Matrices & Determinants
Sets S₁ and S₂ — Unique and Infinite Solutions
nta_pyq_2023_jan
Grade None
Question:
Let $S_1$ and $S_2$ be the sets of all $a\in\mathbb{R}-\{0\}$ for which the system $ax+2ay-3az=1$, $(2a+1)x+(2a+3)y+(a+1)z=2$, $(3a+5)x+(a+5)y+(a+2)z=3$ has unique solution and infinitely many solutions respectively. Then:
$n(S_1)=2$ and $S_2$ is an infinite set
$S_1$ is an infinite set and $n(S_2)=2$
$S_1=\Phi$ and $S_2=\mathbb{R}-\{0\}$
$S_1=\mathbb{R}-\{0\}$ and $S_2=\Phi$
Step-by-Step Solution
Key Concept: $\Delta=a(15a^2+31a+36)=0\Rightarrow a=0$ only real root. For $a\in\mathbb{R}-\{0\}$: $\Delta\neq0$, so system always has unique solution.
$S_1=\mathbb{R}-\{0\}$, $S_2=\Phi$.
Correct Answer: 4