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JEE Main20231 Feb 2023Morning ShiftMathematicsProperties of TrianglesActual

For a triangle A B C , the value of cos 2 A + cos 2 B + cos 2 C is least. If its inradius is 3 and incentre is M , then which of the following is NOT correct?

Options

  1. APerimeter of ∆ A B C is 18 3
  2. Bsin 2 A + sin 2 B + sin 2 C = sin A + sin B + sin C
  3. CMA → · MB → = - 18
  4. Darea of ∆ A B C is 27 3 2

Correct answer

D. area of ∆ A B C is 27 3 2

Step-by-step solution

We know that, cos 2 A + cos 2 B + cos 2 C ≥ - 3 2 i.e., min cos 2 A + cos 2 B + cos 2 C = - 3 2 So, cos 2 A = cos 2 B = cos 2 C = - 1 2 ⇒ A = B = C = 60 ° So, triangle A B C is an equilateral triangle. Let the side of triangle be a . Also, r = ∆ s = 3 ⇒ 2 ∆ 2 s = 3 ⇒ 2 3 4 a 2 3 a = 3 ⇒ a = 18 3 = 6 3 So, perimeter is = 3 a = 18 3 units. Area = 3 4 a 2 = 81 3 = 27 3 sq. units

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