Let A = {n $ \in $ N: n is a 3-digit number} B = {9k + 2: k $ \in $ N} and C = {9k + $l$: k $ \in $ N} for some $l ( 0 < l < 9)$ If the sum of all the elements of the set A $ \cap $ (B $ \cup $ C) is 274 $ \times $ 400, then $l$ is equal to ________.

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $\mathrm{m}+\mathrm{n}$ is equal to ___________.

Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more than the total number of subsets of B, then the value of m.n is ______.

Let $A = {2, 3, 6, 8, 9, 11}$ and $B = {1, 4, 5, 10, 15}$. Let $R$ be a relation on $A \times B$ defined by $(a, b)R(c, d)$ if and only if $3ad - 7bc$ is an even integer. Then the relation $R$ is

Let $\mathrm{A}=\{-4,-3,-2,0,1,3,4\}$ and $\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.$ or $\left.b^{2}=a+1\right\}$ be a relation on $\mathrm{A}$. Then the minimum number of elements, that must be added to the relation $\mathrm{R}$ so that it becomes reflexive and symmetric, is __________