What if the segments could intersect each other but must, Computer Engineering

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What if the segments could intersect each other but must

Question 1: Prove or disprove: a countable set of parabolas (arbitrarily oriented and placed) can completely cover (every point inside) the unit square in the plane (i.e., the interior and boundary of a square of side 1)

Question 2: Prove or disprove: an uncountable set of pairwise-disjoint line segments can completely cover (every point in) the unit disk in the plane (i.e., the interior and boundary of a circle of diameter 1).

What if the segments could intersect each other, but must all have unique slopes?

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