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TMUA 2017 · Paper 2 · Question 13 of 20

TMUA 2017 Paper 2 Question 13

Inequalities and reasoning — Standard form · what must follow. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 2Inequalities and reasoningStandard form · what must follow8 options
The positive real numbers a × 10−3, b × 10−2 and c × 10−1 are each in standard form, and (a × 10−3) + (b × 10−2) = (c × 10−1). Which of the following statements (I, II, III, IV) must be true?

I    a > 9
II   b > 9
III  a < c
IV  b < c

  1. AI only
  2. BII only
  3. CI and II only
  4. DI and III only
  5. EI and IV only
  6. FII and III only
  7. GII and IV only
  8. HI, II, III and IV
Show the answer and worked solution
answer · B
  1. AI only
  2. BII only
  3. CI and II only
  4. DI and III only
  5. EI and IV only
  6. FII and III only
  7. GII and IV only
  8. HI, II, III and IV
Standard form means 1 ≤ a,  b,  c < 10. Multiplying the equation by 103 gives a + 10b = 100c. Since c ≥ 1, the left side is at least 100, and since a < 10 this forces 10b > 90, so b > 9: statement II must hold. Nothing similar pins down a — take c = 1, b = 9.5, a = 5, which satisfies everything while a > 9 and a < c both fail. And IV is never true, since b > 9 while c < 10 forces c < 1.1.