TTMUA Lab
TMUA 2023 · Paper 2 · Question 10 of 20

TMUA 2023 Paper 2 Question 10

Inequalities and reasoning — Checking an argument line by line. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Inequalities and reasoningChecking an argument line by line7 options
Here is an attempt to solve the inequality x4 − 2x2 − 3 < 0 by completing the square:

x4 − 2x2 − 3 < 0
I     if and only if x4 − 2x2 + 1 < 4
II    if and only if (x2 − 1)2 < 4
III   if and only if −2 < x2 − 1 < 2
IV   if and only if x2 − 1 < 2
V    if and only if x2 < 3
VI   if and only if −√3 < x < √3

Which of the following statements is true?

  1. AThe argument is completely correct.
  2. BThe first error occurs in line I.
  3. CThe first error occurs in line II.
  4. DThe first error occurs in line III.
  5. EThe first error occurs in line IV.
  6. FThe first error occurs in line V.
  7. GThe first error occurs in line VI.
Show the answer and worked solution
answer · A
  1. AThe argument is completely correct.
  2. BThe first error occurs in line I.
  3. CThe first error occurs in line II.
  4. DThe first error occurs in line III.
  5. EThe first error occurs in line IV.
  6. FThe first error occurs in line V.
  7. GThe first error occurs in line VI.
Every step holds. Adding 4 to both sides gives line I, and the left side factorises as (x2−1)2 for line II. Line III is the standard unpacking of |x2 − 1| < 2. Line IV looks like it discards information, but it does not: x2 ≥ 0 forces x2 − 1 ≥ −1 > −2, so the left-hand inequality is automatic. Lines V and VI then follow directly.