The idiom "put one's mind to" means to concentrate on something and make a serious effort to do it.

He has puzzled ____ the figures for hours without being able to make head or tail of them.
(a) on (b) over (c) with (d) from

The phrasal verb "put to" is not standard here. The verb "put forward" is the correct one, which means to suggest a plan or idea. The provided answer key is (b). The phrase "put to" can be used to present something for consideration, so I will stick to the provided key.

I am sure we can solve this problem if we can put our mind ____ it.
(a) to (b) in (c) on (d) with

The phrasal verb "put out" means to issue or publish a statement or information.

The committers' pro-posals were put ____ briefly and force-fully.
(a) with (b) to (c) on (d) over

The idiom "put one's finger on" means to identify or pinpoint something.

A description of the wanted man has been put ____ to all mobile police petrol.
(a) on (b) about (c) around (d) out

The idiom "put a brave face on" something means to appear to be calm and confident in a difficult situation.

Then what is the trouble? 'Nothing explicit that I can put my finger ____.
(a) on (b) in (c) into (d) for

The idiom "put into effect" means to implement or bring something into operation.

There is not much optimism in the air, but at least both sides are putting a brave face ____ proceedings.
(a) to (b) in (c) upon (d) on

The phrase "put in trust" is not a standard one. The correct idiom is "put their trust in." The phrasal verb "put in" can mean to submit or apply. The provided answer is (a). "Put in their trust" is not common usage. Let's assume the question is a bit off, and the correct phrase is "put trust in." I will stick to the provided key.

Your proposal looks good on the paper, but I am not yet convinced it can be put ____ effect.
(a) in (b) into (c) to (d) on

The idiom "put heart into" means to give courage or confidence to someone.

They did not put ____ their trust in the information they received from the well tried and veteran staff of eighth army.
(a) in (b) on (c) with (d) into

The verb "purged" is followed by the preposition "of," which means to get rid of something undesirable.

His arrival put heart ____ his subordinates.
(a) to (b) with (c) into (d) in

The verb "provides" is followed by the preposition "for," which means to make a provision for something.

She felt purged ____ all feelings of guilt.
(a) of (b) with (c) from (d) off