Sets, Relations & Functions
Floor Function — Intersection of Sets A and B
nta_pyq_2023_apr
Grade 11
Question:
Let $A=\{x\in\mathbb{R}:[x+3]+[x+4]\leq3\}$, $B=\left\{x\in\mathbb{R}:\ 3^x\left(\displaystyle\sum_{r=1}^\infty\frac{3}{10^r}\right)^{x-3}<3^{-3x}\right\}$. Then,
$B\subset A,\ A\neq B$
$A\cap B=\emptyset$
$A\subset B,\ A\neq B$
$A=B$
Step-by-Step Solution
Key Concept: For $A$: $[x+3]+[x+4]=2[x]+7\leq3\Rightarrow[x]\leq-2\Rightarrow A=(-\infty,-1)$. For $B$: geometric series sum $=\frac{1}{3}$; condition simplifies to $3^{x+1}<1\Rightarrow B=(-\infty,-1)$.
$A=B=(-\infty,-1)$.
Correct Answer: 4