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KVPY2017MathematicsQuadratic Equation

Let S = a ²+ b ²+ c ² ab + bc + ca : a , b , c R , ab + bc + ca 0 where R is the set of real numbers. Then S equals [2017]

Options

  1. A(- ,-1] [1, )
  2. B(- , 0) (0, )
  3. C(- ,-1] [2, )
  4. D(- ,-2] [1, )

Correct answer

D. (- ,-2] [1, )

Step-by-step solution

Case-I array l (a-b)²+(b-c)²+(c-a)² 0 a²+b²+c²-a b-b c-c a 0 a²+b²+c² a b+b c+c a If a b+b c+c a>0 array Then a²+b²+c² a b+b c+c a 1 Case-II Let a b+b c+c a < 0 array l (a+b+c)² a b+b c+c a 0 a²+b²+c²+2(a b+b c+c a) a b+b c+c a 0 a²+b²+c² a b+b c+c a +2 0 a²+b²+c² a b+b c+c a -2 array So, Range is (- ,-2] [1, )

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