Relation between Coefficient and Indices of X and Y

Q.

Coefficient of x in the expansion of $(x^2+\frac{a}{x}{)}^5$ is

1. 

2. 

3. 

4. 

Q. If in the expansion of $(1+x{)}^m(1–x{)}^n$ the coefficients of x and $x^2$ are 3 and –6 respectively, then m is

1. 6
2. 9
3. 12
4. 24

Q.

The coefficient of three consecutive terms in the expansion of $(1+x{)}^n$ are in the ratio 1 : 7 : 42. Find r and n. Also, determine the square root of n + 1-r.

Or

Find the coefficient of the term independent of x in the expansion of ${(\frac{x+1}{x^{\frac{2}{3}}-x^{\frac{1}{3}}+1}-\frac{x-1}{x-x^{\frac{1}{2}}})}^{10}$.

Q.

Two successive terms in the expansion of $(1+x{)}^{24}$, whose coefficients are in the ratio 4:1 are $(r+1{)}^{th}$ and $(r{)}^{th}$, r <15. Find the value of r

___

Q.

If $(1+x+2x^2{)}^{20}=a_0+a_{1}x+a_2{x}^2.....a_{40}{x}^{40}$, then $a_0+a_2+a_4....a_{38}$ =

1. 

2. 

3. 

4. None of these

Q.

The coefficient of two consecutive terms in the

expansion of $(1+x{)}^n$ will be equal, if

1. n is an odd integer

2. n is an even integer

3. None of these

4. n is any integer

Q. If the numerical coefficient of the $p^{th}$ term in the expansion of $(2x+3{)}^6$ is 4860, then the value(s) of p is /are

1. 3
2. 4
3. 5
4. 2

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.

The sum of coefficients in the expansion of $(1+x-3x^2{)}^{171}$ is

1. 2

2. 0

3. 1

4. -1

Q.

Find the coefficient of $x^{11}$ in the expansion $(1-6x^6)(1-x{)}^{-7}$

1. -

2. -

3. -

4. -

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

Q.

Find ${}^n{C}_0$ - 2${}^n{C}_1$ + 3${}^n{C}_2$ - 4${}^n{C}_3$ .....................+ $(-1{)}^n$ (n+1) ${}^n{C}_n$

1. 0

2. (n+2)

3. (n+1)

4. (n+1)

Q.

The coefficients of three successive terms in the

expansion of $(1+x{)}^n$ are 165, 330 and 462

respectively, then the value of n will be

1. 10

2. 12

3. 8

4. 11

Q. The coefficients of $x^r$ in the expansion of
$1+(1+x)+(1+x{)}^2+.....+(1+x{)}^n$ is ${}^{n+1}{C}_{r+1}$

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

Q.

If the coefficients of $a^{r-1},a^r$ and $a^{r+1}$ in the binomial expansion $(1+a{)}^n$ are in the arithmetic progression, prove that $n^2-n(4r+1)+4r^2-2=0$

Or

The 2nd, 3rd and 4th terms in the expansion of $(x+y{)}^n$ are 240, 720 and 1080, respectively. Find the values of x, y and n.

Q.

(i) Find the value of r, if the coefficients of (2r + 2)th and (r-2)th terms in the expansion of (1+x)^{18} are equal.

(ii) Find the coefficient of $x^{-}17$ in the expansion of ${(x^4-\frac{1}{x^3})}^{15}$

(iii) Find the term independent of x in the expansion of ${(2x-\frac{1}{x})}^{10}$

Q.

In the expansion $(1+x{)}^5=1+5x+a{x}^2......x^5$, find the value of a.

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

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

Q. If ${(x^2+2x+4)}^n={\sum }_{r=0}^{2n}{a}_r{x}^r,then{\sum }_{r=n}^{2r}\left(\frac{{}^{a}2n-r}{a_r}\right)$ is

1. 
2. 
3. 
4. 

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

3. $\sqrt{2}a$

4. $2\sqrt{2}a$

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

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

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

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 ___