Sets, Relations & Functions
Counting Elements in Relation via Quadratic
nta_pyq_2023_apr
Grade 11
Question:
Let $A=\{1,2,\ldots,10\}$ and $B=\{0,1,2,3,4\}$. The number of elements in the relation $R=\{(a,b)\in A\times A:\ 2(a-b)^2+3(a-b)\in B\}$ is __________.
Step-by-Step Solution
Key Concept: Let $a-b=n$. Need $2n^2+3n\in\{0,1,2,3,4\}$. For $n=0$: $0\in B$ — 10 pairs. For $n=-2$: $2\in B$ — 8 pairs.
18 elements.
Correct Answer: 18