Explanation:
The general rule here is to use that, with no comma preceding it, if the adjective clause following is essential to the sentence. Here, the fact that the projects were undertaken to improve government services is the point of the sentence. If we wanted to introduce a little tidbit of inessential information, we could use which with a comma preceding it:

Einstein is the product of government projects, which were conducted over a period of three years.

If you follow AP style, that’s the rule. However, rules change and this one seems to be a case in point.

Here’s an example from World Wide Words:

Another cause of stress is a traumatic event that is out of the ordinary and has a major impact on the person’s life.

The argument here is that the clause “that is out of the ordinary and has a major impact on the person’s life” modifies and constrains “event”. It’s not just any event but one specific type of event, to the extent that the whole block from “event” onwards forms one idea. The clause is restrictive.

Older grammar books make two firm points about the difference between the two types of clause:

* Restrictive clauses are introduced by that and are not separated from the rest of the sentence by commas.
* Non-restrictive clauses are introduced by which and must be separated by commas from the rest of the sentence to indicate parenthesis.

Author Michael Quinion goes on to discuss the way those rules are changing and when you can — and can’t — bend the rules. His conclusion:

If you wish to write naturally, don’t fuss too much about the usage of that versus which. Obsessive correction (sarcastically called a which hunt) is best avoided. If your sense of the language is not strong enough to be sure of the right pronoun, use that for the restrictive cases and which for the others and you won’t go wrong.

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Answer: a.

Explanation:

Grammar Girl Mignon Fogarty explains that vs. which and restrictive vs. non-restrictive clauses:

Restrictive Clause–That
A restrictive clause is just part of a sentence that you can’t get rid of because it specifically restricts some other part of the sentence. Here’s an example:

* Gems that sparkle often elicit forgiveness.

The words that sparkle restrict the kind of gems you’re talking about. Without them, the meaning of the sentence would change. Without them, you’d be saying that all gems elicit forgiveness, not just the gems that sparkle. (And note that you don’t need commas around the words that sparkle.)

Nonrestrictive Clause–Which
A nonrestrictive clause is something that can be left off without changing the meaning of the sentence. You can think of a nonrestrictive clause as simply additional information. Here’s an example:

* Diamonds, which are expensive, often elicit forgiveness.

Alas, in Grammar Girl’s world, diamonds are always expensive, so leaving out the words which are expensive doesn’t change the meaning of the sentence. (Also note that the phrase is surrounded by commas.)

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The new company camcorders, _______ are smaller and take better video, cost half as much as the camcorders we already had.
a. which
b. that

Answer:a

This is a tricky one because you have to look at the meaning of the sentence. In the sentence above, the fact that the new cameras are smaller and take better video is not essential to the meaning of the sentence.  ( The essential fact is that they cost half as much.) Use which.

In a statement, that introduces essential information in what is called a restrictive clause. Which introduces extra information in what’s called a nonrestrictive clause.

Here’s an example of how choosing which or that can change the meaning of the sentence.

• The computers are new.
• The computers, which have 2 gigs of RAM, are new. (you are providing extra information about the new computers)
• The computers that have 2 gigs of RAM are new. (you are saying that only the computers with 2 gigs of RAM are new — the others are old)

Need an easy way to remember? Use which if you’re providing nonessential information.

Answer: b

Explanation:
Typically, you would use which in this sentence. The use of which indicates that there was a single contract AND it was difficult to follow.

The use of that would indicate that one contract (of some unspecified number) was difficult to follow and the client complained about it.