TMUA Official 2021 Paper 2 · Question 11

A student attempts to solve the following problem, where $a$ and $b$ are non-zero real numbers:

Show that if $a^2-4b^3\ge0$ then there exist real numbers $x$ and $y$ such that $a=xy(x+y)$ and $b=xy$ .

Consider the following attempt:

\begin{aligned} (x-y)^2&\ge0 &&\text{(I)}\\ \text{so}\quad x^2+y^2-2xy&\ge0 &&\text{(II)}\\ \text{so}\quad (x+y)^2-4xy&\ge0 &&\text{(III)}\\ \text{so}\quad x^2y^2(x+y)^2-4x^3y^3&\ge0 &&\text{(IV)}\\ \text{so}\quad a^2-4b^3&\ge0 &&\text{(V)} \end{aligned}

Which of the following best describes this attempt?

AIt is completely correct.

BIt is incorrect, but it would be correct if written in the reverse order.

CIt is incorrect, but the student has correctly proved the converse.

DIt is incorrect because there is an error in line (II).

EIt is incorrect because there is an error in line (III).

FIt is incorrect because there is an error in line (IV).

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