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TMUA 2016 · Paper 2 · Question 15 of 20

TMUA 2016 Paper 2 Question 15

Algebra and functions — Quadratics · translating a condition on the gap between the roots. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Algebra and functionsQuadratics · translating a condition on the gap between the roots5 options
Consider the quadratic f(x) = x2 − 2px + q and the statement:

(*) f(x) = 0 has two real roots whose difference is greater than 2 and less than 4.

Which one of the following statements is true if and only if (*) is true?

  1. Aq < p2 < q+4
  2. B√q+1 < p < √q+4
  3. Cq − 3 ≤ p2 − 4 ≤ q
  4. Dq < p2 − 1 < q+3
  5. Eq − 2 < p2 − 3 < q+2
Show the answer and worked solution
answer · D
  1. Aq < p2 < q+4
  2. B√q+1 < p < √q+4
  3. Cq − 3 ≤ p2 − 4 ≤ q
  4. Dq < p2 − 1 < q+3
  5. Eq − 2 < p2 − 3 < q+2
If the roots are α and β then α + β = 2p and αβ = q, so (α−β)2 = (α+β)2 − 4αβ = 4p2 − 4q. The condition 2 < |α − β| < 4 squares to 4 < 4p2 − 4q < 16, that is 1 < p2 − q < 4, or q + 1 < p2 < q + 4. Rearranging, this is q < p2 − 1 < q + 3, which is option D. Option B says the same thing only when p > 0, so it is not an equivalence, and A and E have the wrong endpoints.