GATE CSE 2023 | GA Question: 7

Question

Consider two functions of time $(t),$
$$
\begin{gathered}
f(t)=0.01 t^2 \\
g(t)=4 t
\end{gathered}
$$
where $0<t<\infty.$

Now consider the following two statements:

1. For some $t>0, g(t)>f(t)$.
2. There exists a $T,$ such that $f(t)>g(t)$ for all $t>T$.

Which one of the following options is $\text{TRUE}?$

• A. only (i) is correct
• B. only (ii) is correct
• C. both (i) and (ii) are correct
• D. neither (i) nor (ii) is correct

Answer 1

Given,

$f(t)=0.01t^{2}$

$g(t)=4t$

and $0< t< \infty$

for option $(i)$,

take $t=1$,Now,

$f(1)=0.01$

$g(1)=4$,

So, here exists some $t$ ,t=1 , for which $g(t)>f(t).$ So $(i)$ is true .

for option $(ii)$,There exists some $ T=1000$(say) .Now any $t>T$ , $f(t)>g(t)$ .

So option (II) is also true .

This question is nothing but Big oh definition .

 $g(t)\leq cf(t)$ for all $t>T$ and $c>0$ .

https://en.wikipedia.org/wiki/Big_O_notation

So correct option is (C) .

Answer 2

For (i) :

f(1) = 0.01   and   g(1) = 4

So, there exists a t = 1, such that (i) is true.

(ii) matches the definition of small oh notation

Here, g(t) = o(f(t)) is true.

So, f(t) > g(t)  is always true for all t > T

Hence, (ii) is also correct. Option D is the correct answer.

Comments

Edit the answer, Option C is correct.

Answer 3

C option is correct

Answer 4

Since t>0 and less than infinity both f(t) and g(t)

Take $\frac{f(t)}{g(t)}$ = 0.01t/4

this would be greater than 1 if t>400 that implies f(t)>g(t)

 thus T is 400

and for all t<400 g(t)<f(t).

Thus answer is C.