Question #40546

This question tests the 'Cross Multiplication Trap' in rational inequalities and its intersection with Cramer's Rule constraints.

Students commonly cross-multiply the denominator without considering domain restrictions, incorrectly simplifying $\frac{\lambda^2 - 7\lambda + 10}{\lambda - 5} \le 0$ to $\lambda^2 - 7\lambda + 10 \le 0$.

This yields $\lambda \in [2, 5]$, leading to an incorrect maximum value of $\alpha = 5$.

Substituting $\alpha = 5$ into the system yields a non-zero determinant ($D = 3$), resulting in a unique solution where $y+z=1$, which perfectly matches a distractor option.

Recognizing the domain constraint $\lambda \neq 5$ correctly identifies $\alpha = 2$, which makes the system's matrix singular ($D=0$) with $D_y \neq 0$, correctly identifying the system as inconsistent.