Fill in the Blanks

This is a case of 'seeking permission', and no other option than 'may' fits to the requirements of the blank. But when it is a case of 'speaking', we prefer 'can'.

A piece of advice. It shows that it is our responsibility to pay our debts. Thus, option 2.

This is a case of 'hope'. In this case there is a hope that landing of the plane can happen at any time. For conditions like this, we prefer 'may'.

Case of seeking formal permission. In such cases, we use 'may'.

In cases of ethics or morality, 'ought to' is a preferable choice. Moreover, helping the poor is a sort of responsibility (social responsibility). Modal 'should' is the best options from the given ones.

'May' is used in case of seeking or giving permissions.

Whether one keeps one's words or not depends on many factors, of which 'personal will' is most crucial. When we say 'rest assured',  we mean 'if everything else taken care of'. In such cases, keeping our words is just a function of our 'willingness' and when we are sure, we use modal 'will'.

This is a case of 'advice'. Whenever we advise somebody, we use modal 'should'.

Occurence of rain is always a matter of probability/possibility, and never a matter of certainity. For showing probability, we use either 'may' or 'might'. In this case, the information that 'the sky is overcast' add to the probability. Thus, 'may' is preferred to 'might'.

The sentence implies that for the crops to survive, rain is urgently required. This is a sort of compulsion. Modal 'must' best fits to the blank.

It is a case of advice. 'Need not' is the best usage in such cases. It means 'you are not supposed/required to'.

Solving a problem depends on one's capability. For capabilities, we use either 'can'  (present tense) or 'could' (past tense). Verb 'tried' means past tense, so 'could' is the preferred usage.

To pass an exam is a matter of probability, and for probability we use either 'may' or 'might'. But conjunction 'so that' is followed by the first form of the verb. Hence, 'may' is the correct usage.

The sentence is in the past tense ('did'). For past ability, we use 'could'.

Usage of exclamatory mark at the end shows that this is a case of 'a wish'. In cases like this, modal 'may' is used.