$1.$Consider the language, $L=L_1{L}_2$

$L_1=\{a^{2i}{b}^{2j}|i>0,j0\}$

$L_2=\{c^{3k+1}{d}^2|k>2\}$

a$.$ Construct an automaton that accepts the language L$.(5$ marks$)$

b$.$ Write a grammar that generates all strings in L$.(5$ marks$)$

Design a deterministic finite automaton to recognise the following language

$2.L=\{0^{2m}+1^{3n}+\frac{2^k}{m}>0,n>0,k>2\},$ where $=\{0,1,2,+\}.$

$3.$Write regular expression for each of the following languages over the alphabet ${\displaystyle \underset{?}{\overset{?}{}}}=\{a,b\}.$

i$.$ The set of strings containing $ab$ as a substring. $(2\mathrm{.}5$ marks$)$

ii$.$ The set of strings having at most one pair of consecutive $a^{\text{'}}s$ and at most one pair of consecutive $b\text{'}$s$.(2\mathrm{.}5$ marks $)$

iii. The set of strings whose length is divisible by $6.$

iv$.$ The set of strings whose $5$th symbol from last is b$.$