Evaluation of Limit

Q. $\underset{n\to \infty }{lim}\frac{1+5+5^2+...+5^n}{1-{25}^n}$

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

Q.

If $X=1+a+a^2+....\infty$, where |a| < 1 and $y=1+b+b^2+.....\infty$ where |b| <1,

prove that $1+ab+a^2{b}^2+.....\infty \frac{xy}{x+y-1}$

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. 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 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 $\alpha -\beta$.___