General Term of Binomial Expansion

Q.

Write last two digits of the number $3^{400}$

Q. The sixth term in the expansion of ${[\sqrt{\left\{2^{\log (10-3^x)}\right\}}+\sqrt[5]{5\left\{2^{(x-2)\log 3}\right\}}]}^m$ is equal to $21$. If it is known that the binomial coefficient of the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion represents respectively the first, third and fifth terms of an A.P. (the symbol log stands for logarithm to the base $10$) then sum of possible values of $x$ is

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

Q. The smallest natural number $n,$ such that the cofficient of $x$ in the expansion of ${(x^2+\frac{1}{x^3})}^n$is ${}^n{C}_{23}$, is :

1. $23$
2. $38$
3. $35$
4. $58$

Q.

Let $S_k=$$\underset{r=1}{\overset{k}{\sum {\tan }^{-1}\left(\frac{6^r}{2^{r+1}+3^{2r+1}}\right)}}$. Then $\underset{k\to \infty }{lim}{S}_k=$

1. ${\tan }^{-1}\left(\frac{3}{2}\right)$

2. $co{t}^{-1}\left(\frac{3}{2}\right)$

3. $\frac{\mathrm{\pi }}{2}$

4. ${\tan }^{-1}\left(3\right)$

Q. Let $X=({(}^{10}{C}_1{)}^2+2{(}^{10}{C}_2{)}^2+3{(}^{10}{C}_3{)}^2)+...+10{(}^{10}{C}_{10}{)}^2),$ where ${}^{10}{C}_r,r\in \{1,2,...,10\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430}X$ is

Q. The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+(1+x{)}^{23}+\cdots +(1+x{)}^{30}$

1. ${}^{31}{C}_6-{}^{21}{C}_6$
2. ${}^{31}{C}_6+10\cdot {}^{21}{C}_6$
3. ${}^{31}{C}_6+{}^{21}{C}_6$
4. $2.{}^{31}{C}_6+3.{}^{21}{C}_6$

Q. If ${17}^{\text{th}}$ and ${18}^{\text{th}}$ terms in the expansion of $(2+a{)}^{50}$ are equal, then the value of $a$ is:

1. $2$
2. $3$
3. $0$
4. $1$

Q. If the third term in the binomial expansion of ${(1+x^{{\log }_{2}x})}^5$ equals $2560$, the a possible value of $x$ is :

1. $\frac{1}{8}$
2. $\frac{1}{4}$
3. $2\sqrt{2}$
4. $4\sqrt{2}$

Q. If some three consecutive coefficients in the binomial expansion of $(x+1{)}^n$ in powers of $x$ are in the ratio $2:15:70$, then the average of these three coefficients is :

1. $227$
2. $232$
3. $625$
4. $964$

Q.

If ${}^{n}P_r=840$ and ${}^{n}C_r=35$, then $n$ is equal to?

1. $1$

2. $3$

3. $5$

4. $7$

Q.

If the coefficient of x in ${(x^2+\frac{k}{x})}^5$ is 270, then k =

1. 3

2. 4

3. 5

4. None

Q. The $6^{\text{th}}$ term in the expansion of ${(2x^2-\frac{1}{3x^2})}^{10}$ is

1. $-\frac{35840}{9}$
2. $\frac{560}{243}$
3. $-\frac{896}{27}$
4. $-\frac{764}{25}$

Q. The positive value of $\lambda$ for which the co-efficient of $x^2$ in the expression $x^2{(\sqrt{x}+\frac{\lambda }{x^2})}^{10}$ is $720$, is :

1. $2\sqrt{2}$
2. $\sqrt{5}$
3. $3$
4. $4$

Q. In the expansion of ${\left(\right(5{)}^{\frac{1}{2}}+\left(7{)}^{\frac{1}{8}}\right)}^{1024},$ the number of integral terms is

1. $128$
2. $129$
3. $131$
4. $130$

Q. If the fourth term in the binomial expansion of ${(\sqrt{\frac{1}{x^{1+{\log }_{10}x}}}+x^{1/12})}^6$ is equal to $200$, and $x>1$, then the value of $x$ is

1. ${10}^4$
2. ${10}^3$
3. None of these
4. $100$
5. $10$

Q.

Total number of terms in the expansion of (1-x-x^2)^5 is ________

Q. If the last term in the binomial expansion of ${(2^{1/3}-\frac{1}{\sqrt{2}})}^n$ is ${\left(\frac{1}{3^{5/3}}\right)}^{{\log }_{3}8},$ then the $5^{\text{th}}$ term from the beginning is

1. $420$
2. $210$
3. $105$
4. $84$

Q. If the ${21}^{\text{st}}$ and ${22}^{\text{nd}}$ terms in the expansion of $(1-x{)}^{44}$ are equal, then the value of $|8x|$ is

Q.

If ${\left(81\right)}^{\frac{5}{x}}=243$, then $x=.............$

Q.

The number of odd proper divisors of $3^p·6^m·{21}^n$ is equal to$?$

Q. If the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion of $(x+a{)}^n$ are $240,720$ and $1080$ respectively, then the value of least term in the expansion is

1. $16$
2. $32$
3. $64$
4. $81$

Q.

Let $S_n\left(x\right)={\log }_{a^{\frac{1}{2}}}x+{\log }_{a^{\frac{1}{3}}}x+{\log }_{a^{\frac{1}{6}}}x+{\log }_{a^{\frac{1}{11}}}x........$ up to $n-$terms, where $a>1$ If $S_{24}\left(x\right)=1093and{S}_{12}\left(2x\right)=265$ then value of $a$ is equal to

Q.

The value of x in the expression $[x+x^{lo{g}_{10}\left(x\right)}{]}^5$, if the third term in the expansion is 10, 00, 000

1. 10

2. 11

3. 12

4. None of these

Q.

If $y={(x^2-1)}^m$, then the ${\left(2m\right)}^{th}$ differential coefficient of $y$ with respect to $x$ is:

1. $m$

2. $\left(2m\right)!$

3. $m!$

4. $2m$

Q. The number of all numbers having $5$ digits, with distinct digits is

1. $99999$
2. $9\times {}^9{P}_4$
3. ${}^{10}{P}_5$
4. ${}^9{P}_4$

Q. If the second term of the expansion ${[a^{\frac{1}{13}}+\frac{a}{\sqrt{a^{-1}}}]}^n$ is $14a^{\frac{5}{2}}$, then the value of $\frac{{}^n{C}_3}{{}^n{C}_2}$ is

Q. The total number of irrartional terms in the binomial expansion of ${(7^{1/5}-3^{1/10})}^{60}$ is :

1. $48$
2. $54$
3. $49$
4. $55$

Q.

Find the term independent of x in the expansion of (3x²/2-1/3x)⁹

Q. If the coefficients of $x^{–2}$ and $x^{–4}$ in the expansion of ${(x^{\frac{1}{3}}+\frac{1}{2x^{\frac{1}{3}}})}^{18},(x>0),$ are $m$ and $n$ respectively, then $\frac{m}{n}$ is equal to :

1. $\frac{4}{5}$
2. $182$
3. $27$
4. $\frac{5}{4}$

Q. The ratio of the coefficient of $x^2$ to the coefficient of $x^{10}$ in the expansion of ${(x^5+4\cdot 3^{-{\log }_{\sqrt{3}}\sqrt{x^3}})}^{10}$ is

1. $4:7$
2. $10:3$
3. $3:10$
4. $7:4$