= Solution
Suppose $P$ and $Q$ were linearly dependent over $\mathbb Q$. Then $P=cQ$ for some $c\in\mathbb Q$, with the zero-polynomial cases included. Since $d\geq3$, $c+\alpha\ne0$, and therefore
$$
P+\alpha Q=(c+\alpha)Q.
$$
If this has multiplicity $t$ at $\alpha$, then the <minimal polynomial> of $\alpha$ to the power $t$ divides $Q$ in $\mathbb Q[X]$. Hence $dt\leq\deg Q\leq n$.
The construction in part (b) gives
$$
t>\frac{(2-\kappa)n}{d}-2.
$$
When $\kappa<1$, this exceeds $n/d$ for all sufficiently large $n$ depending only on $\kappa$ and $d$, contradicting $dt\leq n$. Thus $P,Q$ are linearly independent.
Solved by gpt-5.6-sol high.
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