There are two algorithms A and B for solving the same problem.

Algorithm A has a worst$-$case time complexity

that satisfies the recurrence relation

$,$ where

is the input size.

Algorithm B has a worst$-$case time complexity

that satisfies the recurrence relation

$,$ where

is the input size and

is a positive integer.

Answer the following questions.

In the dropdown lists, $\backslash$Theta stands for

$,\backslash$log$\_$b a stands for

$,$ and n$^$x stands for

$.$

$($a$)$ Which case of the Master method applies when solving for

$?$

Case $1$

$($b$)$ What is the asymptotic notation of

$?$

$\backslash$Theta$($n$^\{\backslash$log$\_27\}\backslash$log$\_2$ n

$($c$)$ Assume that

$.$ Which case of the Master method applies when solving for

$?$

$[$ Select $]$

$($d$)$ Assume that

$.$ What is the asymptotic notation of

$?$

$[$ Select $]$

$($e$)$ Assume that

$.$ Which case of the Master method applies when solving for

$?$

$[$ Select $]$

$($f$)$ Assume that

$.$ What is the asymptotic notation of

$?$

$[$ Select $]$

$($g$)$ Assume that

$.$ Which case of the Master method applies when solving for

$?$

$[$ Select $]$

$($h$)$ Assume that

$.$ What is the asymptotic notation of

$?$

$[$ Select $]$

$($i$)$ What is the largest integer a such that Algorithm B is asymptotically faster than Algorithm A$?$

$[$ Select $]$

$($j$)$ What is the smallest integer a such that Algorithm B is asymptotically slower than Algorithm A$?$