Binomial Expression

Q.

What is ${\left(x-y\right)}^3$ formula?

Q.

If $X=\left\{8^n–7n–1:n\in N\right\}$ and $Y=\left\{49(n–1):n\in N\right\}$, then

1. $X\subseteq Y$

2. $Y\subseteq X$

3. $X\subset Y$

4. None of these

Q.

${}^{n}C_r+{}^{n}C_{r-1}$ is equal to?

1. ${}^{n+1}C_r$

2. ${}^{n}C_{r+1}$

3. ${}^{n+1}C_{r+1}$

4. ${}^{n-1}C_{r-1}$

Q.

$\alpha$, $\beta$, $\gamma$ are the roots of $x^3-3x^2$ + 3x + 7 = 0(w is cube root of unity) then $(\frac{\alpha -1}{\beta -1}+\frac{\beta -1}{\gamma -1}+\frac{\gamma -1}{\alpha -1})$ is

1. 

2. 

3. 

4. 

Q. The coefficient of $x^{50}$ in the expansion of $(1+x{)}^{1000}+2x(1+x{)}^{999}+3x^2(1+x{)}^{998}+.....+1001x^{1000}$

1. 
2. 
3. 
4. 

Q.

The approximate value of $(7.995{)}^{\frac{1}{3}}$ correct to four decimal places is

1. 1.9995

2. 1.9996

3. 1.9990

4. 1.9991

Q. A possible value of $x$, for which the ninth term in the expansion of ${\{3^{{\log }_3\sqrt{{25}^{x-1}+7}}+3^{(-\frac{1}{8}){\log }_3(5^{x-1}+1)}\}}^{10}$ in the increasing powers of $3^{(-\frac{1}{8}){\log }_3(5^{x-1}+1)}$ is equal to $180$, is

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

Q.

The smallest positive integer $n$ for which ${\left(1+i\right)}^{2n}={\left(1-i\right)}^{2n}$ is

1. $1$

2. $2$

3. $3$

4. $4$

Q. If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of $(x+\sqrt{x^2-1}{)}^6+(x-\sqrt{x^2-1}{)}^6$, then :

1. $\alpha -\beta =-132$
2. $\alpha +\beta =-30$
3. $\alpha +\beta =60$
4. $\alpha -\beta =60$

Q.

$Expand:(a+b{)}^3$.

Q. If $n$ is a positive integer and $C_k={}^n{C}_k,$ then
the value of $\sum _{k=1}^n{k}^3{\left(\frac{C_k}{C_{k-1}}\right)}^2$ is

1. $\frac{n(n+1)(n+2)}{12}$
2. None of these
3. $\frac{n(n+2)(n+1{)}^2}{12}$
4. $\frac{n(n+1)(n+2{)}^2}{12}$

Q.

The coefficient of $x^5$ in the expansion of $(2-x+3x^2{)}^6$ is

1. 4692

2. 5052

3. - 4692

4. - 5052

Q.

If the seventh term from the beginning and end in the binomial expansion of ${(\sqrt[3]{32}+\frac{1}{\sqrt[3]{33}})}^n$ are equal, find n.

Q.

Find $(x+1{)}^6+(x-1{)}^6$. Hence, or otherwise evaluate $(\sqrt{2}+1{)}^6+(\sqrt{2}-1{)}^6$.

Q.

Factorize the expression $x^3-\frac{1}{x^3}-2x+\frac{2}{x}$.

Q.

Find the sum of the series
${\sum }_n^{r=0}(-1{)}^r{}^n{C}_r[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+\frac{{15}^r}{2^{4r}}\cdots uptomterms]$

1. $\frac{2^n-1}{2^{mn}(2^n-1)}$

2. $\frac{2^m-1}{2^n(2^n-1)}$

3. $\frac{2^{mn}-1}{2^{mn}(2^n-1)}$

4. $\frac{2^{mn}-1}{2^{mn}(2^m-1)}$

Q. If ${25}^{x-1}=5^{2x-1}-100$, then the value of $x$ is ____.

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

Q.

The third term from the end in the expansion of ${(\frac{3x}{5}-\frac{5}{2x})}^8$ is

1. 

2. 

3. 

4. 

Q. If the coefficients of $x^{2}and{x}^3$ in the expansion of $(3+kx{)}^9$ are equal, then the value of k is

1. 
2. 
3. 
4. 

Q. If the function $g\left(x\right)=\frac{x^{200}}{200}+\frac{x^{199}}{199}+\frac{x^{198}}{198}+\cdots +\frac{x^2}{2}+x+5,$ then $g^{\prime }\left(0\right)=$

1. $200$
2. $1$
3. $5$
4. $100$

Q. Let $a=4^{1/401}-1$ and for each $n\ge 2,$ let $b_n={}^n{C}_1+{}^n{C}_2\cdot a+{}^n{C}_3\cdot a^2+\cdots +{}^n{C}_n\cdot a^{n-1}.$ If the value of $b_{2006}-b_{2005}$ is $4^k,$ where $k\in N,$ then the value of $k$ is

Q. Let $a=3^{1/223}+1$ and for all $n\ge 3$, let $f\left(n\right)={}^n{C}_0\cdot a^{n-1}-{}^n{C}_1\cdot a^{n-2}+{}^n{C}_2\cdot a^{n-3}-\dots +(-1{)}^{n-1}\cdot a^0$. If the value of $f\left(2007\right)+f\left(2008\right)=3^k$ where $k\in N$, then the value of $k$ is

Q. If $\sum _{r=0}^{44}{}^{49-r}{C}_5={}^{50}{C}_x$, then the value(s) of $x$ is/are

1. $6$
2. $5$
3. $44$
4. $45$

Q.

Integrate the following functions w.r.t. x.

$\int \frac{1}{x-x^3}dx$

Q.

If ${}^{16}{C}_{2r}{=}^{16}{C}_{r+2^{\prime }}$ find ${}^r{C}_4$

Q. Number of irrational terms in the binomial expansion of $(3^{1/5}+7^{1/3}{)}^{100}$ is

1. $93$
2. $90$
3. $94$
4. $88$
5. $95$

Q. If $\underset{x\to -3}{lim}\frac{x^2+x-6}{x+3}=k$, then the value of $|k|$ is

Q. The value of $\underset{x\to -5}{lim}\frac{x^2-25}{x^2+2x-15}$ is

1. $\frac{2}{3}$
2. $\frac{-2}{3}$
3. $\frac{5}{4}$
4. $\frac{-5}{4}$

Q.

If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that $\frac{b^2-ac}{c^2-bd}=\frac{4a}{3c}.$

Q. R=$(5\sqrt{5}+11{)}^{2n+1}$ and $f=R-\left[R\right]$, where [] denotes the greatest integer function. Then the value of $Rf$ is

1. $4^{2n+1}$
2. $2^{2n+1}$
3. $2^n$
4. $8^{n+1}$