UPSC NDA 2 2022 Mathematics Solved Paper

Concept:Use the inclusion-exclusion principle for three sets and Venn diagram analysis to find the number of students who received medals in exactly two sports.Explanation:Let $A$, $B$, $C$ denote students who got medals in basketball, football, and volleyball respectively.Given: $n(A)=5x$, $n(B)=4x+15$, $n(C)=x+25$, $n(A \cup B \cup C)=15x$, and $n(A \cap B \cap C)=x$ with $x<6$.Apply the formula:$n(A \cup B \cup C)=n(A)+n(B)+n(C)-n(A \cap B)-n(B \cap C)-n(C \cap A)+n(A \cap B \cap C)$$\Rightarrow 15x = 5x+(4x+15)+(x+25) - [n(A \cap B)+n(B \cap C)+n(C \cap A)] + x$Simplify: $15x = 11x+40 - [\text{sum of pairwise intersections}]$So $n(A \cap B)+n(B \cap C)+n(C \cap A)=40-4x$. (1)In a Venn diagram, let $p,q,r$ denote the number of students in exactly two sports (excluding the triple overlap), and $s=x$ denote those in all three. Then:$n(A \cap B)+n(B \cap C)+n(C \cap A) = (p+s)+(q+s)+(r+s)=p+q+r+3s$Thus, number in exactly two sports = $p+q+r = [n(A \cap B)+n(B \cap C)+n(C \cap A)] - 3x$Substitute from (1): $= (40-4x)-3x = 40-7x$.Answer:$40-7x$ students received medals in exactly two of the three sports. (Option C)