Step-by-Step Solution
Key Concept: For a quadratic with roots in a given interval, check discriminant positivity and function values at interval endpoints.
Given $D > 0$, $f(2) < 0$ and $f(3) < 0$. From $D = (1-2\lambda)^2 - 4(\lambda^2 - \lambda - 2) = 1 + 3\lambda^2 - 4\lambda + 8 = 9 > 0$ (always true). From $f(2) < 0$: $4 - 2(1-2\lambda) + (\lambda^2 - \lambda - 2) < 0 \Rightarrow \lambda^2 - 5\lambda + 4 < 0 \Rightarrow (\lambda - 1)(\lambda - 4) < 0 \Rightarrow 1 < \lambda < 4$. From $f(3) < 0$: $9 - 3(1-2\lambda) + (\lambda^2 - \lambda - 2) < 0 \Rightarrow \lambda^2 - 7\lambda + 10 < 0 \Rightarrow (\lambda - 2)(\lambda - 5) < 0 \Rightarrow 2 < \lambda < 5$. Taking intersection: $\lambda \in (2, 4)$.
Correct Answer: 2