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TMUA 2020 · Paper 2 · Question 10 of 20

TMUA 2020 Paper 2 Question 10

Inequalities and reasoning — Combining inequalities. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Inequalities and reasoningCombining inequalities8 options
The real numbers a, b, c and d satisfy both 0 < a + b < c + d and 0 < a + c < b + d Which of the following inequalities must be true?

I    a < d
II   b < c
III  a + b + c + d > 0

  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 · F
  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
Add the right-hand inequalities: a + b + a + c < c + d + b + d, so 2a < 2d and a < d — statement I holds. For III, the two conditions give a + b > 0 and c + d > a + b > 0, so a+b+c+d is a sum of two positive numbers and is positive. Statement II fails because the setup is symmetric in b and c, so nothing can force one below the other: a = 1, b = 2, c = 1, d = 5 satisfies 0 < 3 < 6 and 0 < 2 < 7, yet b > c.