Evaluation of Limit

Q. If for all real triplets $(a,b,c)$, $f\left(x\right)=a+bx+c{x}^2$; then $\underset{0}{\overset{1}{\int }}f\left(x\right)dx$ is equal to :

1. $\frac{1}{3}\left(f\right(0)+f\left(\frac{1}{2}\right))$
2. $\frac{1}{6}\left(f\right(0)+f(1)+4f\left(\frac{1}{2}\right))$
3. $2\left(3f\right(1)+2f\left(\frac{1}{2}\right))$
4. $\frac{1}{2}\left(f\right(1)+3f\left(\frac{1}{2}\right))$

Q.

Write the value of ${lim}_{x\to \infty }\frac{n!+(n+1)!}{(n+1)!+(n+2)!}$

Q.

${lim}_{x\to 0}\frac{3x+1}{x+3}$

Q. $y^r=\frac{n!{\cdot }^{n+r-1}{C}_{r-1}}{r^n}$ where $n=2r$. If
$\underset{n\to \infty }{lim}y$ is equal to $\left(\frac{a}{e^b}\right)$ then value of $a+b$ is

Q.

The value of ${lim}_{n\to \infty }{\left\{\frac{(n^3+1)(n^3+2^3)(n^3+3^3)........(n^3+n^3)}{n^3}\right\}}^{\frac{1}{n}}$ is equal to $\alpha .e^{\beta \int \frac{x^3}{1+x^3}dx}$, where $\alpha ϵN,\beta ϵR$, then find $\alpha -\beta$.___

Q. Let the first term $a$ of an infinite G.P. is the value of $x$, where the function $f\left(x\right)=7+2x{\log }_{e}25-5^{x-1}-5^{2-x}$ has the greatest value and the common ratio $r$ is equal to $\underset{x\to 0}{lim}\underset{0}{\overset{x}{\int }}\frac{t^2}{x^2\tan (\pi +x)}dt$. Also, let $S$ be the sum of infinite terms of G.P.

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& a& \text{(P)}& 4\\ \text{(B)}& \frac{1}{r}& \text{(Q)}& 3\\ \text{(C)}& S& \text{(R)}& 2\\ \text{(D)}& a-rS& \text{(S)}& 1\\ & & \text{(T)}& 5\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(R)},\text{(D)}\to \text{(S)}$
2. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(P)}$
3. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(S)}$
4. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(T)}$

Q.

${lim}_{x\to \frac{\pi }{2}}\frac{sin2x}{cosx}$

Q. If for all real triplets $(a,b,c)$, $f\left(x\right)=a+bx+c{x}^2$; then $\underset{0}{\overset{1}{\int }}f\left(x\right)dx$ is equal to :

1. $2\left(3f\right(1)+2f\left(\frac{1}{2}\right))$
2. $\frac{1}{3}\left(f\right(0)+f\left(\frac{1}{2}\right))$
3. $\frac{1}{2}\left(f\right(1)+3f\left(\frac{1}{2}\right))$
4. $\frac{1}{6}\left(f\right(0)+f(1)+4f\left(\frac{1}{2}\right))$

Q. If $f\left(x\right)=\{\begin{array}{c}2x+b\text{if}x<a\\ x+d\text{if}x\ge a\end{array}$ and $\underset{x\to a}{lim}f\left(x\right)=l,$ then $l$ is equal to

1. $b+d$
2. $b-d$
3. $2d+b$
4. $2d-b$

Q. ${\int }_{-2}^{10}sgn(x-[x\left]\right)dx$ equals, where $\left[\right]$ denotes greatest integer function

1. $-12$
2. $10$
3. $12$
4. $8$

Q.

The value of ${lim}_{n\to \infty }{\left\{\frac{(n^3+1)(n^3+2^3)(n^3+3^3)........(n^3+n^3)}{n^3}\right\}}^{\frac{1}{n}}$ is equal to $\alpha .e^{\beta \int \frac{x^3}{1+x^3}dx}$, where $\alpha ϵN,\beta ϵR$, then find $\alpha -\beta$.___

Q. Using differential, find the approximate values of the following:
$\left(xxii\right)$. ${\left(\frac{17}{81}\right)}^{1/4}$

Q. Let the first term $a$ of an infinite G.P. is the value of $x$, where the function $f\left(x\right)=7+2x{\log }_{e}25-5^{x-1}-5^{2-x}$ has the greatest value and the common ratio $r$ is equal to $\underset{x\to 0}{lim}\underset{0}{\overset{x}{\int }}\frac{t^2}{x^2\tan (\pi +x)}dt$. Also, let $S$ be the sum of infinite terms of G.P.

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& a& \text{(P)}& 4\\ \text{(B)}& \frac{1}{r}& \text{(Q)}& 3\\ \text{(C)}& S& \text{(R)}& 2\\ \text{(D)}& a-rS& \text{(S)}& 1\\ & & \text{(T)}& 5\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(R)},\text{(B)}\to \text{(Q)}$
2. $\text{(A)}\to \text{(P)},\text{(B)}\to \text{(R)}$
3. $\text{(A)}\to \text{(S)},\text{(B)}\to \text{(P)}$
4. $\text{(A)}\to \text{(Q)},\text{(B)}\to \text{(T)}$

Q. The sum of the series $1+\frac{1}{4.2!}+\frac{1}{16.4!}+\frac{1}{64.6!}+.....$ ad inf. is

1. $\frac{e-1}{\sqrt{e}}$
2. $\frac{e-1}{2\sqrt{e}}$
3. $\frac{e+1}{\sqrt{e}}$
4. None of these

Q.

${lim}_{n\to infty}\left[\frac{1+2+3+\cdots n}{n+2}\frac{n}{2}\right]$ is

1. $-\frac{1}{2}$
   

2. $\frac{1}{2}$
   

3. 1

4. -1

Q.

The value of ${lim}_{n\to \infty }{\left\{\frac{(n^3+1)(n^3+2^3)(n^3+3^3)........(n^3+n^3)}{n^3}\right\}}^{\frac{1}{n}}$ is equal to $\alpha .e^{\beta \int \frac{x^3}{1+x^3}dx}$, where $\alpha ϵN,\beta ϵR$, then find $\alpha -\beta$.___