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TMUA 2018 · Paper 2 · Question 11 of 20

TMUA 2018 Paper 2 Question 11

Logic and arguments — Exponential meets a line · if, only if, if and only if. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 2Logic and argumentsExponential meets a line · if, only if, if and only if8 options
Consider the equation 2x = mx + c, where m and c are real constants.

Which of the following statements is/are true?

I    The equation has a negative real solution only if c > 1.
II   The equation has two distinct real solutions if c > 1.
III  The equation has two distinct positive real solutions if and only if c ≤ 1.

  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
All three can be knocked down by taking m = 0, which makes the right-hand side a horizontal line. With m = 0 and c = 12 the equation 2x = 12 has the negative solution x = −1 even though c ≤ 1, so I fails. With m = 0 and c = 2 the equation 2x = 2 has the single solution x = 1, so c > 1 does not force two solutions and II fails. With m = 0 and c = 0 the equation 2x = 0 has no solutions at all, yet c ≤ 1, so the "if" half of III fails. None of them.