Question $1$ In a programming language a variable is represented as a string of at least four characters with the following requirements. The variable must start with a $$,$ and end with the digit $7,$ and it must contain at least one letter, and all letters must be before digits. For example, $$1237$ and $$7$ are valid ways to represent two variables but $$5$$ and $1$ are not. Give a regular expression over alphabet $\{,,,,,0,1,2,3,4,5,6,7,8,9,$ $$,\%\}$ that defines all variables with no even digits or vowels.$[5$ marks$]$Question $2$ Construct an NFA over the alphabet $\{0,1,2,3,4,5\}$ that accepts all numbersthat are either divisible by $3$ or are by $5.$ Explain your construction.$[25$ marks$]$Question $3$ Build a deterministic finite state automaton $($DFA$)$ overt the alphabet $\{$a$,$b$\},$ with only three states accepting the following language: $\{$w$|$ w ends with bb$\}.[5$ marks$]$Question $4$ a$-$ Find a non$-$deterministic automaton accepting the language represented by the regular following expression: e$=($ab$)*($ab $|$ c$)$ in a systematic way as used by lexical analysers generators. $[10$ marks$]]$b$-$ Change obtained non$-$deterministic automaton to a deterministic one in a systematic way? $[15$ marks$][25$ marks$]$Question $5$ Let L$=\{$an bn $|$ cn dn $\},$ such that n$>=0\},$ a$-$ give three strings belonging to L$.$ What is the category of this language? $(3$ marks$)$b$-$ Find a grammar$-$recognizing language L$.(3$ marks$)$c$-$ What is the explicit expression of language L$2?(5$ marks$)$ Find a grammar$-$recognizing language L$2.$d$-$ Select the simplest category of automata that should be used to recognize a string belonging to L$2.$ Explain how it works while trying to recognize the following string: cdaabb. $(4$ marks$)[15$ marks$]$