Solution — 2024 Question 1 Quadratic Equations & Inequalities

Equating the two equations,

$\begin{matrix} k x + k = k - 1 + 3 k x - x^{2} \\ x^{2} - 2 k x + 1 = 0 \end{matrix}$

For the line to intersect the curve at two distinct points,

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

$k < - 1 ∎ or k > 1 ∎$