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$9$ grows slower than g$($n$)=($log$($n$))9,$ where n is a positive integer greater than $1.$

$(2)$ h$($n$)=7$n grows slower than q$($n$)=11$n where n is a positive integer.

$(3)2222$n$2,2222$n$2+222$n$,$ and $2222$n$2+222$n $+22,$ where n is a positive integer, are all O$($n$).$

$(4)77777$n$3,77777$n$3+7,$ and $77777$n$3+7777$n$2+777$n $+7,$ where n is a positive integer, are all O$($n$).$

$(5)22$log$($n$2),333$log$($n$3),444$log$($n$4),$ and $6666$log$($n$6),$ where n is a positive integer, are all O$($log$($n$)).$

$(6)$ It is possible to have a recursive implementation of some algorithms.

$(7)$ It is possible to write a recurrence equation for some algorithms.

$(8)$ Dijkstra's single$-$source$-$shortest$-$path algorithm can be used on any graph with finite number of edges and vertices such that edge costs are positive.

$(9)$ Mergesort cannot be used at all in ordering people based on the number of their connections on a social$-$media platform.

$(10)$ Dijkstra's single$-$source$-$shortest$-$path algorithm cannot be applied to a graph in which the weight of each edge is only $1.$