CDS 2026 (I) Elementary Mathematics, question 19

Topic: Algebra

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Q19

Let Q=x2+bx+cQ=x^{2}+bx+c. If the sum of the roots is equal to product of the roots of the equation Q=0Q=0, then which of the following statements is/are correct ?

I. Q can be a perfect square.

II. Q is positive for all real values of xx.

Select the answer using the code given below :

  1. (a) I onlyCorrect answer
  2. (b) II only
  3. (c) Both I and II
  4. (d) Neither I nor II

Solution.

Answer: (a) I only

  1. By the root–coefficient relations, −b=c- b = c, so Q=x2+bx−bQ = x^{2} + bx - b.
  2. Taking b=−4b = - 4 gives Q=(x−2)2Q = (x - 2)^{2}, proving I.
  3. The same example is zero at x=2x = 2, so Q need not be strictly positive for every real x.
  4. For an example with negative values, b=1b = 1 gives Q(0)=−1Q(0) = - 1.