Greatest Binomial Coefficients

Q.

Find the numerically greatest term in the expansion of $(2+3x{)}^9$, when x=$\frac{3}{2}$

1. 

2. 

3. 

4. 

Q.

Given that the 4th term in the expansion of ${(2+\frac{3x}{8})}^{10}$ has the maximum numerical value, then x lies in the interval

1. $(2,\frac{64}{21})$

2. $(-\frac{60}{23},-2)$

3. $(-\frac{64}{21},-2)$

4. $(2,-\frac{60}{23})$

Q.

The sum of the rational terms in the expansion of $(\sqrt{2}+5\sqrt{3}{)}^{10}$ is:

1. 41

2. 42

3. 39

4. 45

Q.

The total number of local maxima and local minima of the function $f\left(x\right)={(2+x)}^3$, when $-3<x\le -1$ and $x^{\frac{2}{3}}$, when $-1<x<2$ is

1. $0$

2. $1$

3. $2$

4. $3$

Q.

Numerically greatest term in the expansion of $(3-5x{)}^{11}whenx=\frac{1}{5}$ is:

1. 55x3⁹

2. 55x3⁸

3. 55x3¹⁰

4. None

Q. The greatest integer less than or equal to ${(\sqrt{2}+1)}^6$ is?

1. 196
2. 197
3. 198
4. 199

Q. Prove that ${\sum }_{r=0}^n{3}^r{n}_{C_r}=4^n$.

Q. The Greatest co-efficient in the expansion of $(1+x{)}^{2n+2}$ is

1. 
2. 
3. 
4. 

Q.

The coefficient of middle term in the expansion of $(1+x{)}^{10}$ is:

1. 10!/(5!)²

2. 10!.5!.7!

3. None of these

4. 10!/{5!.6!}

Q.

If ${}^n{P}_r=720$ and ${}^n{C}_r=120$, find the value of r.

Q.

Determine n if

$\left(i\right){}^{2n}{C}_3:{}^n{C}_2=12:1\left(ii\right){}^{2n}{C}_3:{}^n{C}_3=11:1$

Q.

The numerically greatest term in the expansion of $(3x+5y{)}^{24}$, when x=4 and y=2 is:

1. 

2. 

3. 

4. 

Q.

r × $\frac{{}^n{C}_r}{{}^n{C}_{r-1}}$ When n = 1000 and r = 500

___

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.

How many terms of G.P. $3,3^2,3^3,$ ............ are needed to give the sum 120 ?

Q.

The sum ${\sum }_{i=0}^m(\frac{10}{i})(\frac{20}{m-i}),where(\frac{p}{q})=0ifp>q,$ is maximum when m is equal to

1. 15

2. 20

3. 5

4. 10

Q. The sum of the third from the beginning and the third from the end of the binomial coefficients in the expansion of ${(\sqrt[4]{43}+\sqrt[3]{34})}^n$ is equal to $9900.$ The number of rational terms contained in the expansion is:

1. $8$
2. $9$
3. $10$
4. $11$

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

Q. Prove that the coefficient of $x^n$ in the expansion of $(1+x{)}^{2n}$ is twice the coefficient of $x^n$ in the expansion of $(1+x{)}^{2n-1}$

1. True
2. False

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. If $p,q,r$are in A.P., then the value of determinant $\left|\begin{array}{ccc}{a}^2+2^{n+1}+2p& b^2+2^{n+2}+3q& c^2+p\\ 2^n+p& 2^{n+1}+q& 2q\\ a^2+2^n+p& b^2+2^{n+1}+2q& c^2-r\end{array}\right|$ is

1. $1$
2. $a^2{b}^2{c}^2-2^n$
3. $0$
4. $(a^2+b^2+c^2)-2^{n}q$

Q.

Given that the 4th term in the expansion of ${(2+\frac{3x}{8})}^{10}$ has the maximum numerical value, then x lies in the interval

1. $(2,\frac{64}{21})$

2. $(-\frac{60}{23},-2)$

3. $(-\frac{64}{21},-2)$

4. $(2,-\frac{60}{23})$