Is truth a necessary condition for knowledge?

You can't know something which is false. You can believe it, mistakenly.

Before the Copernican revolution, it was known that the "heavens" appear to revolve around the earth. It was not known that the earth actually was spinning and revolving around the sun. It was believed that the earth was stationary and the "heavens" revolved around it. It was not known that this belief was, in fact, false, and therefore not knowledge.

How do we tell the difference between false belief and true knowledge? By omniscience? No. Just by additional observation.

Truth is definitely a necessary condition for knowledge, and I think you should be able to see that in the definition of knowledge from AR that you quote, which says knowledge is a mental grasp of a FACT.

Of course, we don't learn much simply by deducing from definitions. Consider what would happen if everything we believed with some justification also counted as knowledge. Then the pre-Copernican belief that the earth is at the center of the heavens would be knowledge, as would the belief that the earth is not at the center. But these are contradictory propositions. What we know when we know is reality. If we know P at one time, but we know non-P at a succeeding time, then either the universe has actually changed, or reality is contradictory.

But there's no reason to think that the universe changed between the time of Ptolemy and Copernicus. And there's every reason to think that the universe is not contradictory. So when and if a theory is false, like pre-Copernican systems of astronomy, these theories do not constitute knowledge. That doesn't mean that people who believe these theories don't have any knowledge at all. Doubtless they have a lot, and this includes the knowledge they have about the evidence for their false theories. So they might have justified belief in their theory--justified in virtue of some knowledge--but the theory itself is not knowledge.

It is important not to take the idea that knowledge is contextual to imply that every theory that is succeeded by another in the history of science is "true in its context." You will search in vain for references in the Objectivist literature that actually say this, and it's a good thing, because it's hard to make sense of how this could imply anything other than outright cultural relativism.

Incidentally, the fact that knowledge implies truth (and justification, and perhaps belief) doesn't mean that contemporary philosophers are correct to think that the concept of knowledge should be analyzed as a kind of justified, true belief. It's one thing for a concept to imply other concepts; it's another for the one concept to be definable in terms of those concepts. (If you're interested in reading a contemporary philosopher who critiques the JTB analysis along these lines, please consult the works of Timothy Williamson.)

You're probably studying this issue because of the Gettier problem, which involves finding counterexamples to the JTB definition. In my view, at least, many of these counterexamples are successful, but it's no problem, because knowledge shouldn't be defined as justified true belief in the first place. It's a primitive concept, close to being an axiomatic concept like "consciousness," and therefore close to needing no definition at all. (I say "close," because knowledge is one particular kind of consciousness, distinguished from others that are not retained over time.)