1/3 < x < y < 2 के लिए निम्नलिखित में से कौन-सा/कौन-से कथन सही है/हैं?
I. x + 1/x < y + 1/y
II. $\sqrt{1+y^2}/y < \sqrt{1+x^2}/x$

नीचे दिए गए कूट का प्रयोग कर उत्तर चुनिए।

Correct Answer: (B) II only

For Statement I, the function $f(t) = t + 1/t$ decreases on $(1/3, 1)$ and increases on $(1, 2)$. For instance, taking $x = 0.5$ and $y = 1$ yields $x + 1/x = 2.5 > y + 1/y = 2$, showing Statement I is not always correct. For Statement II, let $g(t) =$$\frac{\sqrt{1+t^2}}{t}$ = $\sqrt{\frac{1}{t^2} + 1}$. As $t > 0$ increases, $1/t^2$ strictly decreases, which means $g(t)$ is strictly decreasing. Therefore, for any $x < y$, it is always true that $g(y) < g(x)$, making Statement II always correct.