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

Practise Edexcel IAL Mathematics P2.1.1 by stating valid assumptions, expanding expressions, using positivity or squares, and finishing with a precise proof conclusion.

Exam points

represent odd or natural numbers in algebraic form before expanding
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