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KVPY2013MathematicsQuadratic Equation

Let f ( x ) be a non-constant polynomial with real coefficients such that f ( 1 2 )=100 and f ( x ) 100 for all real x . Which of the following statements is NOT necessarily true?

Options

  1. AThe coefficient of the highest degree term in f ( x ) is negative
  2. Bf(x) has at least two real roots
  3. CIf x 1 / 2 then f(x) < 100
  4. DAt least one of the coefficients of f ( x ) is bigger than 50.

Correct answer

C. If x 1 / 2 then f(x) < 100

Step-by-step solution

Coefficient of highest degree term must be negative becuase ifit is positive, then x , y and it is not possible, since f(x) 100 . Now, graph will be like at least two real roots will be there, & if x 1 2 , then f(x) Now, let the highest coefficients, it can have is 49 then, f ( 1 2 )=49+ 49 2 + 49 2² + But the sum cannot be equal to 100 .

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