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TMUA 2022 · Paper 2 · Question 5 of 20

TMUA 2022 Paper 2 Question 5

Logic and arguments — Converse and contrapositive of a statement about a line. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Logic and argumentsConverse and contrapositive of a statement about a line8 options
A straight line L passes through (1,  2).

Let P be the statement

if the y-intercept of L is negative, then the x-intercept of L is positive.

Which of the following statements must be true?

I   P
II  the converse of P
III the contrapositive of P

  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
Write the line as y = m(x−1) + 2, so its y-intercept is 2 − m and its x-intercept is 1 − 2m. A negative y-intercept means m > 2, and then 2m < 1, so the x-intercept is positive — P is true, and so is its contrapositive. The converse fails: m = −1 gives y = −x + 3, whose x-intercept is 3 > 0 while its y-intercept is also positive.