Matrices & Determinants
Inconsistent System — Sum over Parameter Set
nta_pyq_2023_jan
Grade 12
Question:
Let $S$ denote the set of all real values of $\lambda$ such that the system $\lambda x+y+z=1$, $x+\lambda y+z=1$, $x+y+\lambda z=1$ is inconsistent. Then $\displaystyle\sum_{\lambda\in S}(|\lambda|^2+|\lambda|)$ is equal to:
Step-by-Step Solution
Key Concept: $\Delta=(\lambda+2)(\lambda-1)^2=0\Rightarrow\lambda=-2$ or $\lambda=1$. At $\lambda=1$: infinite solutions (not inconsistent). At $\lambda=-2$: inconsistent. So $S=\{-2\}$.
$6$.
Correct Answer: 4