Summation Using Sigma

Q.

Integrate $\int \frac{1}{\left[\left(\sin x-2\cos x\right)\left(2\sin x+\cos x\right)\right]}dx$

Q. $\text{If}\underset{n\to \infty }{lim}\frac{(1^{\lambda }+2^{\lambda }+3^{\lambda }+--+n^{\lambda })}{(1^4+2^4+--+n^4)(1^3+2^3+--+n^3)}=F\left(\lambda \right),\lambda ϵN,\text{then}$

1. $F\left(\lambda \right)$ is finite for $\lambda \le 8$
2. $F\left(8\right)=\frac{20}{11}$
3. $F\left(8\right)=\frac{20}{9}$
4. $F\left(7\right)=0$

Q. The value of $\sum _{r=2}^n(-2{)}^r\left|\begin{array}{ccc}{}^{n-2}{C}_r& {}^{n-2}{C}_{r-1}& {}^{n-2}{C}_{r-2}\\ -3& 1& 1\\ 2& -1& 0\end{array}\right|,(n>2)$ is

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