Integration to Solve Modified Sum of Binomial Coefficients

Q. If $\underset{x\to 0}{lim}\frac{\alpha x{e}^x-\beta {\log }_e(1+x)+\gamma x^2{e}^{-x}}{x{\sin }^{2}x}=10,\alpha ,\beta ,\gamma \in R,$
then the value of $\alpha +\beta +\gamma$ is

Q. The lowest integer which is greater than ${(1+\frac{1}{{10}^{100}})}^{{10}^{100}}$ is

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

Q. The integer $n$ for which $\underset{x\to 0}{lim}\frac{(\cos x-1)(\cos x-e^x)}{x^n}$ is a finite non-zero number is

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

Q.

The left−handed derivative of $\mathrm{f}\left(\mathrm{x}\right)=\left[\mathrm{x}\right]\sin \left(\mathrm{\pi x}\right)$ at $\mathrm{x}=\mathrm{k},\mathrm{k}$ is an integer, is

1. ${\left(-1\right)}^{\mathrm{k}}\left(\mathrm{k}-1\right)\mathrm{\pi }$

2. ${\left(-1\right)}^{\mathrm{k}-1}\left(\mathrm{k}-1\right)\mathrm{\pi }$

3. ${\left(-1\right)}^{\mathrm{k}}\mathrm{k\pi }$

4. ${\left(-1\right)}^{\mathrm{k}-1}\mathrm{k\pi }$

Q.

If $C_r$ means ${}^n{C}_r$ then $\frac{C_0}{1}+\frac{C_2}{3}+\frac{C_4}{5}+\cdots =$

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

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

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

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

Q. The value of $2c_0+\frac{2^2}{2}{C}_1+\frac{2^3}{3}{C}_2+\frac{2^4}{4}{C}_3+....+\frac{2^{11}}{11}{C}_{10}is$

1. 
2. 
3. 
4. 

Q.

What is the integration of $\ln x$?

Q. If the function $f\left(x\right)=x^5+e^{\frac{x}{5}}$ and $g\left(x\right)=f^{-1}\left(x\right)$, then the value of $\frac{1}{g^{{}^{\prime }}(1+e^{\frac{1}{5}})}$ is -

1. $5$
2. $5+\frac{5}{e}$
3. $5+\frac{e^{\frac{1}{5}}}{5}$
4. $1$

Q. The sum of the series
$S=1+\frac{4}{5}+\frac{9}{25}+\frac{16}{125}+\cdots \infty$
is $\frac{15m}{32}$, then the value of $m$ is

Q. The value of $\underset{n\to \infty }{lim}\frac{(1^2+2^2+3^2+\cdots +n^2)(1^3+2^3+3^3+\cdots +n^3)}{1^6+2^6+3^6+\cdots +n^6}$ is

1. $\frac{7}{12}$
2. $\frac{5}{7}$
3. $\frac{6}{13}$
4. $\frac{9}{17}$

Q. If $f\left(x\right)=\{\begin{array}{cc}3(1+|\tan x|{)}^{\frac{\alpha }{|\tan x|}},& -\frac{1}{2}<x<0\\ \beta ,& x=0\\ 3{(1+\left|\frac{\sin x}{3}\right|)}^{\frac{6}{|\sin x|}},& 0<x<\frac{2}{3}\end{array}$ is continuous at $x=0,$ then the ordered pair $(\alpha ,\beta )$ is equal to

1. $(2,e^2)$
2. $(2,\frac{2}{e^2})$
3. $(2,3e^2)$
4. $(2,\frac{3}{e^2})$

Q. The value of $\underset{n\to \infty }{lim}(\frac{\sqrt{n}}{{(3+4\sqrt{n})}^2}+\frac{\sqrt{n}}{\sqrt{2}{(3\sqrt{2}+4\sqrt{n})}^2}+\frac{\sqrt{n}}{\sqrt{3}{(3\sqrt{3}+4\sqrt{n})}^2}+\cdots +\frac{1}{49n})$
is of the form $\frac{1}{p},$ where $p\in N.$ Then possible factors of $p$ is/are

1. $7$
2. $2$
3. $3$
4. $5$

Q. Let $S_n=\sum _{k=1}^n\frac{k}{(k-1{)}^{4/3}+(k^2-1{)}^{2/3}+(k+1{)}^{4/3}}$ and $\underset{n\to \infty }{lim}\frac{S_n}{n^{2/3}}=\frac{1}{p}.$ Then the value of $p$ is

Q.

$1+\frac{3}{2!}+\frac{7}{3!}+\frac{15}{4!}+...\infty =?$

1. $e(e+1)$

2. $e(1-e)$

3. $e(e-1)$

4. $3e$

Q. Let $\left[T\right]$ denote the greatest integer less than or equal to $T$. Then the value of ${\int }_1^2|2x-\left[3x\right]|dx$ is

Q. The value of $\frac{C_1}{2}+\frac{C_3}{4}+\frac{C_5}{6}+\cdots$ equals to

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

Q. $\underset{n\to \infty }{\text{lim}}\frac{1\cdot n^2+2{(n-1)}^2+3{(n-2)}^2+\cdots +n\cdot 1^2}{1^3+2^3+\cdots +n^3}$ is equal to

1. $\frac{8}{3}$
2. $\frac{4}{3}$
3. $\frac{2}{3}$
4. $\frac{1}{3}$

Q. For $a\in R$ (the set of all real numbers), $a\ne -1,$ $\underset{n\to \infty }{lim}\frac{(1^a+2^a+\cdots +n^a)}{{(n+1)}^{a-1}\left[\right(na+1)+(na+2)+\cdots +(na+n\left)\right]}=\frac{1}{60}.$
Then $a=$

1. $5$
2. $7$
3. $\frac{-15}{2}$
4. $\frac{-17}{2}$

Q. If $\left[\begin{array}{cc}2x+1& 5x\\ 0& y^2+1\end{array}\right]=\left[\begin{array}{cc}x+3& 10\\ 0& 26\end{array}\right]$, find the value of (x + y).

Q. If $f\left(x\right)=1+x^2-x^3+...-x^{15}+x^{16}-x^{17}$, then the coefficient of $x^2$ in $f(x-1)$ is

1. $826$
2. $816$
3. $822$
4. none of these

Q. If $\overrightarrow{a}=2\hat{i}+2\hat{j}+3\hat{k},\overrightarrow{b}=-\hat{i}+2\hat{j}+\hat{k}\mathrm{and}\overrightarrow{c}=3\hat{i}+\hat{j}$ are such that $\overrightarrow{a}+\lambda \overrightarrow{b}$ is perpendicular to $\overrightarrow{c}$, then find the value of λ.

Q. If $f^{\prime }$ is differentiable function and $f^{\prime \prime }\left(x\right)$ is continuous at $x=0$ and $f^{\prime \prime }\left(0\right)=a$, the value of $\underset{x\to 0}{lim}\frac{2f\left(x\right)-3f\left(2x\right)+f\left(4x\right)}{x^2}$ is

1. $a$
2. $2a$
3. $3a$
4. none of these

Q. The coefficient of $x^{15}$ in the product
$(1-x)(1-2x)(1-2^2.x)(1-2^3.x)...(1-2^{15}.x)$
is equal to

1. $2^{105}-2^{121}$
2. $2^{121}-2^{105}$
3. $2^{120}-2^{104}$
4. none of these

Q. Express $2\hat{i}-\hat{j}+3\hat{k}$ as the sum of a vector parallel and a vector perpendicular to $2\hat{i}+4\hat{j}-2\hat{k}.$

Q. A relation ϕ from C to R is defined by x ϕ y ⇔ | x | = y. Which one is correct?
(a) (2 + 3 i) ϕ 13
(b) 3 ϕ (−3)
(c) (1 + i) ϕ 2
(d) i ϕ 1

Q. For $\left|x\right|>1,$ if $\underset{n\to \infty }{lim}\prod _{k=0}^n(1+\frac{2}{x^{2^k}+x^{-2^k}})=f\left(x\right),$ then

1. $\underset{2}{\overset{5}{\int }}f\left(x\right)dx=3+\ln 16$
2. $\underset{x\to \infty }{lim}f\left(x\right)=1$
3. $f\left(x\right)=0$ has exactly one solution
4. $f\left(x\right)$ is a decreasing function

Q. $\sum _{r=0}^{r=n}\frac{\left(}{\begin{array}{c}n\\ r\end{array}}$

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

Q.

The value of $\frac{C_1}{2}$ + $\frac{C_3}{4}$ + $\frac{C_5}{6}$ + ...... is equal to

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

2. n.$2^n$

3. $\frac{2^n}{n}$

4. $2^{n-1}$

Q. The value of
$100\underset{n\to 200}{lim}[C_1-(1+\frac{1}{2})C_2+(1+\frac{1}{2}+\frac{1}{3})C_3-\cdots +(-1{)}^{n-1}(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n})C_n]$ is (where $C_r={}^n{C}_r$)

Q. $\int \left(\frac{m}{x}+\frac{x}{m}+m^x+x^m+mx\right)dx$