Relation between Coefficient and Indices of x and y

Q. Coefficient of $x^{11}$in the expansion of $(1+x^2{)}^4(1+x^3{)}^7(1+x^4{)}^{12}$ is

1. 1120
2. 1051
3. 1106
4. 1113

Q. The coefficients of the (r-1)th, rth, (r+1)th terms in the expansion of $(x+1{)}^n$ are in the ratio of 1:3:5. Find both n and r ?

Q. If the coefficients of $r^{\text{th}},(r+1{)}^{\text{th}}\text{and}(r+2{)}^{\text{th}}$ terms in the expansion of $(1+x{)}^{14}$ are in A.P., then value $r$ can be

1. $5$
2. $12$
3. $10$
4. $9$

Q. Coefficient of $t^{24}$ in $(1+t^2{)}^{12}(1+t^{12}\left)\right(1+t^{24})$ is

1. ${}^{12}{C}_6+3$
2. ${}^{12}{C}_6+1$
3. ${}^{12}{C}_6$
4. ${}^{12}{C}_6+2$

Q.

If the coefficients of 2nd, 3rd and 4th terms in the expansion of $(1+x{)}^n$ are in A.P., then find the value of

Q.

The coefficient of $x^8{y}^{10}$ in the expansion of $(x+y{)}^{18}$ is

1. $2^{18}$

2. None of these

3. ${}^{18}{P}_{10}$

4. ${}^{18}{C}_8$

Q.

The coefficients of 2nd, 3rd and 4th terms in the expansion of $(1+x{)}^{2n}$ are in A.P., show that $2n^2-9n+7=0.$

Q. If the coefficients of second, third and fourth term in the expansion of $(1+x{)}^{2n}$ are in A.P, then $2n^2-9n+7$ is equal to

1. -1
2. 0
3. 1
4. 3/2

Q. If the coefficients of $5^{th},6^{th}$ and $7^{th}$ terms in the expansion of $(1+x{)}^n$ be in A.P, then n =

1. 7 only
2. 14 only
3. None of these
4. 7 or 14

Q. The quotient when $1+x^2+x^4+x^6+\cdots +x^{34}$ is divided by $1+x+x^2+x^3+\cdots +x^{17}$

1. $x^{17}-x^{15}+x^{13}-x^{11}+\cdots +x$
2. $x^{17}+x^{15}+x^{13}+x^{11}+\cdots +x$
3. $x^{17}+x^{16}+x^{15}+x^{14}+\cdots +1$
4. $x^{17}-x^{16}+x^{15}-x^{14}+\cdots -1$

Q. If coefficients of 2, 3 and 4 terms in the binomial expansion of $(1+x{)}^n$ are in A.P, then $n^2-9n$ is equal to

1. -7
2. 7
3. 14
4. -14

Q. If [.] denotes the greatest integer function, then $\underset{n\to \infty }{lim}\frac{\left[x\right]+\left[2x\right]+\left[3x\right]+....+\left[nx\right]}{n^2}$ is

1. $0$
2. $\frac{x}{2}$
3. $x$
4. $\frac{x^2}{2}$

Q. The coefficient of $x^6$ in the expansion of ${(1+\frac{x}{\angle 1}+\frac{x^2}{\angle 2}+\frac{x^3}{\angle 3}+\frac{x^4}{\angle 4}+\frac{x^5}{\angle 5})}^2$ is

1. 
2. 
3. 
4. 

Q. The coefficients of three consecutive terms of $(1+x{)}^{n+5}$ are in the ratio $5:10:14$. Then $n=$ ___

Q.

In the coefficients of (2r+4)th and (r-2)th terms in the expansion of $(1=x{)}^{18}$ are equal, find r.

Q. If coefficients of 2, 3 and 4 terms in the binomial expansion of $(1+x{)}^n$ are in A.P, then $n^2-9n$ is equal to

1. -7
2. 7
3. 14
4. -14

Q. If $C_0,C_1,C_2,.......C_n$ denote the coefficients in the expansion of ${(1+x)}^n$, prove that
$C_1+2C_2+3C_3+....n{C}_n=n.{\left(2\right)}^{n-1}$

Q. Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x{)}^2+(1+x{)}^3+...+(1+x{)}^{49}+(1+mx{)}^{50}\text{is}(3n+1{)}^{51}{C}_3$ for some positive interger $n$. Then the value of $n$ is ___

Q. Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+9y^2=9,$ then the ratio $a^2:b^2$ equals

1. $8:1$
2. $1:8$
3. $9:1$
4. $1:9$

Q. If the coefficients of $r^{\text{th}},(r+1{)}^{\text{th}}\text{and}(r+2{)}^{\text{th}}$ terms in the expansion of $(1+x{)}^{14}$ are in A.P., then value $r$ can be

1. $5$
2. $12$
3. $10$
4. $9$

Q. Let $P\left(n\right):n^2+n$ is an odd integer. It is seen that truth of $P\left(n\right)\Rightarrow$ the truth of P(n + 1). Therefore, P(n) is true for all

1. n
2. n > 1
3. n > 2
4. none of these

Q. $\frac{2x+1}{(x-1)(x^2+1)}=\frac{A}{x-1}+\frac{Bx+C}{x^2+1}\Rightarrow C=$

1. 1/2
2. 5/2
3. -1/2
4. 0

Q.

Which term in the expansion of ${\{{\left(\frac{x}{\sqrt{y}}\right)}^{1/3}+{\left(\frac{y}{x^{1/3}}\right)}^{1/2}\}}^{21}$ contains x and y to one and the same power?

Q. Let $f\left(n\right)=[\frac{1}{3}+\frac{3n}{100}]n$, where [n] denotes the greatest integer less than or equal to n. Then ${\sum }_{n=1}^{56}f\left(n\right)$ is equal to

1. $56$
2. $689$
3. $1287$
4. $1399$

Q. If coefficients of $5^{\text{th}},6^{\text{th}}$ and $7^{\text{th}}$ terms in the expansion of $(1+x{)}^n$ are in A.P., then the value(s) of $n$ is/are

1. $7$
2. $14$
3. $8$
4. $16$

Q.

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is 'a', the closest approach between two atoms in metallic crystal will be

1. $2a$

2. $2\sqrt{2}a$

3. $\sqrt{2}a$

4. $\frac{a}{\sqrt{2}}$

Q.

If ${}^{35}{C}_{n+7}={}^{35}{C}_{4n-2}$, then write the values of n.

Q. If the sum of the coefficients in the expansion of $(1+2x{)}^n$ is $2,187$, the greatest term in the expansion, if $x=\frac{1}{2}$ is/are

1. $5^{th}$
2. $4^{th}$ and $5^{th}$
3. $4^{th}$
4. None of these

Q. Find the coefficient of $x^{12}$ in $\frac{4+2x-x^2}{{(1+x)}^3}$

Q. Write the coefficient of $x^3$ in $\frac{3}{4}{x}^3+2x-3$.

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