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MHT CET20255 May 2025Evening ShiftMathematicsProperties of TrianglesActual

In a triangle ABC , with usual notations, the sides a, ~b , c are such that they are roots of the equation x^3-11 x^2+38 x-40=0 then A a + B b + C c =

Options

  1. A9 16
  2. B3 4
  3. C1
  4. D5 16

Correct answer

A. 9 16

Step-by-step solution

Let a , b , and c be the roots of x^3-11x^2+38x-40=0 , which represent the sides of triangle ABC . Using Vieta's formulas: a+b+c=11 , ab+bc+ca=38 , and abc=40 . Testing integer divisors of 40 reveals x=2 is a root. Factoring gives (x-2)(x-4)(x-5)=0 , so the sides are 2 , 4 , and 5 . Evaluate A a + B b + C c using the identity: A a + B b + C c = a^2+b^2+c^2 2abc . Substitute values: a^2+b^2+c^2=4+16+25=45 and abc=40 , yielding 45 80 = 9 16 . The result is 9 16 .

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