TTMUA Lab
TMUA 2023 · Paper 2 · Question 19 of 20

TMUA 2023 Paper 2 Question 19

Proof and counterexample — Polynomials · comparing root counts. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Proof and counterexamplePolynomials · comparing root counts8 options
In this question, f(x) is a non-constant polynomial, and g(x) = x f'(x).

f(x) = 0 for exactly M real values of x.
g(x) = 0 for exactly N real values of x.

Which of the following statements is/are true?

I   It is possible that M < N
II  It is possible that M = N
III It is possible that M > N

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · H
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Each case needs only one example. Take f(x) = x2 + 1: it has no real roots, so M = 0, while g(x) = 2x2 vanishes only at x = 0, so N = 1 and M < N. Take f(x) = x2: then M = 1 and g(x) = 2x2 gives N = 1, so M = N. Take f(x) = x2 − 1: then M = 2 while g(x) = 2x2 gives N = 1, so M > N. All three are possible.