$1$A$)$ Design a deterministic and non$-$deterministic finite automaton $($dfa and nfa$)$ which accept a string containing "ion" at the end of a string in a string of $\{$a$-$z$\},$ e$.$g$.,$ "imagination" but not "image". Mentio the quintuple representation of both dfa and nfa you designed.

$1$B$)$ Consider the language $L=\{(10{)}^n$:$n0\}.$ Prove that Deterministic Finite Automata $($DFA$)$ is equivalen to non$-$Deterministic Finite Automata $($NFA$)$ on the language L$.$

$1$C$)$ The language, $L=\{a^n{b}^n{c}^n$:$n1\}$ is a context sensitive language. Design context sensitive grammar for the language.

$2$A$)$ The Grammar $G,$ with productions.

$S{S}_{1}B,$

$S_{1}a{S}_{1}b,$

$bBbbbB,$

$a{S}_{1}baa,$

$B$

Identify whether this is unrestricted grammar or not. Supplement your answer with proper technical justifications.

$2$B$)$ Generate the regular expressions for the sets given below.

i$)\{\begin{array}{c}{a}^n\text{:}n\end{array}$ is divisible by $2$ or $3$ or $\{$:$n=5\}$

ii$)\{a^2.a^5,a^8,$dots$\}$

iii$)$ The set of all strings on $\{a,b\}$ terminated by either $\text{'}$a$\text{'}$ or 'abb'

$2$C$)$ Identify the languages given below is regular or not and support your answer with proper justifications.

i$)L=\{a^{n}bmck$:$n,m,k0\}$

ii$)L=\{aib2j$:$i,j0\}$

iii$)L=\{\begin{array}{c}{a}^n\text{:}n\end{array}$ is prime