WEBVTT

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Hi I'm Richard.

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We're looking at TMUA 2021 paper two,
question 11.

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Let's read it through.

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So it says a student attempts to solve
the following problem where A&

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B are non 0 real numbers.

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We have show that if a ^2 -,
4 B cubed is greater than or equal to 0

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then there exists real numbers X&
Y such that a is XY times X + y and B

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equals XY.

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And here we have the students attempt.

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We have to pick out the best description
of the attempt from the six options A to

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F over on the left here.

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So let's just read through the attempt.

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So we start with this line which says
that X -,

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y ^2 is greater than or equal to 0.

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So we can certainly say that will be true
for any two real numbers.

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X&Y.

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We're squaring a real number and that
will give us a number greater than or

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equal to 0.

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The next line is just expanding the left
hand side.

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So because this line is true,
it will follow.

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This line will follow because it's just
expanding that bracket X -, y ^2.

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In the next line,
we have replaced X ^2 + y ^2 - 2 XY with

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X + y ^2 - 4 XY.

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Now this is perfectly fine because if we
you can see if we were to expand this,

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we would have X ^2 + y ^2 + 2 XY and then
subtracting the four XY,

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we would be left with what we had on the
previous line.

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So all we've done here is we've replaced
this expression with an identical

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expression,
an expression which is identically equal

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to it.

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So this is fine as a deduction.

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In the next line,
we seem to have multiplied both sides by

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X ^2, y ^2.

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Now we know that X ^2,
y ^2 is going to be greater than or equal

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to 0.

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It's the product of 2 squares and so that
line will be fine as well.

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We're taking an inequality when
multiplying both sides by value,

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which we know is greater than or equal to
0,

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so this will follow from the line above.

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Now in the next line,
we seem to have replaced X ^2 y ^2 * X +

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y ^2 with a ^2 and also X ^3 y ^3 with b
^3.

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So in this line we are assuming that a is
equal to XY times X + y and B is equal to

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X * y in order to do that.

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And if that were true,
then this would follow.

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So what's actually happened in this line
is we have assumed that what we need.

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We have assumed that a is XY times X + y
and that B is X * y.

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So what we're looking at here is a proof
that if A&B do satisfy this,

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then a ^2 -,
4 B cubed will be greater than or equal

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to 0.

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Now this is exactly the opposite of what
the student has been asked to prove.

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The student has been asked to prove that
if a ^2 -,

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4 B cubed is greater than or equal to 0,
then there will be solutions X&

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Y to these two relationships.

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So since the student has assumed that
these solutions exist and and has a valid

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argument that H ^2 - 4 BQ is greater than
or equal to 0,

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they have proved what we call the
converse.

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So to emphasise what that means is in our
question the state.

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The student has been asked to show that a
statement P implies a statement Q.

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So P was the a ^2 -, 4,
B cubed is greater or equal to 0,

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and Q was that there exist X&
Y such that this and this is true.

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The student has actually proved that Q
implies P,

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and so they have proved the converse of
what they were required to prove.

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And so the best description of this
response is that it is incorrect,

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but that the student has proved the
converse, which is option C here.