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Squares and cubes of real numbers · 3 marks

If x = y, then x2 = y2.

Answer: Converse: If x2 = y2, then x = y. The proposition is true; the converse is false: x = 3, y = −3 give x2 = y2 = 9 but x ≠ y.

Step-by-step solution

Given: x and y are real numbers
To find: Frame the converse of this proposition. Then, determine if each of the two statements is true or not. Justify the true statements and give a counterexample for each false statement.

Idea: Squaring loses the sign: a number and its negative have the same square. So equal squares only tell us x = y or x = −y.

  1. Converse: If x2 = y2, then x = y.½ mark
  2. Proposition — true. If x = y, multiplying the equal numbers by themselves gives x × x = y × y, i.e. x2 = y2.1 mark
  3. Converse. x2 = y2 means x2 − y2 = (x − y)(x + y) = 0, so x = y or x = −y. The second case allows x ≠ y.½ mark
  4. Counterexample: x = 3, y = −3. Then x2 = 9 and y2 = 9, so x2 = y2, but 3 ≠ −3.½ mark
  5. So the proposition is true and its converse is false.½ mark
Converse: If x2 = y2, then x = y. The proposition is true; the converse is false (counterexample: x = 3, y = −3).

Answer to write in the exam

Converse: If x2 = y2, then x = y.

Proposition: x = y ⇒ x × x = y × y ⇒ x2 = y2. True.

Converse: x2 − y2 = (x − y)(x + y) = 0 ⇒ x = y or x = −y

Counterexample: x = 3, y = −3: x2 = y2 = 9, but 3 ≠ −3.

∴ Proposition true; converse false.

Common mistakes that cost marks

  • Taking square roots of both sides and writing x = y, forgetting the negative root.
  • Offering x = y = 0 or x = y = 2 as a counterexample; there x = y, so the converse holds.

How this can come in the exam

MCQ (1 mark)

x and y are real numbers with x2 = 49 and y2 = 49. Which must be true?

  1. x = y
  2. x = −y
  3. x = y or x = −y
  4. x = 7 and y = 7
Show answer

(C) x = y or x = −y
Each of x, y is 7 or −7, so either they are equal or one is the negative of the other.

Short answer (2 marks)

Is ‘If x2 = y2 and x, y are both positive, then x = y‘ true? Justify.

Show answer(x − y)(x + y) = 0 (1 mark). x + y > 0 because both are positive, so x − y = 0 and x = y. True (1 mark).

Try one yourself

Find a counterexample to ‘If x2 > y2, then x > y‘ (x, y real).

Show answer

x = −5, y = 2: x2 = 25 > 4 = y2, but −5 < 2.

More questions like this

All Mathematical reasoning questions · All maths questions

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