Sets, Relations & Functions
Set Operations / Geometric Interpretation
nta_pyq_2025_apr
Grade 11
Question:
Let $A = \{(x,y) \in \mathbb{R} \times \mathbb{R} : |x+y| \geq 3\}$ and $B = \{(x,y) \in \mathbb{R} \times \mathbb{R} : |x|+|y| \leq 3\}$. If $C = \{(x,y) \in A \cap B : x = 0 \text{ or } y = 0\}$, then $\sum_{(x,y) \in C} |x+y|$ is:
Step-by-Step Solution
Key Concept: Region $A$ is the exterior (or boundary) of the strip $|x+y|<3$; region $B$ is the diamond $|x|+|y|\leq 3$. Their intersection consists of only the four vertices of the diamond where the axes meet the boundary.
$A\cap B$ on the axes gives $C = \{(-3,0),(3,0),(0,3),(0,-3)\}$. $\sum |x+y| = 3+3+3+3 = 12$.
Correct Answer: 12