The lengths of the sides of a triangle are 10 + x 2 , 10 + x 2 and 20 - 2 x 2 . If for x = k , the area of the triangle is maximum, then 3 k 2 is equal to

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Area of triangle, Δ = 20 20 - 10 - x 2 20 - 10 - x 2 20 - 20 + 2 x 2 Δ = 2 10 x 10 - x 2 For maxima, d Δ d x = 0 ⇒ 2 10 10 - x 2 - 4 10 x 2 = 0 ⇒ x 2 = 10 3 Here, d 2 ∆ d x 2 < 0 Hence 3 k 2 = 10

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