Sum of Coefficients of All Terms

Q.

What is the integral of $\cos \left(2\theta \right)$?

Q.

The sum of $n$ terms of the series $2^2+4^2+6^2+....$is

1. $\frac{n(n+1)(2n+1)}{6}$

2. $\frac{2n(n+1)(2n+1)}{3}$

3. $\frac{n(n+1)(2n+1)}{3}$

4. none of these

Q. The middle term in the expansion of ${(x^2+\frac{1}{x})}^n$ is $924x^6$. If $n$ is even, then $n$ is equal to

1. $10$
2. $12$
3. $24$
4. $6$

Q.

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

1. 

2. 

3. 

4. 

Q. The sum of the coefficients of even power of $x$ in the expansion of ${(1+x+x^2+x^3)}^5$ is

1. $256$
2. $128$
3. $512$
4. $64$

Q.

If$a,b$ and $c$ are the greatest values of ${}^{19}C_p,{}^{20}C_q,{}^{21}C_r$, then:

1. $(a/11)=(b/22)=(c/42)$

2. $(a/10)=(b/11)=(c/42)$

3. $(a/11)=(b/22)=(c/21)$

4. $(a/10)=(b/11)=(c/21)$

Q. If $C_0,C_1,C_2,\cdots C_n$, are the binomial coefficients of the expansion $(1+x{)}^n$, where $n$ is even, then $C_0+(C_0+C_1)+(C_0+C_1+C_2)+\cdots \cdots +(C_0+C_1+C_2+\cdots +C_{n-1})=$

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

Q.

If $(n+2)!=2550\times n!$then the value of $n$ is equal to

1. $48$

2. $49$

3. $50$

4. $51$

5. $52$

Q.

For $a\in R$ (the set of all real numbers), $a\ne -1$, $\underset{\to \infty }{\lim }\frac{(1^a+2^a+3^a+.....+n^a)}{{(n+1)}^{a-1}\left[\right(na+1)+(na+2)+....+(na+n\left)\right]}=\frac{1}{60}$. Then $a=$

1. $5$

2. $7$

3. $\frac{-15}{2}$

4. $\frac{-17}{2}$

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. The value of ${}^{20}{C}_0-{}^{20}{C}_1+{}^{20}{C}_2-{}^{20}{C}_3+\dots +{}^{20}{C}_{10}$ is

1. $-\frac{{}^{20}{C}_{10}}{2}$
2. ${}^{20}{C}_{10}$
3. $-{}^{20}{C}_{10}$
4. $\frac{{}^{20}{C}_{10}}{2}$

Q.

Let $\left[x\right]$ denotes the greatest integer less than or equal of $x$. If $x={(\sqrt{3}+1)}^5,$ then $\left[x\right]$ is equal to

1. $75$

2. $50$

3. $76$

4. $51$

5. $152$

Q. If ${}^1{P}_1+2\cdot {}^2{P}_2+3\cdot {}^3{P}_3+\cdots +15\cdot {}^{15}{P}_{15}={}^q{P}_r-s,0\le s\le 1,$ then ${}^{q+s}{C}_{r-s}$ is equal to

Q.

$\sum _{i=1}^n\sum _{j=1}^1\sum _{k=1}^{j}1$ is equal to

1. $\frac{n(n+1)}{2}$

2. $\frac{n(n+1)(n+2)}{6}$

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

4. None of these

Q.

If the coefficients of $T_r,T_{r+1},T_{r+2}$ terms of $(1+x{)}^{14}$ are in A.P., then r =

1. 7

2. 6

3. 8

4. 9

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

Q.

The coefficient of the term independent of $x$ in the expansion of ${\left\{\left[\frac{\left(x+1\right)}{\left(x^{{\displaystyle \frac{2}{3}}}-x^{{\displaystyle \frac{1}{3}}}+1\right)}\right]-\left[\frac{\left(x-1\right)}{\left(x-x^{{\displaystyle \frac{1}{2}}}\right)}\right]\right\}}^{10}$

1. $210$

2. $105$

3. $70$

4. $112$

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.

The sum of the series $1·3^2+2·5^2+3·7^2+........$upto $20$ terms is

1. $188090$

2. $189080$

3. $199080$

4. None of these

Q. The value of ${}^{20}{C}_0-{}^{20}{C}_1+{}^{20}{C}_2-{}^{20}{C}_3+\dots +{}^{20}{C}_{10}$ is

1. $\frac{{}^{20}{C}_{10}}{2}$
2. $-\frac{{}^{20}{C}_{10}}{2}$
3. ${}^{20}{C}_{10}$
4. $-{}^{20}{C}_{10}$

Q.

Let r and n be positive integers such that $1\le r\le n.$ Then prove the following :

(i) $\frac{{}^n{C}_r}{{}^n{C}_{r-1}}=\frac{n-r+1}{r}$
(ii) $n{}^{n-1}{C}_{r-1}=(n-r+1{)}^n{C}_{r-1}$
(iii) $\frac{{}^n{C}_r}{{}^{n-1}{C}_{r-1}}=\frac{n}{r}$
(iv) ${}^n{C}_r+2^n{C}_{r-1}{+}^n{C}_{r-2}{=}^{n+2}{C}_r$

Q.

If the sum of the coefficients in the expansion of $(a+b{)}^n$ is 4096, then the greatest coefficient in the expansion is

1. 1594

2. 792

3. 924

4. 2924

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.

If the function $f:N\to N$ is defined by $f\left(x\right)=\sqrt{x}$, then $\left[\frac{f\left(25\right)}{f\left(16\right)+f\left(1\right)}\right]$ is equal to:

1. $\frac{5}{6}$

2. $\frac{5}{7}$

3. $\frac{5}{3}$

4. $1$

Q. The coefficient of $x^n$ in the polynomial $(x+{}^n{C}_0)(x+3\cdot {}^n{C}_1)\cdots (x+(2n+1)\cdot {}^n{C}_n)$ is

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

Q.

Suppose $A$ is a $3\times 3$ matrix. If for every column vector $X$, $X^{T}AX=0$ and $a_{23}=-1008$ , then the sum of the digits of $a_{32}$ is:

Q. If $C_0,C_1,C_2,\cdots ,C_n$ denote the binomial coefficients in the expansion of $(1+x{)}^n$ and $k=\sum _{r=0}^n(-1{)}^r{}^n{C}_r\frac{1+r{\log }_{e}10}{(1+{\log }_e{10}^n{)}^r},$ then $|k+3|$ is equal to

Q. Sum upto $n$ terms for series $\frac{C_0}{1\cdot 2}+\frac{C_1}{2\cdot 3}+\frac{C_2}{3\cdot 4}+\cdots$ is
( where $C_r={}^n{C}_r$)

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

Q. The absolute value of the sum of all the coefficients in the binomial expansion of ${(x^2+x-3)}^{319}$

Q. The value of $\sum _{k=0}^7[\frac{\left(\begin{array}{c}7\\ k\end{array}\right)}{\left(\begin{array}{c}14\\ k\end{array}\right)}\cdot \sum _{r=k}^{14}\left(\begin{array}{c}r\\ k\end{array}\right)\left(\begin{array}{c}14\\ r\end{array}\right)],$ where $\left(\begin{array}{c}n\\ r\end{array}\right)$ denotes ${}^n{C}_r$ is $a^b.$ Then the value of $a+b$ is (where $a$ and $b$ are coprime numbers)