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TMUA 2022 · Paper 1 · Question 18 of 20

TMUA 2022 Paper 1 Question 18

Graphs and transformations — Polynomial intersections · counting sign changes. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Graphs and transformationsPolynomial intersections · counting sign changes6 optionshard
It is given that f(x) = x2(x−1)2(x−2) g(x) = −p(x−q)2(x−r)2 where p, q and r are positive and q < r.

Find the set of values of q and r that guarantees the greatest number of distinct real solutions of the equation f(x) = g(x) for all p.

  1. Aq < 1 and r < 1
  2. Bq < 1 and 1 < r < 2
  3. Cq < 1 and r > 2
  4. D1 < q < 2 and 1 < r < 2
  5. E1 < q < 2 and r > 2
  6. Fq > 2 and r > 2
Show the answer and worked solution
answer · B
  1. Aq < 1 and r < 1
  2. Bq < 1 and 1 < r < 2
  3. Cq < 1 and r > 2
  4. D1 < q < 2 and 1 < r < 2
  5. E1 < q < 2 and r > 2
  6. Fq > 2 and r > 2
Let ϕ(x) = f(x) − g(x) = x2(x−1)2(x−2) + p(x−q)2(x−r)2, a quintic, so at most five roots. Since x2(x−1)2 ≥ 0, the first term has the sign of x − 2; the second is never negative. Evaluating, ϕ is negative at x = 0,  1,  2 (the first term vanishes, the second is positive — wait, note g ≤ 0 so −g ≥ 0): more simply, ϕ → +∞ as x → −∞ and → −∞ as x → +∞, is negative at x = 0,  1,  2 and positive at x = q and x = r whenever those sit strictly between the integers. Five sign changes therefore need q and r interleaved as 0 < q < 1 < r < 2.