Mathematics - Definite Integration Question with Solution | TestHub

Column-IColumn-II

(A) ${\lim }_{n\to \infty }{\sum }_{i=1}^n\frac{3}{n}\cdot \sin \left(2\pi +\frac{3\pi i}{n}\right)$ is equal to(P) $\frac{1}{\mathrm{e}}$

(B) ${\int }_0^{\infty }[x]e^{-x}dx$ equals (where [ ] denotes the (Q) $\frac{2}{\mathrm{e}}$

greatest integer function)(Q) $\frac{2}{e}$

(C) If ${\lim }_{\mathrm{n}\to \infty }{\sum }_{\mathrm{i}=1}^{\mathrm{n}}\left(\frac{\ln 2\mathrm{i}-\ln \mathrm{n}}{\mathrm{n}}\right)=\ln \mathrm{k}$, then ' k ' equals(R) $\frac{1}{\pi }$

(D) ${\mathrm{Lim}}_{n\to \infty }{\sum }_{i=1}^n\frac{i}{n^2}\sin \left(\frac{i^2\pi }{n^2}\right)$ is equal to(S) $\frac{2}{\pi }$

(T) $\frac{1}{\mathrm{e}-1}$