Do you understand the distinction between countable and uncountable?
Every subset of a countable set is countable.
Every superset of an uncountable set is uncountable.
The set of integers, Z, is countable.
The set of reals, R, is uncountable.
Given those facts, answer #1 & #4.
If define the image function .
Now show that if is bijective then is bijective.
That proves #2.
For #3 note that is countable and every quadratic equation has at most two real roots.