State for each of the following statements whether it is true or false, by entering a T for the statements that are true and an F for the statements that are false. Please mention the statement number before each answer, so that answers can be related to the statements.

$(1)$ f$($n$)=$ n$3$ grows faster than g$($n$)=$ log$($n$),$ where n is a positive integer.

$(2)$ h$($n$)=$ n$!$ grows faster than g$($n$)=$ n$*$ where n is a positive integer.

$(3)2$n$2,3$n$2+20,$ and $5$n$2+7$n $-6,$ where n is a positive integer, are all O$($n$?).$

$(4)7$n$3,32$n$3+200,$ and $25$n$3+17$n$?+11$n $-6,$ where n is a positive integer, are all O$($n$3).$

$(5)$ log$($n$?),2$log$($n$3),3$log$($n$4),$ and $5$log$($nS$),$ where n is a positive integer, are all O$($log$($n$)).$

$(6)$ It is not possible to have a recursive implementation of Mergesort.

$(7)$ It is not possible to write a recurrence equation for Mergesort.

$(8)$ Each node in any graph searched by Dijkstra's single$-$source$-$shortest$-$path algorithm must be a geographical location and the weight of each edge in such a graph must be the shortest distance between the corresponding nodes.

$(9)$ Diikstra's single$-$source$-$shortest$-$bath algorithm accepts a pair of nodes and a graph and finds all paths from the first node in the pair to the second node in the pair, and orders these paths based on their costs.

$(10)$ Each shortest path found by Dijkstra's single$-$source$-$shortest$-$path algorithm has two or more edges.