Binomial Coefficients

Q.

The coefficient of $x^9$ in the expansion of $1+x\left)\right((1+x^2)$ $(1+x^3)$ $\cdots$ $(1+x^{100})$ is

1. 6

2. 7

3. 8

4. 9

Q.

What is the expansion of ${\left(x-1\right)}^4$?

Q. In the expansion of $(1+x+x^3+x^4{)}^{10},$ the coefficient of $x^4$ is

1. ${}^{10}{C}_5$
2. $210$
3. ${}^{40}{C}_4$
4. $310$

Q. If the coefficients of $(2r+4{)}^{th}$ and $(r-2{)}^{th}$ terms in the expansion of $(1+x{)}^{18}$ are equal, then the value of $r$ is

Q. The coefficient of $x^7$ in the expression $(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\dots +x^{10}$ is :

1. $420$
2. $330$
3. $210$
4. $120$

Q. If ${}^{n-1}{C}_r=(k^2-3{)}^n{C}_{r+1}$, then $kϵ$

1. $(-\infty ,-2]$
2. $[2,\infty )$
3. $[-\sqrt{3},\sqrt{3}]$
4. $(\sqrt{3},2]$

Q.

Find: $0.5÷0.25$

Q. The coefficient of $x^{13}$ in the expansion of $(1-x{)}^5(1+x+x^2+x^3{)}^4$ is

Q. The coefficient of $x^4$ in the expansion of $(2-x+3x^2{)}^6$ is

Q. Let $S=\left\{\begin{array}{c}n\in N|{\left(\begin{array}{cc}0& i\\ 1& 0\end{array}\right)}^n\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\forall a,b,c,d\in R\end{array}\right\}$, where $i=\sqrt{-1}$. Then the number of $2-$digit numbers in the set $S$ is

Q. Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x{)}^2+(1+x{)}^3+...+(1+x{)}^{49}+(1+mx{)}^{50}\text{is}(3n+1{)}^{51}{C}_3$ for some positive integer $n$. Then the
value of $n$ is

Q.

Find the coefficient of a4 in the product $(1+2a{)}^4(2-a{)}^5$ using binomial theorem.

Q. The sum of the coefficients of all the integral powers of $x$ in the expansion of $(1+2\sqrt{x}{)}^{40}$ is

1. $3^{40}+1$
2. $3^{40}-1$
3. $\frac{1}{2}(3^{40}-1)$
4. $\frac{1}{2}(3^{40}+1)$

Q. If the coefficient of $x^2$ and $x^3$ are both zero, in the expansion of the expression $(1+ax+b{x}^2)(1-3x{)}^{15}$ in powers of $x$, then the ordered pair $(a,b)$ is equal to :

1. $(-21,714)$
2. $(-54,315)$
3. $(28,861)$
4. $(28,315)$

Q. The coefficient of $x^4$ in the expansion of ${(\frac{x}{2}-\frac{3}{x^2})}^{10}$ is

1. $-\frac{405}{256}$
2. $\frac{405}{16}$
3. $\frac{405}{256}$
4. $-\frac{405}{16}$

Q. If the constant term in the binomial expansion of ${(x^2-\frac{1}{x})}^n$ is $15,$ then $n=$

Q.

The value of ${}^{20}{P}_2$ is ?

1. $20$

2. $19$

3. $380$

4. None of these

Q. What is binomial theorem ?

Q.

If x + y = 1, then ${\sum }_{r=0}^{n}r{}^n{C}_r{x}^r{y}^{n-r}$ equals:

1. n

2. None

3. ny

4. nx

Q. The largest non-negative integer $k$ such that ${24}^k$ divides $13!$ is

1. $2$
2. $3$
3. $4$
4. $5$

Q. If the coefficients of $x^3$ and $x^4$ in the expansion of $(1+ax+b{x}^2$) $(1–2x{)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to

1. $(16,\frac{272}{3})$
2. $(16,\frac{251}{3})$
3. $(14,\frac{251}{3})$
4. $(14,\frac{272}{3})$

Q.

If the coefficients of $r^{th},(r+1{)}^{th}$ and $(r+2{)}^{th}$ terms in the binomial expansion of $(1+y{)}^m$are in $A.P.$, then $m$ and $r$ will satisfy the equation

1. $m^2-m(4r+1)+4r^2+2=0$
2. $m^2-m(4r-1)+4r^2-2=0$
3. $m^2-m(4r-1)+4r^2+2=0$
4. $m^2-m(4r+1)+4r^2-2=0$

Q. If $(2x-1{)}^{20}-(ax+b{)}^{20}={(x^2+px+q)}^{10}$ holds true $\forall x\in R$ where $a,b,p$ and $q$ are real numbers, then which of the following is (are) CORRECT?

1. $2p+3q=1$
2. $a+2b=0$
3. $a=\sqrt[20]{202^{20}-1}$
4. $4q+p=0$

Q.

The complete set of values of 'a' such that $x^2+ax+a^2+6a$ < 0 $\forall$ x $ϵ$ [-1, 1] is:

1. 

2. 

3. 

4. None of these

Q.

If $(p+q)th$ term of a G.P be m and $(p-q)th$ term be $n$, then the $pth$ term will be

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

2. $mn$

3. $\sqrt{\left(mn\right)}$

4. $0$

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

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

Q. The value of
$1^2.{}^n{C}_1+2^2.{}^n{C}_2+3^2.{}^n{C}_3+...+n^2.{}^n{C}_n$ is

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

Q.

Factorize the following expression:

$x^3+8y^3+64z^3-24xyz$

Q. If $p$ and $q$ be positive, then the coefficients of $x^p$ and $x^q$ in the expansion of $(1+x{)}^{p+q}$ will be:

1. Equal
2. Reciprocal to each other
3. Unequal
4. None of the above

Q.

If in the expansion ${\left[3x-\frac{2}{x^2}\right]}^{15}$ $r$th term is independent of $x$, then the value of $r$ is

1. $6$

2. $10$

3. $9$

4. $12$