Question 1
His illness got worse and worse. In the end he _____ go into hospital for an operation.
The story is in the past (got worse), so we need the past form of must/have to: had to. Must has no past form. Not ought to have, because that means he should have gone but didn't.
Question 2
You _____ any more aspirins; you’ve had four already.
Mustn't + base verb means it is not allowed or is dangerous: You mustn't take any more aspirins. After had better (not) we use the base verb without to, so had better not to take is wrong. Needn't have taken and shouldn't have taken talk about the past, but this warning is about now.
Question 3
I’ve searched everywhere for Bob but I _____ to find him.
I've searched is present perfect and links the past to now, so the second part also uses the present perfect: haven't been able to. Not couldn't, because it is followed by to here and is past simple, which doesn't match I've searched.
Question 4
A: The window is broken and there's a football on the kitchen floor.
B: It _____ broken by the children next door.
We use must have been + past participle for a strong deduction about the past in the passive: It must have been broken by the children. The football is the clue. Not ought to have been, because that means something was expected or right, not a logical conclusion.
Question 5
You _____ punished her. She didn't deserve it.
Shouldn't have + past participle criticises a past action: it happened, but it was wrong. The clue is She didn't deserve it. Not needn't have, which only means the action was unnecessary, not wrong.
Question 6
A: I wonder who took my alarm clock.
B: It _____ Julia. She _____ supposed to get up early.
The clock was taken in the past, so we use must have been for a past deduction. Was supposed to means she had to get up early, which is the reason for our conclusion. Might be / is and could be / is are present, so they don't fit a past event.
Question 7
In a hundred years’ time we _____ out of water to drink.
We use may have + past participle (future perfect meaning) for something that will possibly be completed by a future time: In a hundred years' time we may have run out of water. Not must have run, which is a deduction about the past. Not might have been run, because it is passive and makes no sense here. Not had run, which is past perfect and cannot describe the future.
Question 8
If you’d explained your problem to me, I _____ to help you.
This is a third conditional: If + past perfect (you'd explained), then would have + past participle. So would have been able to. Not could have, because could have to help is not correct English; could have helped would be needed.
Question 9
Joe ran all the way. It wasn’t necessary. Joe _____ run all the way.
Needn't have + past participle means someone did something, but it wasn't necessary. Joe ran, but it wasn't necessary, so: Joe needn't have run. Mustn't have is not used for past necessity, and doesn't need to is present.
Question 10
Tom was serious when he said he wanted to be an actor when he grew up. We _____ at him. We hurt his feelings.
Shouldn't have + past participle says a past action was a mistake. We laughed and hurt his feelings, so: We shouldn't have laughed at him. Not needn't have laughed, which only means it wasn't necessary.
Question 11
He _____ hungry. He isn’t eating his food.
Must be makes a strong deduction about the present. The clue He isn't eating is present, and we feel sure. Not should be, which is about what is expected or right, not a conclusion from evidence.
Question 12
All the lights in Terry’s room are turned off. He _____ sleeping.
The lights are off now, so we make a present deduction: He must be sleeping (must be + -ing). Not must have been sleeping, because that is about the past, and the evidence is about now.
Question 13
Let’s go to the seminar. It _____ interesting.
Be supposed to be means 'people say' or 'it is expected to be': It is supposed to be interesting. That is a good reason to say Let's go. Not used to be, which is about the past and doesn't explain why we should go now.
Question 14
Children _____ taught to respect their elders.
We need a modal + passive: should be + past participle (taught). It gives advice about what is right. Has to be doesn't agree with the plural Children, and had better and must have can't be followed by taught here.
Question 15
This application _____ last week.
For a past duty that wasn't done, we use ought to have been + past participle in the passive: This application ought to have been sent last week. Not should have sent, which is active, but the application can't send itself.
Question 16
We can’t wait any longer! Something _____ immediately.
Must be done is a present modal passive that shows strong necessity now. The clue is We can't wait any longer and immediately. Not had to be done, which is past.
Question 17
A: I overslept this morning.
B: You _____ your alarm clock.
Should have + past participle criticises something that didn't happen in the past: You should have set your alarm clock. Not had better set, which is advice for now or the future, but the oversleeping has already happened.
Question 18
Rice _____ have water in order to grow.
Must expresses something that is absolutely necessary: rice cannot grow without water. Should and ought to are only advice, and had better is a warning about a specific situation.
Question 19
I _____ to go now. I have a class in ten minutes.
The gap is followed by to, so only have got to (= have to) fits: I have got to go now. Had better, must and should are all followed by the base verb without to.
Question 20
“Doctors are supposed to help sick people” means: _____.
Be supposed to means what someone is expected to do, like should: Doctors are supposed to help sick people = They should help sick people. Not They help sick people, which is a simple fact and doesn't show expectation.
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