Middle Terms

Q.

Show that the middle term in the expansion of $(1+x{)}^{2n}$ is $\frac{1.3.5\cdots (2n-1)}{n!}.2^n.X^n,\text{where}nϵN.$

Q. For natural numbers m, n if ${(1-y)}^m{(1+y)}^n=1+a_{1}y+a_2{y}^2+...and{a}_1=a_2=10,then(m,n)$ is

1. 
2. 
3. 
4. 

Q.

${16}^{th}$ term in the expansion of $(\sqrt{x}-\sqrt{y}{)}^{17}$ is

1. 

2. 

3. 

4. 

Q.

The middle term in the expansion of (x + y)¹⁰ is the

1. 6^th term

2. 7^th term

3. average of 5^th and 6^th terms

4. 5^th term

Q. The sum of middle terms in the expansion of ${(2a-\frac{a^2}{4})}^9$ is

1. 
2. 
3. 
4. 

Q.

The coefficient of the middle term in the binomial expansion in powers of $x$ in the expansion of ${(1+\alpha x)}^4$ and ${(1-\alpha x)}^6$ is the same if $\alpha$ equals

1. $-\left(\frac{5}{3}\right)$

2. $\frac{10}{3}$

3. $-\left(\frac{3}{10}\right)$

4. $\left(\frac{3}{5}\right)$

Q.

The middle term in the expansion of $(x^2+\frac{1}{x^2}+2{)}^n$ is:

1. 

2. 

3. 

4. 

Q.

Middle term in the expansion of

$(1+3x+3x^2+x^3{)}^6$ is

1. 

2. 

3. 

4. None of these

Q.

Find the middle terms in the expansion of ${(3x-\frac{x^3}{6})}^7$

Or

Find the fourth term from the end in the expansion of ${(\frac{3}{x^2}-\frac{x^3}{6})}^7$

Q.

If $x^p$ occurs in the expansion of ${(x^2+\frac{1}{x})}^{2n}$
Prove that its coefficient is
$\frac{\left(2n\right)!}{\{\left(\frac{4n-p}{3}\right)!\}\times \left\{\left(\frac{2n+p}{3}\right)\right\}!}$

Q.

Find the number of terms in the expansion of $(x_1+x_2+x_3....x_k{)}^n$

1. 

2. 

3. 

4. 

Q.

$\underset{x\to 0}{lim}[\frac{1}{x}-\frac{lo{g}_e(1+x)}{x^2}]=$

1. 

2. 

3. -1

4. 1

Q.

Find the value of ${}^n{C}_0$ 4 + $4^2$ × $\frac{{}^n{C}_1}{2}$ + ......... $\frac{4^{n+1}}{n+1}$ × ${}^n{C}_n$

1. 

2. 

3. 

4. 

Q. The coefficient of the middle term in the binomial expansion of $(1+\alpha x{)}^4$ and $(1-\alpha x{)}^6$ is same if $\alpha$ equals .

1. $-\frac{5}{3}$
2. $-\frac{3}{5}$
3. $-\frac{3}{10}$
4. $-\frac{10}{3}$

Q.

Find the middle term(s) in the expansion of $(a+b{)}^{21}$

1. 

2. 

3. 

4. 

Q.

The middle term in the expansion of $(1+x{)}^{2n}$ is

1. 

2. 

3. 

4. 

Q. If the coefficient of the middle term in the expansion of $(1+x{)}^{2n+2}$ is $\alpha$ and the coefficients of middle terms in the expansion of $(1+x{)}^{2n+1}$ are $\beta$ and $\gamma$, then relation between $\alpha ,\beta$ and $\gamma$ is-

1. $\alpha +\beta +\gamma =0$
2. $\alpha =\beta +\gamma$
3. $\beta =\alpha +\gamma$
4. $\gamma =\beta +\alpha$

Q. If the sum of $n$ terms of an A.P. is ${3n}^2+5n$ and its $m^{th}$ term is $164$. Find the value of $m$.

Q.

The value of $i^{1+3+5+......+(2n+1)}$ is

1. i if n is even, - i if n is odd

2. 1 if n is even, - 1 if n is odd

3. i if n is even, - 1 if n is odd

4. 1 if n is odd, - 1 if n is even

Q. Find the middle terms in the expansions of:
${(3-\frac{x^3}{6})}^7$

Q.

$\underset{x\to \infty }{lim}(1+\frac{1}{x}{)}^x=$

1. e

2. 

3. 1

4. 

Q.

The coefficient of $x^7$ in the expansion of $\frac{(1-x-x^2+x^3{)}^6}{(1-x{)}^{12}}$ is

___

Q.

The middle term in the expansion of $(1+x{)}^{2n}$ is

1. 

2. 

3. 

4. 

Q. Find the middle terms in the expansion of ${(\frac{x}{3}+9y)}^{10}$.

Q.

Find the middle term(s) in the expansion of $(a+b{)}^{20}$

1. 

2. 

3. None of these

4. 

Q.

$x^2+a^{2}x+b=0and{x}^2+x+1=0$ have a common roots which of the following is true.

1. 

2. 

3. 

4. 

Q. The coefficient of the middle term in the binomial expansion in powers of x of ${(1+\alpha x)}^4$ and of ${(1-\alpha x)}^4$is the same if α =

1. 
2. 
3. 
4. 

Q.

The coefficient of the middle term in the binomial expansion in powers of x of $(1+\alpha x{)}^4$ and of $(1-\alpha x{)}^6$ is the same if $\alpha$ equals

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

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

3. $-\frac{3}{10}$

4. $-\frac{10}{3}$

Q.

Find the number of middle terms in the expansion of $(a+b{)}^{20}$.

__

Q. The middle term of ${(1-3x+3x^2-x^3)}^{2n}$ is

1. 
2. 
3. 
4.