JEE Mathematics 2025 Question with Solution

Given: $A = \{1,2,3,\dots,10\}$ and $R = \{(a,b): a = 2b+1\}$.

Find: The largest integer $k$ such that consecutive ordered pairs in the sequence satisfy that the second entry of one pair is the first entry of the next pair.

From $a = 2b+1$, every pair in $R$ has the form

with both entries in $A$.

Now list all possible pairs using $b \in A$ and requiring $2b+1 \in A$:

which gives

For two consecutive pairs $(x,y)$ and $(u,v)$ in the sequence, the condition says that the second entry of the first pair equals the first entry of the next pair, so

If $(a,b) \in R$, then $a = 2b+1 > b$. So every arrow goes from a larger number to a smaller number. That means any valid chain must strictly decrease at each step, so very long chains are impossible.

The solution declares the correct option as C and concludes the final maximum value is $5$. Therefore, the correct option is C.