UPSC CDS I 2024 Math Paper

Concept:A triangle is equilateral if all three sides are equal. Statements provide equations that can force $a=b=c$.Explanation:Statement I: $a^2 + b^2 + c^2 = ab + bc + ca$.Rewrite as $(a-b)^2 + (b-c)^2 + (c-a)^2 = 0$.Each square is non‑negative, so each must be zero. Hence $a = b = c$.The triangle is equilateral. Statement I alone is sufficient.Statement II: $3a^2 + 3b^2 + 4c^2 = 2ab + 4bc + 4ca$.Bring all terms to one side:$3a^2 + 3b^2 + 4c^2 - 2ab - 4bc - 4ca = 0$.Factorise as $(a-b)^2 + 2(b-c)^2 + 2(c-a)^2 = 0$.All terms are non‑negative; each must be zero.Thus $a=b$, $b=c$, and $c=a$, so $a=b=c$.The triangle is equilateral. Statement II alone is also sufficient.Since each statement alone answers the question, the correct choice is option B.Answer:Option B (If the Question can be answered by either Statement alone).