TMUA 2021 Paper 2 Q11 — Worked Solution

P2·Algebra & Surds, Logic & Quantifiers·7.0Official · 2021 · Q11

proof-errorconversedirection-of-argument

A student attempts to solve the following problem, where

a

and

b

are non-zero real numbers:Show that if

a^{2} - 4 b^{3} \geq 0

then there exist real numbers

x

and

y

such that

a = x y (x + y)

and

b = x y

.Consider the following attempt:

(I)

(x - y)^{2} \geq 0

(II)

x^{2} + y^{2} - 2 x y \geq 0

(III)

(x + y)^{2} - 4 x y \geq 0

(IV)

x^{2} y^{2} (x + y)^{2} - 4 x^{3} y^{3} \geq 0

(V)

a^{2} - 4 b^{3} \geq 0

Which of the following best describes this attempt?

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