Matrices & Determinants
No-Solution Conditions — Sum Over Solution Set
nta_pyq_2023_apr
Grade 12
Question:
Let $S$ be the set of values of $\lambda$ for which the system $6\lambda x-3y+3z=4\lambda^2$, $2x+6\lambda y+4z=1$, $3x+2y+3\lambda z=\lambda$ has no solution. Then $12\displaystyle\sum_{\lambda\in S}|\lambda|$ is equal to _______.
Step-by-Step Solution
Key Concept: No solution requires $\Delta=0$ but $\Delta_i\neq0$. Compute $\Delta=9\lambda^3-7\lambda-2$ and factor.
$S=\{1,-\frac{1}{3},-\frac{2}{3}\}$. $12\sum|\lambda|=12(1+\frac{1}{3}+\frac{2}{3})=24$.
Correct Answer: 24