$(20$ points$)$

$($a$)$ For each of the following, indicate whether the statement is true or false $($no explanations necessary$)$:

i$.10n^5$ is $O(2^n).$

ii$.lo{g}_3(n)$ is $O(nlog(n)).$

iii. $2n^2+3n$ is $O(n).$

iv$,2^{2+n}$ is $O(2^n)$

v$.2^{2n}$ is $O(2^n)$

$($b$)$ Suppose a sorting algorithm takes $3$ seconds to sort $10$ items. How much time in seconds will the same algorithm take to sort $100$ items if the number of operations the algorithm performs is exactly $n^2.$

$($c$)$ Suppose algorithm A takes $30nlogn$ time and algorithm $B$ takes $3n^2$ time. What is the smallest value of $n($assuming $n>2)$ for which A will be faster than $B?$

$($d$)$ If $p(n)$ and $q(n)$ are two functions such that $p(n)$ is $O(q(n)),$ does this always mean that for every value of $n,p(n)q(n)?$ If you think the answer is yes, give a justification. If you think the answer is no$,$ give a counterexample i$.$e$.$ find two functions $p(n)$ and $q(n)$ and a number $y$ such that $p(n)$ is $O(q(n))$ and $p(y)>q(y)$