{n=1}^{\infty }\frac{1}{f\left(n\right)}=\frac{3}{8}$
3. largest value of $p$ for which $f\left(5\right)$ is divisible by $2^p$ is $11$
4. largest value of $p$ for which $f\left(5\right)$ is divisible by $2^p$ is $9$

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

1. $\frac{1}{2}[2^{2n}-{}^{2n}{C}_n]$
2. $\frac{1}{4}[2^{2n}-{}^{2n}{C}_n]$
3. $\frac{1}{2}[2^{2n}+{}^{2n}{C}_n]$
4. $\frac{1}{2}[2^n-{}^{2n}{C}_n]$

Q. The sum $1\cdot {}^{20}{C}_1-2\cdot {}^{20}{C}_2+3\cdot {}^{20}{C}_3-\cdots -20\cdot {}^{20}{C}_{20}$ is equal to

Q. In the expansion of ${(1-x-x^2+x^3)}^6,$ the sum of the coefficients of $x$ is

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

Q. Let $(1+x)(1+x+x^2)(1+x+x^2+x^3)......(1+x+x^2+....+x^{30})=a_0+a_{1}x+a_2{x}^2+........+a_{465}{x}^{465}$

then sum of $a_0+a_2+a_4+........+a_{464}$ is

1. $\left(31\right)!$
2. $\frac{\left(31\right)!}{2}$
3. $\left(30\right)!$
4. $\frac{\left(60\right)!}{2}$

Q. If $a=3^{1/223}+1$ and $f\left(n\right)=\sum _{r=1}^n(-1{)}^{r-1}{}^n{C}_{r-1}\cdot a^{n-r}$ for all $n\ge 3$, then the value of $f\left(2007\right)+f\left(2008\right)$ is

1. $3^6$
2. $3^9$
3. $3^{12}$
4. $3^{15}$

Q. If ${}^{2n+1}{C}_0+{}^{2n+1}{C}_1+{}^{2n+1}{C}_2+\cdots +{}^{2n+1}{C}_n=4^{11}$ then which of the following is/are correct ?

1. ${}^n{C}_0+\frac{{}^n{C}_1}{2}+\frac{{}^n{C}_2}{2^2}+\cdots \frac{{}^n{C}_n}{2^n}={\left(\frac{3}{2}\right)}^{11}$
2. ${}^n{C}_0+\frac{{}^n{C}_1}{2}+\frac{{}^n{C}_2}{2^2}+\cdots \frac{{}^n{C}_n}{2^n}={\left(\frac{2}{3}\right)}^{11}$
3. ${}^n{C}_0-\frac{{}^n{C}_1}{2}+\frac{{}^n{C}_2}{2^2}-\cdots +(-1{)}^n\frac{{}^n{C}_n}{2^n}={\left(\frac{1}{2}\right)}^{11}$
4. ${}^n{C}_0-\frac{{}^n{C}_1}{2}+\frac{{}^n{C}_2}{2^2}-\cdots +(-1{)}^n\frac{{}^n{C}_n}{2^n}={\left(\frac{2}{3}\right)}^{11}$