Is it a necessary and sufficient condition that invalid arguments have a false conclusion?
1) An invalid argument may have a true conclusion:
Consider the invalid argument: "Because Socrates is mortal, all humans are mortal." Nevertheless the conclusion is true: "All humans are mortal."
2) An invalid argument may have a false conclusion:
Because Socrates is Greek, all humans are Greek.
3) A valid argument may have a false conclusion:
Because all humans live in Africa, also Socrates lives in Africa. (this kind of valid argument is named "ex falso quodlibet")
4) A valid argument may have a true conclusion:
Because all humans are mortal, also Socrates is mortal.
You may want to look at the answer to The validity of the definition of a valid argument
but having a false conclusion is neither a necessary nor sufficient condition for an invalid argument.
The definition of validity (assuming we're not working with Tarski's model theory) is that
an argument is valid if it is the case that it cannot have all true premises and a false conclusion.
Thus, a valid argument can have a false conclusion if it is not the case all of the premises are true.
Conversely, an invalid argument can have a true conclusion and/or all true premises.
Pigs can fly. Therefore, apples are a fruit.
false premise. True conclusion. Invalid argument. (Because it would be possible for the premises to all be true and the conclusion false).
If pigs can fly, then Obama is president. Therefore, Japan is a country.
True premise. true conclusion. Invalid argument. (Because it would be possible for the premises to all be true and the conclusion false).