I would like to get basketball officials opinions on this senario. It appeared on our state basketball exam.

First quarter: Administrative technical foul on Team B. By accident the officials shoot the 2 Free throws at the wrong basket. A1 makes both shots. 2 points are added to team A's score.

Fourth quarter ends. Game tied. Team B coach calls the officials over and points out the error in the first quarter. He says that since the shots we administered at his basket, the 2 points should have been given to his team (Team B) and not to Team A (the team who shot the 2 Free throws into the wrong basket).

Official rules that:
1- It is a correctable error but that it is too late to correct.
2- Even though the shots went into Team B's basket they were shot by A1 and they count for Team A.
Game tied. Overtime will be played.

Is the official correct?

Yep. It's correctable but too late. Also, the correction would be to make A shoot the throws again, not give the points to B. Move on with the overtime, and don't forget to take a timeout away from B.

Let me try this one,

Yes the official is correct per 2-10.2

I've got that it is too late to do anything about it. Bonus basketball

Clarification note
 

I probably need to add a little clarification....

We all know that this is a correctable error.
We all know that it is too late to correct.

This is not the point of the question.

The point of the question is that the coach of Team B thinks the 2 points should count for HIS team since they went into HIS basket.

The referee says that the 2 points (even though they were shot into the wrong basket) count for THE TEAM WHO SHOT (and made) THE TWO FREE THROWS.

Is the referee correct on this point?

Re: Clarification note
 

yes

Yes, the ref is absolutley correct.

2-10.5

"Points scored, consumed time and additional activity, which may occur prior to the recognition of an error, shall not be nullified"

Good enough for me.

Let's also add that any coach that tries to win a game this way has no class.

Since the officials screwed up by lining the shooter up incorrectly, you can't penalize the team for making their own shots. Even if they caught this in time, you still wouldn't grant the points to B. The most you would do is remove the points and make A reshoot at the right basket. I don't even think I would do this.
Is this really a CE?

See 2-10.

Yes, if it would be if it was caught in time. 2-10-1d & 2-10-4

So you make A reshoot then? (If caught in time?)

Yes

An ugly situation in any case but the rules indicate to me that this a correctable error, with a remedy, if discovered in time.

What if they missed the shots, then do you let them shoot them at the right basket? What if they make one of two?

I say you swallow your medicine, follow the rules and shoot them over at the proper basket.

Yep, it's one of those 5 famous CE's - and 4 of the 5 have to do with free throws (2-10-1). And it does say that the correction (if caught in time) is to re-shoot. That's one of those things that's good to include in pre-game: get together with your partners in the event of something strange, such as T's. In this case, since the game started with a T, the partners should've gotten together and communicated who got the T, who's shooting (and where), who's got the ball after, what direction is the arrow going to point, etc. etc. This way, you don't have that issue of explaining to a coach why you're taking points off the board and making them re-shoot.

Also, I agree completely that the coach who waits until the end of the game to bring this up has no class.