Greatest Binomial Coefficients

Q.

Find the answer for $64$ power $\frac{1}{6}$.

Q. The term independent of $x$ in the expansion of $(\frac{1}{60}-\frac{x^8}{81})\cdot {(2x^2-\frac{3}{x^2})}^6$ is equal to:

1. $-108$
2. $-72$
3. $-36$
4. $36$

Q.

The greatest term in the expansion of $\sqrt{3}{\left[1+\left(\frac{1}{\sqrt{3}}\right)\right]}^{20}$ is

1. $\frac{26840}{9}$

2. $\frac{24840}{9}$

3. $\frac{25840}{9}$

4. None of these

Q.

If $n\ge 2$is a positive integer, then the sum of the series ${}^{n+1}C_2+2\left({}^{2}c_2+{}^{3}c_2+{}^{4}c_2+..+{}^{n}c_2\right)$ is

1. $\frac{n{\left(n+1\right)}^2\left(n+2\right)}{12}$

2. $\frac{n\left(n-1\right)\left(2n+1\right)}{6}$

3. $\frac{n\left(n+1\right)\left(2n+1\right)}{6}$

4. $\frac{n\left(3n+1\right)\left(2n+1\right)}{6}$

Q.

If $n$is even and the value of ${}^n{C}_r$ is maximum, then $r=$

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

2. $\frac{n+1}{2}$

3. $\frac{n-1}{2}$

4. None of these

Q.

The $1057$ number is not a perfect square. Give reason.

Q.

What is the shortcut to finding the determinant of a $5\times 5$ matrix?

Q.

The maximum and minimum values of $x^3–18x^2+96x$ in the interval $(0,9)$ are

1. $160,0$

2. $60,0$

3. $160,128$

4. $120,28$

Q.

The natural number $m$, for which the coefficient of $x$ in the binomial expansion of ${\left(x^m+\frac{1}{x^2}\right)}^{22}$ is $1540$ is

Q.

If $\underset{n\to \infty }{lim}{r}^n=0$, then $r=?$

1. $\frac{4}{5}$

2. $\frac{5}{4}$

3. $2$

4. $1$

Q. If $y=\log \left(\frac{1-x^2}{1+x^2}\right),$ then $\frac{dy}{dx}$ is equal to

1. $\frac{-4x}{1-x^4}$
2. $\frac{4x^3}{1-x^4}$
3. $\frac{1}{4-x^4}$
4. $\frac{-4x^3}{1-x^4}$

Q. $If{s}_n={\sum }_{r=0}^n\frac{1}{n_{C_r}}and{t}_n={\sum }_{r=0}^n\frac{r}{n_{C_r}},then\frac{t_n}{s_n}=$

1. n -1
2. 
3. 
4. 

Q.

The largest coefficient in the expansion of ${(1+x)}^{24}$ is

1. ${}^{24}{C}_{24}$

2. ${}^{24}{C}_{13}$

3. ${}^{24}{C}_{12}$

4. ${}^{24}{C}_{11}$

Q.

Given positive integers $r>1,n>2$and that the coefficient of ${\left(3r\right)}^{th}$and ${\left(r+2\right)}^{th}$terms in the binomial expansion of ${\left(1+x\right)}^{2n}$are equal. Then

1. $n=2r$

2. $n=2r+1$

3. $n=3r$

4. None of these

Q.

The sum $1(1!)+2(2!)+3(3!)+........+n(n!)$ equals

1. $3\left(n!\right)+n-3$

2. $\left(n+1\right)!-(n-1)!$

3. $(n+1)!-1$

4. $2(n!)-2n-1$

Q.

The coefficient of $x^8$in the product $(x-1)(x-2)(x-3)...(x-10)$ is

1. $1024$

2. $1300$

3. $1320$

4. $1360$

Q. The interval in which x must lie so that the greatest term in the expression of $(1+x{)}^{2n}$ has the greatest coefficient, is

1. 
2. 
3. 
4. None of these

Q.

The $23453$ number is not a perfect square. Give reason.

Q. Maximum value of ${}^{15}{C}_r$ is

Q. If the coefficient of $5^{\text{th}}$ term is numerically the greatest coefficient in the expansion of $(1-x{)}^n,$ then the positive integral value of $n$ is

1. $9$
2. $8$
3. $7$
4. $10$

Q. Maximum value of ${}^{12}{C}_r$ is

Q. Find the number of positive integers less than 101 that cannot be written as the difference of two squares of integers.

Q.

The sum of the coefficient in the expansion of ${\left(\frac{1}{x}+2x\right)}^n$ is $6561$. Find the coefficient independent of $x$

1. $16{}^8{C}_4$

2. ${}^8{C}_4$

3. ${}^8{C}_5$

4. None of these

Q.

$\frac{1}{2^1}-\frac{1}{2.2^2}+\frac{1}{3.2^3}-\frac{1}{4.2^4}+....isequalto$

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

2. ${\log }_e\left(\frac{3}{4}\right)$

3. ${\log }_e\left(\frac{3}{2}\right)$

4. ${\log }_e\left(\frac{2}{3}\right)$

Q. The number of terms in the expansion of $(x^2+y^2{)}^{25}–(x^2–y^2{)}^{25}$ after simplification is

1. $0$
2. $50$
3. $13$
4. $26$

Q.

The coefficient of $x^3{y}^4{z}^5$ in the expansion of ${(xy+yz+xz)}^6$ is

1. $70$

2. $60$

3. $50$

4. none of these

Q.

The term independent of $x$ in the expansion of ${\left[\frac{2\sqrt{x}}{5}-\frac{1}{2x\sqrt{x}}\right]}^{11}$ is

1. $5^{\mathrm{th}}$ term

2. $6^{\mathrm{th}}$ term

3. ${11}^{\mathrm{th}}$ term

4. No term

Q. The expression $\frac{(1+i{)}^n}{(1-i{)}^{n-2}}$ equals

1. $-i^{n+1}$
2. $i^{n+1}$
3. $-2i^{n+1}$
4. $1$

Q. The coefficient of $\frac{1}{y^2}$ in ${(y+\frac{c^3}{y^2})}^{10}$ is $3360$. If the value of $y=2$, then

1. $c=\sqrt[3]{32}$
2. greatest term will be $2880$
3. $5^{\text{th}}$ term will be coefficient of $y^{-2}$
4. $3^{\text{rd}}$ term will have greatest numerical value

Q. If $n$ is any natural numbers, then $6^n-5^n$ always ends with_________