Coefficients of Terms Equidistant from Beginning and End

Q. The number of rational terms in the binomial expansion of ${(4^{\frac{1}{4}}+5^{\frac{1}{6}})}^{120}$ is

Q. If $\left(\frac{3^6}{4^4}\right)k$ is the term, independent of $x,$ in the binomial expansion of ${(\frac{x}{4}-\frac{12}{x^2})}^{12},$ then $k$ is equal to

Q. The ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of ${(2^{\frac{1}{3}}+\frac{1}{2(2{)}^{\frac{1}{3}}})}^{10}$ is:

1. $1:2(6{)}^{\frac{1}{3}}$
2. $2(36{)}^{\frac{1}{3}}:1$
3. $4(36{)}^{\frac{1}{3}}:1$
4. $1:4(16{)}^{\frac{1}{3}}$

Q. If the number of integral terms in the expansion of ${(3^{1/2}+5^{1/8})}^n$ is exactly $33,$ then the least value of $n$ is:

1. $128$
2. $248$
3. $256$
4. $264$

Q. The term independent of $x$ in the expansion of ${(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}})}^{10}$, where $x\ne 0,1$ is equal to

Q. If $a,b,c$ and $d$ are any four consecutive coefficients in the expansion of $(1+x{)}^n$, then $\frac{a+b}{a},\frac{b+c}{b},\frac{c+d}{c}$ are in

1. A.P.
2. G.P.
3. H.P.
4. A.G.P.

Q. The coefficient of $x^m$ in $(1+x{)}^m+(1+x{)}^{m+1}+\cdots +(1+x{)}^n,$ where $m,n\in N$ and $m<n$, is

1. ${}^n{C}_m$
2. ${}^n{C}_{m+1}$
3. ${}^{n+1}{C}_{m+1}$
4. ${}^{n+2}{C}_{m+2}$

Q. If the ratio of the $5^{\text{th}}$ term from the beginning to the $5^{\text{th}}$ term from the end in the expansion of ${(\sqrt[4]{42}+\frac{1}{\sqrt[4]{43}})}^n$ is $\sqrt{6}:1,$ then the value of $n$ is

1. $12$
2. $8$
3. $14$
4. $10$

Q. For a positive integer $n$ the binomial expression ${(1+\frac{1}{x})}^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $2:5:12,$ then $n$ is equal to

Q. The coefficient of $t^8$ in ${(1+t)}^2{(1+t+t^2+....+t^9)}^3$ is

1. $144$
2. $145$
3. $146$
4. $147$

Q. If the constant term in the binomal expansion of ${(\sqrt{x}-\frac{k}{x^2})}^{10}$ is $405$, then $\left|k\right|$ equals:

1. $1$
2. $3$
3. $2$
4. $9$

Q. Find the value of ${}^n{C}_r{+}^n{C}_{r-1}$.

1. 
2. 
3. 
4. 

Q. Find $r$ if (i)${{}^{5}P}_r=2{{}^{6}P}_{r-1}$(ii)${{}^{5}P}_r={{}^{6}P}_{r-1}$

Q. If the term independent of $x$ in the expansion of ${(\frac{3}{2}{x}^2-\frac{1}{3x})}^9$ is $k$, then $18k$ is equal to:

1. $5$
2. $9$
3. $7$
4. $11$

Q.

Find $n$ in the binomial

$(\sqrt[3]{32}+\frac{1}{\sqrt[3]{33}}{)}^n$ if the ratio of $7^{th}$ term from the beginning to the $7^{th}$ term from the end is $\frac{1}{6}$

Q. If the $5^{th}$ term in the expansion of ${(3\sqrt{x}+\frac{1}{x})}^n$ is independent of $x$, then $n$ $=$

1. $8$
2. $12$
3. $16$
4. $20$

Q. Value of $(1+\frac{1}{3})(1+\frac{1}{3^2})(1+\frac{1}{3^4})(1+\frac{1}{3^8})...(1+\frac{1}{3^{2n}})$ is equal to

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

Q. The number of integral terms in the expansion of ${(\sqrt{3}+\sqrt[8]{85})}^{256}$ is

1. $40$
2. $33$
3. $26$
4. $42$

Q. The ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of ${(2^{\frac{1}{3}}+\frac{1}{2(2{)}^{\frac{1}{3}}})}^{10}$ is:

1. $1:2(6{)}^{\frac{1}{3}}$
2. $2(36{)}^{\frac{1}{3}}:1$
3. $4(36{)}^{\frac{1}{3}}:1$
4. $1:4(16{)}^{\frac{1}{3}}$