Modals: They are verbs which are used before the main verb to express ability, possibility, obligation, permission, suggestion, request, certainty or necessity.They indicate the attitude, intention and the mood of the speaker.

• They cannot be used alone in a sentence.
• In case of Questions they are used alone.
• They are not affected by the change in person i.e. first, second or third form.
• Present Participle or Past Participle form of a verb is never used with a modal.

Use of 'Need' : As a model auxiliary verb in negative terms, it indicates absence of obligation. It expresses the speaker's authority or advice and is used for the present and the future. For example :

You needn't type this letter.   (The speaker is the authority; could be his boss.) .

In the question : -

1. He ____________ have written such a detailed essay. 

In this sentence , needn't is the correct modal that can be placed in the blank part as it is used to denotes an unnecessary action which was nevertheless performed.

2. He won't ____________ to attend the court.

In this sentence , need is the correct modal that can be placed in the blank part as it is used as ordinary verb in negative sentence, takes auxiliary 'do'. 'Need', conjugated with will.shall, do/does/did etc, takes the full infinitive.

Thus , the Correct Option is B , which is 1. Needn't 2. Need .

Modals: They are verbs which are used before the main verb to express ability, possibility, obligation, permission, suggestion, request, certainty or necessity.They indicate the attitude, intention and the mood of the speaker.

• They cannot be used alone in a sentence.
• In case of Questions they are used alone.
• They are not affected by the change in person i.e. first, second or third form.
• Present Participle or Past Participle form of a verb is never used with a modal.

Use of 'Need' : As a model auxiliary verb in negative terms, it indicates absence of obligation. It expresses the speaker's authority or advice and is used for the present and the future. For example :

You needn't type this letter.   (The speaker is the authority; could be his boss.) .

In the question : -

1. Cans of soup _____________  be kept in the fridge.

In this sentence , needn't is the correct modal that can be placed in the blank part as it is used to denotes an unnecessary action which was nevertheless performed.

2. I didn’t __________ to buy any books. They were all in the library.

In this sentence , need is the correct modal that can be placed in the blank part as it is used to express no obligation in the past .

3. You___________  waited for me .

 In this sentence , needn't have is the correct modal that can be placed in the blank part as it is used to refer to events which happened but which the speaker considers were unnecessary .

Thus , the Correct Option is B , which is 1. Needn't 2. Need 3. Needn't have .

Modals: They are verbs which are used before the main verb to express ability, possibility, obligation, permission, suggestion, request, certainty or necessity.They indicate the attitude, intention and the mood of the speaker.

• They cannot be used alone in a sentence.
• In case of Questions they are used alone.
• They are not affected by the change in person i.e. first, second or third form.
• Present Participle or Past Participle form of a verb is never used with a modal.

Use of 'Need' : As a model auxiliary verb in negative terms, it indicates absence of obligation. It expresses the speaker's authority or advice and is used for the present and the future. For example :

You needn't type this letter.   (The speaker is the authority; could be his boss.) .

In the question : -

1. Not a thing ___________ change on this page.

In this sentence , need is the correct modal that can be placed in the blank part as it is used to show no obligation or necessity to do something .

2. Let’s forget about it. No one ___________ know about it.

In this sentence , need is the correct modal that can be placed in the blank part as it is used to show no obligation or necessity to do something .

Thus , the Correct Option is A , which is 1. Need 2. Need .