Solution — 2020 Question 1 Quadratic Equations & Inequalities

For the quadratic to be always positive, the graph must lie entirely above the $x$ -axis, which means it must have no real roots. Hence the discriminant must be negative.

\begin{matrix} b^{2} - 4 a c & < 0 \\ {(k - 2)}^{2} - 4 (2 k + 1) (1) & < 0 \\ k^{2} - 4 k + 4 - 8 k - 4 & < 0 \\ k^{2} - 12 k & < 0 \\ k (k - 12) & < 0 \end{matrix}

$0 < k < 12 ∎$