GATE CS 2023 Q.7: Quantitative Aptitude

Consider two functions of time ( $t$ ),

f(t) = 0.01 t^2

g(t) = 4t

where $0 < t < \infty$ .

Now consider the following two statements:
(i) For some $t > 0$ , $g(t) > f(t)$ .
(ii) There exists a $T$ , such that $f(t) > g(t)$ for all $t > T$ .

Which one of the following options is TRUE?

only (i) is correct

only (ii) is correct

both (i) and (ii) are correct

neither (i) nor (ii) is correct

Official answer and solution

Solution

Analyze the inequality $f(t) = 0.01 t^2$ versus $g(t) = 4t$ for $t > 0$ .

Consider the two functions for $t > 0$ :

For statement (i): $g(t) > f(t) \iff 4t > 0.01 t^2$ . Since $t > 0$ , dividing by $t$ gives $4 > 0.01 t \iff t < 400$ . For any $t \in (0, 400)$ , such as $t = 10$ , $g(10) = 40 > f(10) = 1$ . Thus, statement (i) is true.
For statement (ii): $f(t) > g(t) \iff 0.01 t^2 > 4t \iff t > 400$ . Choosing $T = 400$ , for all $t > 400$ , $f(t) > g(t)$ . Thus, statement (ii) is true.

Both statements (i) and (ii) are correct.

Final answer

Option (C)

Common trap

Failing to recognize that quadratic growth eventually dominates linear growth for sufficiently large $t$ .

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