Daniel J Smith Custom TMUA Set A Paper 2 · Question 2

A student is asked to prove the statement:

if $u$ and $v$ are real numbers such that $u^2+v^2=0$ , then $u=0$ and $v=0$

They write the following argument:

I. For all real $w$ , $w^2\geq0$
II. So $u^2\geq0$ and $v^2\geq0$
III. If the sum of two real numbers is $0$ and each is not negative, then each number is $0$
IV. Therefore $u^2=0$ and $v^2=0$
V. Hence $u=0$ and $v=0$

Which of the following best describes this argument?

AIt is a correct direct proof of the statement.

BIt is incorrect, because line I is false.

CIt is incorrect, because line III is false.

DIt is incorrect, because line IV does not follow from the previous lines.

EIt is incorrect, because the reasoning would only work if

u

and

v

were positive.

FIt is a correct proof of the converse, but not of the original statement.

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