Maximum Eggs Bought by Mr. Jose

Let total eggs bought = $9x + 2$. Rotten eggs = $2$, remaining = $9x$. Eggs kept in fridge = $5x$, brought to mother's house = $4x$. Mother cooks 2 eggs, leaving $4x - 2$ eggs in her fridge. Her fridge holds at most 5 eggs: $4x - 2 \le 5 \implies 4x \le 7 \implies x \le 1.75$. Since $x$ must be a positive integer, the maximum value of $x$ is $1$. Maximum eggs bought = $9(1) + 2 = 11$.