If x = y, then x3 = y3.
Step-by-step solution
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: Factorise x3 − y3 = (x − y)(x2 + xy + y2). The second bracket can be written as a sum of squares, so it is zero only when x = y = 0.
- Converse: If x3 = y3, then x = y.½ mark
- Proposition — true. If x = y, then x × x × x = y × y × y, i.e. x3 = y3.½ mark
- Converse. x3 = y3 gives x3 − y3 = 0, i.e. (x − y)(x2 + xy + y2) = 0.½ mark
- Write the second bracket as x2 + xy + y2 = (x + y2)2 + 34y2. Both terms are ≥ 0, so the bracket is 0 only if y = 0 and x + y2 = 0, i.e. x = y = 0.1 mark
- So either x − y = 0, or x = y = 0. In both cases x = y. The converse is true, so both statements are true.½ mark
Check: Expand: (x + y2)2 + 34y2 = x2 + xy + 14y2 + 34y2 = x2 + xy + y2 ✓. Numbers: x3 = 8 has only x = 2 among real numbers, since (−2)3 = −8.
Answer to write in the exam
Converse: If x3 = y3, then x = y.
Proposition: x = y ⇒ x × x × x = y × y × y ⇒ x3 = y3. True.
Converse: x3 − y3 = (x − y)(x2 + xy + y2) = 0
x2 + xy + y2 = (x + y2)2 + 34y2 = 0 only if x = y = 0
∴ x − y = 0 or x = y = 0 ⇒ x = y. True.
∴ Both the proposition and its converse are true.
Common mistakes that cost marks
- Copying the squares case and offering x = 2, y = −2: (−2)3 = −8 ≠ 8, so this is not a counterexample.
- Dividing both sides by x2 + xy + y2 without checking that it is not zero.
- Saying the converse is true ‘because cubes are like squares’. The reason is different: cubes keep the sign of the number.
How this can come in the exam
How many real numbers x satisfy x3 = −27?
- 0
- 1
- 2
- 3
Show answer
(B) 1
Only x = −3: if x3 = (−3)3, then x = −3 by the converse.
Assertion (A): If x and y are real numbers and x3 = y3, then x = y.
Reason (R): If x and y are real numbers and x2 = y2, then x = y.
- Both Assertion (A) and Reason (R) are true and R is the correct explanation of A.
- Both Assertion (A) and Reason (R) are true but R is not the correct explanation of A.
- Assertion (A) is true but Reason (R) is false.
- Assertion (A) is false but Reason (R) is true.
Show answer
(C) Assertion (A) is true but Reason (R) is false.
A is true (proved above). R is false: 32 = (−3)2 but 3 ≠ −3.
Try one yourself
If x is a real number and x3 = 64, find x and explain why there is only one answer.
Show answer
43 = 64, so x3 = 43 and, by the converse, x = 4. No other real number works; for instance (−4)3 = −64.
More questions like this
- If n is divisible by 24, then it is divisible by both 4 and 6.
- If n is divisible by 60, then it is divisible by both 5 and 12.
- If n is the square of a prime number, then it has exactly 3 factors.
- If n is a product of two unequal prime numbers, then it has exactly 4 divisors.
- If n and n + 3 have no factors in common, then n is not a multiple of 3.