Middle Terms

Q. In the expansion of ${(7^{\frac{1}{3}}+{11}^{\frac{1}{9}})}^{6561}$

1. There are exactly $730$ rational terms
2. There are exactly $5831$ irrational terms
3. The term which involves greatest binomial coefficients is irrational
4. The term which involves greatest binomial coefficients is rational

Q. Prove by the Principle of Mathematical Induction:$1+2+2^2+...+2^n=2^{n+1}-1$ for all natural numbers n.

Q.

Fifth term of a GP is $2$, then the product of its $9$ terms is

1. $256$

2. $512$

3. $1024$

4. None of these

Q.

The number of terms in the expansion of ${\left(a+b+c\right)}^{10}$ is

1. $11$

2. $21$

3. $55$

4. $66$

Q. The sum of the real values of $x$ for which the middle term in the binomial expansion of ${(\frac{x^3}{3}+\frac{3}{x})}^8$ equals $5670$ is :

1. $0$
2. $4$
3. $6$
4. $8$

Q. The greatest coefficient in the expansion of $(x+\frac{1}{x}{)}^{2n}$ is

1. $\frac{2n!}{{(\frac{n}{2}!)}^2}$
2. None of these
3. $\frac{2n!}{n!n!}$
4. $\frac{\mathrm{1.3.5....}(2n-1)}{n!}{2}^n$

Q.

Middle term in the expansion of $(1+3x+3x^2+x^3{)}^6$ is

1. $4^{th}$

2. $3^{rd}$

3. ${10}^{th}$

4. None of these

Q. If the middle term in the expansion of ${(\frac{p}{2}+2)}^8$ is $1120$, then the value of $p$ is

1. $\pm 2$
2. $\pm 4$
3. $\pm 1$
4. $\pm 6$

Q.

If rth term is the middle term in the expansion of ${(x^2-\frac{1}{2x})}^{20}$, then $(r+3{)}^{th}$ term is

1. ${}^{20}{C}_{14}\left(\frac{x}{2^{14}}\right)$

2. ${}^{20}{C}_{12}{x}^2{2}^{-12}$

3. ${-}^{20}{C}_{7}x{.2}^{-13}$

4. None of these

Q.

The sum of coefficients of the two middle terms in the expansion of $(1+x{)}^{2n-1}$ is equal to ${}^{2n-1}{C}_n$. State true or false.

Q. If the middle term in the expansion of ${(\frac{p}{2}+2)}^8$ is $1120$, then the value of $p$ is

1. $\pm 2$
2. $\pm 4$
3. $\pm 1$
4. $\pm 6$

Q. If the middle term in the expansion of ${(\frac{1}{x}+x\sin x)}^{10}$ is $\frac{63}{8},$ then the value of $6{\sin }^{2}x+\sin x-2$ is

1. $0$
2. $\frac{3}{2}$
3. $\frac{61}{8}$
4. $5$

Q.

The inventor of the chessboard suggested a reward of one grain of wheat for the first square; $2$ grains for the second; $4$ grains for the third and so on, doubling the number of grains for subsequent squares. How many grins would be given to the inventor?

Q.

Write the coefficient of the middle term in the expansion of $(1+x{)}^{2n}.$

Q. The middle term in the expansion of ${(ax-\frac{b}{x^2})}^{12}$ is:

1. $\frac{984a^6{b}^6}{x^6}$
2. $\frac{924a^5{b}^5}{x^5}$
3. $\frac{628a^6{b}^6}{x^6}$
4. $\frac{924a^6{b}^6}{x^6}$

Q.

Find the middle term in the expansion of:

$\left(i\right){(\frac{2}{3}x-\frac{3}{2x})}^{20}$

$\left(ii\right){(\frac{a}{x}+bx)}^{12}$

$\left(iii\right){(x^2-\frac{2}{x})}^{10}$

$\left(iv\right){(\frac{x}{a}-\frac{a}{x})}^{10}$

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. The coefficient of the term independent of $a$ in the expansion of ${(1+\sqrt{a}+\frac{1}{\sqrt{a}-1})}^{-30}$ is

1. ${}^{30}{C}_{20}$
2. $0$
3. ${}^{30}{C}_{10}$
4. None of these

Q. The middle term(s) in the expansion of ${(3x-\frac{x^3}{6})}^7$ is/are:

1. $\frac{-105x^{13}}{8}$
2. $\frac{35x^{15}}{48}$
3. $\frac{35x^{13}}{73}$
4. $\frac{105x^{15}}{48}$

Q. If the middle term in the expansion of ${(\frac{x}{2}+2)}^8$is $1120$; then $x\in R$ is equal to

1. $-2$
2. $3$
3. $-3$
4. $2$

Q.

The Coefficient of $x^r$ in the expansion of ${\left(1+x\right)}^{-2}$ is

1. $r$

2. $r+1$

3. $r+3$

4. $r-1$

Q. The middle term of expnsion of ${(\frac{10}{x}+\frac{x}{10})}^{10}$

1. ${}^7{C}_5$
2. ${}^8{C}_5$
3. ${}^9{C}_5$
4. ${}^{10}{C}_5$

Q. The coefficient of term independent of $x$ in the expansion of${(\sqrt{\frac{x}{3}}-\frac{\sqrt{3}}{x^2})}^{10},$ is

1. $\frac{5}{3}$
2. $6$
3. $\frac{4}{5}$
4. $\frac{1}{2}$

Q.

If $r$th and $\left(r+1\right)$th terms in the expansion of ${\left(p+q\right)}^n$ are equal, then the value of $\frac{(n+1)q}{r(p+q)}$ is

1. $0$

2. $1$

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

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

Q. For natural number $m$, $n$, if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+...,$ and $a_1=a_2=10$, then

1. $m<n$
2. $m>n$
3. $m+n=80$
4. $m-n=20$

Q. For the natural numbers $m,n,$ if $(1-y{)}^m(1+y{)}^n=$
$1+a_{1}y+a_2{y}^2+...+a_{m+n}{y}^{m+n}$ and $a_1=a_2=10,$ then the value of $(m+n)$ is equal to:

1. $64$
2. $80$
3. $88$
4. $100$

Q.

The expression${\left[x+{\left(x^3-1\right)}^{\frac{1}{2}}\right]}^5+{\left[x-{(x^3-1)}^{\frac{1}{2}}\right]}^5$ is a polynomial of degree.

1. $5$

2. $6$

3. $7$

4. $8$

Q. The coefficient of $x^6.y^{-2}$ in the expansion of ${(\frac{x^2}{y}-\frac{y}{x})}^{12}$ is

1. 
2. 
3. 
4. 

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

1. $10$
2. $12$
3. $14$
4. $6$

Q. The coefficient of the term independent of $x$ in the expansion of ${(\sqrt{\frac{x}{3}}+\frac{\sqrt{3}}{x^2})}^{10}$ is

1. $\frac{5}{9}$
2. $\frac{5}{3}$
3. $\frac{4}{3}$
4. $\frac{1}{3}$