Obeying the law is an obligation. Modal 'must' depicts the obligation.

Following the rules is either an obligation or a determination. For both these cases, modal 'must' is the right choice.

The usage of exclamatory mark itself gives the clue that only 'may' should be used here. 'May', apart from its other usages, is used in case of a 'wish'.

Whenever we ask somebody that whether he/she wants to have something like tea, coffee, etc, we use modal 'would'.

Though 'should' and 'ought to' both can be used, but the word 'to' in the sentence leaves no scope for 'should'. Both of them are used to express a strong obligation politely.

Doing the duty can be either a duty or a responsibility. But here usage of the words 'at all costs' demostrates the obligation, and for obligations we use 'must'.

The sentence implies that the shirt requires to be washed. As the shirt is a singular noun, so the auxilary should also be singular. Therefore, 'needs'.

Passing an exam is always a case of probability. And 'might' and 'may' both are used to depict the probability of happening a thing. But when a person works hard, the probability increases. And 'may' shows a higher probability than 'might'.

Usage of 'will' means that one is 'likely to' pass if one works hard. 'Will', in such cases, neither shows certainity nor probability.

Reading something that is written by someone else depends on the ability of a person. Modals 'could' and 'can' both show the ability, but because the sentence discusses 'general fact of present', so 'can' is the better option. 

Obeying the elders is not a duty or an obligation, rather it is a responsibility that comes from within. Modals 'should' and 'ought to' are used to depict responsibility. In absence of 'ought to', 'should' is the best choice.

This is a case of depiction of a general fact. Here, ability of a person (Mohan) is highlighted. Thus, modal 'can' is the right choice.

Answering the question in an exam is not a matter of choice (except in some cases), rather it is a sort of compulsion or obligation. Modal 'must' is used for the same purpose.

The usage of verb 'hoped' (past of hope) clearifies that the modal should also be in the past form. Among the options, only 'would' and 'might' can be used for past. But 'would' is used in case of past perfect tense, whereas this sentence is in past indefinite. Thus, 'might' is the answer.

Doing a work (like jumping a wall) always depends on the ability of a person. This ability can be of past times or of present times. As this sentence speaks of a general fact, so usage of 'can' is preffered to that of 'could'.

In this case, either modal 'would' or 'needs' can be used. 'Needs' show a sort of situational compulsion, whereas 'would' shows 'personal determination'. Thus, 'would' seems to be a better option.

This is a case of 'advice'. Modal 'should' fits the requirement of the blank.

Obeying the master is an obligation for a servant. Modal 'must' is the best option to depict obligations.

Depositing the fee is always an obligation. Modals 'must' or 'have to' are used for obligations.

The usage of exclamatory mark shows that it is a case of 'a wish'. Modal 'may' is used for 'wishes and hopes'.

The usage of exclamatory mark itself gives the clue for 'may' to be used. 'May' in such cases means 'a wish'.

The answer of this question is to be arrived by using POE i.e., process of elimination. This is a case of present and future ability, and none other than 'can' fits the requirement of the sentence.

Falling of rain is always a matter of probability. The sentence says that 'the sky is overcast'. This means there is more probability of raining. Modal 'may' perfectly fits the requirement of the blank.

This is a case of 'hope' or 'wish'. Except 'would' all the other modals fail to support a wish. Thus, option 4. 

Though this sentence describes a general fact, but the capability of noun (rat) is depicted. As 'rat' does not have the capability to eat a cat, so 'can not' is the perfect choice.