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TMUA 2019 · Paper 2 · Question 3 of 20

TMUA 2019 Paper 2 Question 3

Algebra and functions — Cancelling a factor that may be zero. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 2Algebra and functionsCancelling a factor that may be zero8 options
a, b and c are real numbers.

Given that ab = ac, which of the following statements must be true?

I    a = 0
II   b = 0 or c = 0
III  b = c

  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 · A
  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
Rearranging, ab − ac = 0, so a(b − c) = 0 and therefore a = 0 or b = c. That is a disjunction, so neither branch has to hold on its own. Taking a = 1, b = c = 5 satisfies ab = ac with a ≠ 0 and with b, c both non-zero, killing I and II. Taking a = 0, b = 1, c = 2 satisfies it with b ≠ c, killing III.