GATE CS 2023 Q7 — General Aptitude

We are given two functions of time $t$ for $0 < t < \infty$:

Let us analyze each statement:

Statement (i): For some $t > 0$, $g(t) > f(t)$

We want to find if there exists any $t > 0$ such that:

Since $t > 0$, we can divide both sides by $t$:

Thus, for any $t$ in the interval $(0, 400)$, $g(t) > f(t)$. For example, at $t = 1$, $g(1) = 4$ and $f(1) = 0.01$, which satisfies $g(1) > f(1)$. Therefore, Statement (i) is correct.

Statement (ii): There exists a $T$ such that $f(t) > g(t)$ for all $t > T$

We want to find a threshold $T$ such that for all $t > T$:

Since $t > 0$, dividing both sides by $t$ gives:

If we choose $T = 400$, then for all $t > T$, the inequality $f(t) > g(t)$ holds. Therefore, Statement (ii) is correct.

Since both statements (i) and (ii) are correct, the correct option is C.