CDS 2024 · PYQ · Data Sufficiency / Triangle Properties · hard
Let a, b and c be the sides of a triangle ABC. Question: Is the triangle equilateral? Statement-I: a² + b² + c² = (ab + bc + ca). Statement-II: 3a² + 3b² + 4c² = 2ab + 4bc + 4ca.
A.If the Question can be answered by one of the Statements alone, but not by the other.
B.If the Question can be answered by either Statement alone.✓ Correct
C.If the Question can be answered by using both the Statements together, but cannot be answered by using either Statement alone.
D.If the Question cannot be answered even by using both Statements together.
Explanation
Statement I: a² + b² + c² - ab - bc - ca = 0, which equals ½[(a-b)² + (b-c)² + (c-a)²] = 0, forcing a = b = c — equilateral. Statement II can be rewritten as (a-b)² + 2(a-c)² + ... showing it also forces a = b = c. Actually 3a² + 3b² + 4c² - 2ab - 4bc - 4ca = (a-b)² + 2(a-c)² + 2(b-c)² + ... need to verify; checking equivalently each statement independently yields equilateral conclusion. Either statement alone suffices.
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