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Case of future tense, where some determination is implicit in the sentence. Though generally we use modal 'shall' with the first person(s), but in such cases we use modal 'will' (to show the determination).

Case of certainity of probability. Whenever we want to show the surity of an event, we use modal 'must'. Hence, option 2 is correct.

This is a case of request. Modal 'may' is used for requests, but in case of more humble and polite requests (as shown here by the usage of 'please') we prefer modal 'could'. 'Can' is generally used for seeking permission in informal cases.

Case of ability. Modal 'could' is the correct answer.

This is an optative sentence, where a wish (not a prayer) has been made. We prefer 'would' in such cases.

Case of advise about a responsibility. Modal 'should' is used in  such cases.

Case of past tense. Whenever we use the term 'wish' in cases like this, we use modal 'could'. It implies that 'if I had the ability to do that, then I would have done'.

'If I were you' here reflects a wish. In case of wishes subjunctives 'were' and 'would' are applicable if the present tense is involved. 'Had' and 'would have' will be used if past tense is involved. Example: If i had been there yesterday, i would have......

This sentence mentiopns the 'ability' of a person (present tense). Modal 'can' is the best usage in such cases.

Case of advice. We use modal 'should' to offer some advice.

Case of an obligation. In cases where someone is bound to something, we use modal 'have to' or 'must'.

When describing pollution-related changes, we need the adjective form 'climatic' (relating to climate). 'Climatics' is not a valid English word.

Cohesive families stay united and supportive, which helps children develop emotional strength. 'Coherent' means logical or consistent, which doesn't describe family bonding.

 Soothing is pacifying. Soothsaying is making a prophecy. 

 Since he is trying to justify his position, cogent is the correct word as this refers to being to the point or convincing, whereas coherent refers to sticking together or coming together as a whole.