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TMUA 2022 · Paper 2 · Question 13 of 20

TMUA 2022 Paper 2 Question 13

Logic and arguments — Existential statements · counting valid parameters. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Logic and argumentsExistential statements · counting valid parameters6 options
Consider the statement (×) about a real number x:

(×) There exists a real number y such that x − xy + y is negative.

For how many real values of x is (×) true?

  1. Ano values of x
  2. Bexactly one value of x
  3. Cexactly two values of x
  4. Dall except exactly two values of x
  5. Eall except exactly one value of x
  6. Fall values of x
Show the answer and worked solution
answer · E
  1. Ano values of x
  2. Bexactly one value of x
  3. Cexactly two values of x
  4. Dall except exactly two values of x
  5. Eall except exactly one value of x
  6. Fall values of x
Group the y terms: x − xy + y = x + y(1 − x). If x ≠ 1 the coefficient 1 − x is non-zero, so y can be chosen to drive the expression as negative as we like. If x = 1 the expression collapses to 1, which is never negative. So (×) holds for every real x except that one value.