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TMUA 2021 · Paper 2 · Question 13 of 20

TMUA 2021 Paper 2 Question 13

Inequalities and reasoning — Unbounded regions from simultaneous inequalities. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 2Inequalities and reasoningUnbounded regions from simultaneous inequalities8 options
A region R in the (x, y)-plane is defined by the simultaneous inequalities y − x < 3 y − x2 < 1 Which of the following statements is/are true for every point in R?

I    −1 < x < 2
II   (y−x)(y−x2) < 3
III y < 5

  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
The tempting move is to compare the two curves and read off where they cross, but R is not the small lens between them — it is the unbounded region below both, and one well-chosen point kills all three. Take x = 10, y = 12: then y − x = 2 < 3 and y − x2 = −88 < 1, so the point lies in R, yet x > 2 and y > 5. For II, both bracketed factors can be large and negative: at x = 10, y = −1000 the point is still in R and the product is positive and enormous. None of them.