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Solution
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NOTE: The sentence needs a modal that explains 'necessity'. For example, 'You should have written to him yesterday.'
Option A - 'Shall' is used instead of 'will' with the first and second person. We do not use 'shall' in the present tense. As the given sentence has 'yesterday', we can't use 'shall' which is the future tense. Thus option A is incorrect.
Option C - 'Will' is used in the future tense. As the sentence is in the present perfect tense, we can't use a modal in the future tense here. Thus option C is incorrect.
Option D - 'Can' is used in the simple present tense. As the given sentence is in the present perfect tense, we can't use 'can' here. Thus option D is incorrect.
Option B - 'Ought to' is used to show/explain something that is necessary. As the given sentence requires a modal that explains 'necessity' we can use 'ought to' here, Thus option B is the correct answer.
'You ought to have written to him yesterday.'
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