almost all books have a few errors in them in spite of the care taken to check their proof pages before the final printing. is this sentence correct?
I don't think you need to be so specific:

Almost all books contain errors in spite of the care taken to proof them before printing.
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Is the word "care taken" correct? Why it is not "care is taken" ? I think it is in passive form.
Is the word "care taken" correct?- Yes.
Why it is not "care is taken" -- In fact, the non-finite clause ('the care given') is a reduced version of 'the care that is given'.

It is important that everyone have his car check once or twice a year.

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