(A) : $(0+1)^* 01$
(B) : ∈ + $0$ +$1$ + $(0+1)^* (00+10+11)$
(C) : $(1^*01^*0)^* 1^*$
(E) : I’m not sure about my answer here so I'm Happy to be corrected
Here we can divide this question into 3 parts :
$Part$ $1$ : $All$ $the$ $strings$ $having$ $0$ $occurrence$ $of$ $the$ $substring$ $’00’$
$Part$ $2$ : $All$ $the$ $strings$ $having$ $1$ $occurrence$ $of$ $the$ $substring$ $’00’$
$Part$ $3$ : $All$ $the$ $strings$ $having$ $2$ $occurrences$ $of$ $the$ $substring$ $’00’$ . $Here$ , $we$ $can$ $divide$ $this$ $part$ $into$ $2$ $parts$ :
$3A$ : $Strings$ $that$ $contain$ $one$ $occurrence$ $of$ $’000’$ $as$ $substring$.($Since$ $000$ $will$ $have$ $2$ $occurrences$ $of$ $’00’$)
$3B$ : $Strings$ $that$ $contains$ $2$ $occurrences$ $of$ $the$ $substring$ $00$ ($No$ $000$ $as$ $substring$)
$RegEx$ $for$ $Part$ $1$ : $1^* (01^+)^* (∈ + 0)$
$RegEx$ $for$ $Part$ $2$ : $1^*(01^+)^* 00 (1^+0)^* 1^* $
Look at this to know How I obtained this: https://gateoverflow.in/308132/peter-linz-edition-4-exercise-3-1-question-15-page-no-76?show=399638#a399638
$RegEx$ $for$ $Part$ $3A$ : $1^* (01^+)^* 000 (1^+ 0)^* 1^*$
$RegEx$ $for$ $Part$ $3B$ : $1^* (01^+)^* 00 (1^+0)^* 1^+ 00 (1^+0)^* 1^*$
$Your$ $Regular$ $Expression$ : $Part$ $1$ $+$ $Part$ $2$ $+$ $Part$ $3A$ $+$ $Part$ $3B$
(F) : You can do this question by first making a DFA for all the strings having $101$ as substring.
Now you can convert the Final states into Non Final States and the Non Final states into the Final states ,after this use state removal method to get Regex.