= Solution
A set is countable when it is finite or admits a bijection with a subset of $\mathbb N$. For each degree, the coefficient tuples for <polynomials> in $A[x]$ form a finite Cartesian power of the <countable set> $A$; a countable union over degrees is countable. In particular the algebraic numbers, being roots of countably many integer <polynomials> with finitely many roots each, are countable. Since $\mathbb R$ is uncountable, uncountably many real numbers are transcendental.
The set $Q$ is uncountable. Partition $N$ into four-element blocks and, independently on each block, choose either of two fixed-point-free permutations. Binary <sequences> then inject into $Q$.
Finally choose a line $L$ distinct from and nonparallel to every $L_i$, possible because only countably many directions are excluded. Each $L\cap L_i$ has at most one point, so the union covers only countably many points of the uncountable line $L$. It cannot cover the plane.
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