Binomial Expression

Q. If $S_n=\sum _{r=1}^n\sum _{t=0}^{r-1}\left(\frac{1}{6^n}{}^n{C}_r{}^r{C}_t{4}^t\right),$ then the value of $l,$ where $l=\sum _{n=1}^{\infty }(1-S_n)$ is

Q. The coefficient of $x^7$in the expansion of $(1-x-x^2+x^3{)}^6$ is

1. -132
2. -144
3. 132
4. 144

Q. The set of values of $x,$ which satisfy $x^2-13x+\left[x\right]+36=0$ is

1. $[5,7)$
2. $(6,7)$
3. $[6,7)$
4. $(5,7)$

Q. If $f(x+y,x-y)=xy$, then $\frac{f(x,y)+f(y,x)}{2}$ is

1. $x$
2. $y$
3. $0$
4. $\frac{x+y}{2}$

Q. The coefficient of $x^7$in the expansion of $(1-x-x^2+x^3{)}^6$ is

1. -132
2. -144
3. 132
4. 144

Q. If $X=\{4^n-3n-1:n\in N\}$ and $Y=\left\{9\right(n-1):n\in N\}$, where $N$ is the set of natural numbers, then $X\cup Y$ is equal to:

1. $N$
2. $Y-X$
3. $X$
4. $Y$

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

Q. Sum of the series ${}^n{C}_1+2\cdot 5{}^n{C}_2+3\cdot 5^2{}^n{C}_3+\cdots$ upto $n$ terms is

1. $n\cdot 6^{n-1}$
2. $6^{n-1}$
3. $6^n$
4. $n\cdot 6^n$

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. $S=\sum _{r=0}^n(-1{)}^r{}^n{C}_r[\frac{2^r}{3^r}+\frac{8^r}{3^{2r}}+\frac{{26}^r}{3^{3r}}+...\infty ]$ is

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

Q. Let $a=(4^{1/401}-1)$ and for each $n\ge 2,$ let $b_n={}^n{C}_1+{}^n{C}_2\cdot a+{}^n{C}_3\cdot a^2+{}^n{C}_n\cdot a^{n-1}.$ Then the value of $b_{2020}-b_{2019}$ is

1. $4^{2020/401}$
2. $4^{2019/401}$
3. $-4^{2020/401}$
4. $-4^{2019/401}$

Q. If $(1+x)(1+x+x^2).....(1+x+.....+x^n)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+.....$, then the value of $a_0+a_2+a_4+.....$ is

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

Q. The given expression is ${(x+\sqrt{x^3-1})}^5+{(x-\sqrt{x^3-1})}^5$ is a

1. polynomial of fractional degree.
2. polynomial of degree 7.
3. polynomial with integer degree.
4. not a polynomial.

Q. If the coefficents 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. $(14,\frac{272}{3})$
2. $(16,\frac{272}{3})$
3. $(16,\frac{251}{3})$
4. $(14,\frac{251}{3})$