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KVPY2016MathematicsQuadratic Equation

Consider the quadratic equation n x²+7 n x+n=0 , where n is a positive integer. Which of the following statements are necessarily correct? I. For any n , the roots are distinct. II. There are infinitely many values of n for which both roots are real. III. The product of the roots is necessarily an integer.

Options

  1. AIII only
  2. BI and III only
  3. CII and III only
  4. DI, II and III

Correct answer

B. I and III only

Step-by-step solution

aligned D &=49 n -4 n ² &= n (49-4 n ) aligned D 0 for any n I ⁺ . So roots are distinct For roots to be real D 0 So n 49 4 So n can be 1,2,3, 12 Clearly product of the roots is 1

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