Which of the following statements(s) is/are true?
$$\eqalign{ & {\text{I}}.\frac{3}{{71}} < \frac{5}{{91}} < \frac{7}{{99}} \cr & {\text{II}}.\frac{{11}}{{135}} > \frac{{12}}{{157}} > \frac{{13}}{{181}} \cr} $$

Solution (By Examveda Team)

$$\eqalign{ & {\text{I}}.\,\frac{3}{{71}} < \frac{5}{{91}} < \frac{7}{{99}} \cr & {\text{For }}{{\text{1}}^{{\text{st}}}}{\text{ 2 terms, }}\frac{3}{{71}}{\text{ and }}\frac{5}{{91}} \cr & {\text{3}} \times 91 < 5 \times 71\,\,\left( {{\text{Using cross product}}} \right) \cr & \frac{3}{{71}} < \frac{5}{{91}} \cr & {\text{Now, for }}{{\text{2}}^{{\text{nd}}}}{\text{ and }}{{\text{3}}^{{\text{rd}}}}{\text{ term, }}\frac{5}{{91}}{\text{ and }}\frac{7}{{99}} \cr & 5 \times 99 < 7 \times 91\,\,\left( {{\text{Using cross product}}} \right) \cr & \frac{3}{{71}} < \frac{5}{{91}} < \frac{7}{{99}} \cr & {\text{Hence, statement I is true}} \cr & {\text{II}}.\,\frac{{11}}{{135}} > \frac{{12}}{{157}} > \frac{{13}}{{181}} \cr & {\text{For }}{{\text{1}}^{{\text{st}}}}{\text{ 2 terms, }}\frac{{11}}{{135}}{\text{ and }}\frac{{12}}{{157}} \cr & 157 \times 11 > 12 \times 135\,\,\left( {{\text{Using cross product}}} \right) \cr & \frac{{11}}{{135}} > \frac{{12}}{{157}} \cr & {\text{And for }}{{\text{2}}^{{\text{nd}}}}{\text{ and }}{{\text{3}}^{{\text{rd}}}}{\text{ term}} \cr & \frac{{12}}{{157}} > \frac{{13}}{{181}} \cr & 12 \times 181 > 157 \times 13\,\,\left( {{\text{Using cross product}}} \right) \cr & \therefore \frac{{11}}{{135}} > \frac{{12}}{{157}} > \frac{{13}}{{181}} \cr & {\text{Hence, statement II is true}} \cr} $$