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MHT CET202522 Apr 2025Evening ShiftMathematicsProperties of TrianglesActual

In a triangle ABC with usual notations, if 3 a= b + c , then B 2 C 2 =

Options

  1. A1
  2. B2
  3. C2
  4. D3

Correct answer

C. 2

Step-by-step solution

The product 【cot(B/2) × cot(C/2)】 is evaluated using the half-angle formulas in a triangle: B 2 = s(s − b) (s − a)(s − c) and C 2 = s(s − c) (s − a)(s − b) , where s = a + b + c 2 is the semi-perimeter. Multiplying the expressions gives: B 2 C 2 = s(s − b) (s − a)(s − c) s(s − c) (s − a)(s − b) = s^2 (s − a)^2 = s s − a . Given b + c = 3a , the semi-perimeter simplifies to s = a + b + c 2 = a + 3a 2 = 2a . Substitute into the product: B 2 C 2 = 2a 2a − a = 2a a = 2. Thus, the value is C from the options provided.

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