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TMUA 2022 · Paper 1 · Question 13 of 20

TMUA 2022 Paper 1 Question 13

Algebra and functions — Algebraic manipulation · quadratic in a product. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Algebra and functionsAlgebraic manipulation · quadratic in a product8 optionshard
Given that (a3 + 2b3)(2a3 − b3) = √2 where a and b are real numbers, what is the least value of ab?
  1. A−√2
  2. B√2
  3. C−2√2
  4. D2√2
  5. E−√22
  6. F√22
  7. G−216
  8. H216
Show the answer and worked solution
answer · A
  1. A−√2
  2. B√2
  3. C−2√2
  4. D2√2
  5. E−√22
  6. F√22
  7. G−216
  8. H216
Expanding, the two middle terms are 2 and −2, which cancel, leaving −a3b3 + 4a3b3. Writing t = (ab)3, the equation becomes −t + 4t = √2, that is t2 + √2 t − 4 = 0. Solving, t = −√2 ± 3√22, so t = √2 or t = −2√2. Cube-rooting gives ab = 21/6 or ab = −√2, and the least is −√2.