Edexcel A-Level Mathematics AS P2.1.1 Structure of Mathematical Proof Questions

The direct evidence is narrow but demanding: proofs must start from a valid form, manipulate algebra cleanly and end with the exact divisibility or inequality claim.

Exam points

Represent odd or natural numbers in algebraic form before expanding the expression.
Use positivity, squares or factorisation to justify an inequality conclusion.
Finish divisibility proofs by writing the expression as a multiple plus a remainder.

[Maximum number: 4]

In this question you must show detailed reasoning.

Given that x and y are positive numbers such that

(x-y)^{3}>x^{3}-y^{3}

prove that

y>x

$(x-y)^3=x^3-3x^2y+3xy^2-y^3$ $(x-y)^3>x^3-y^3\Rightarrow -3x^2y+3xy^2>0$ 3xy(y-x)>0
As x and y are positive, (y-x)>0, so y>x.
A1*

M1: Attempts expansion of $(x-y)^3$ .
A1: Correct simplified inequality with cubed terms cancelled.
dM1: Takes out or divides by a common factor of kxy.
A1*: Correct working and reasoning using positivity of x and y.

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Edexcel A-Level Mathematics AS P2.1.1 Structure of Mathematical Proof Questions question 1