WEBVTT

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Hi,
I'm Fiona and let's have a look at Tamua

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2021 paper 2 and question 6.

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We're asked to consider the following two
statements about the polynomial F of X.

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Statement P says that F of X = 0 for
exactly 3 real values of X and statement

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Q says that F-OF X or the derivative of F
of X = 0 for exactly 2 real values of X.

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And then we are asked which of the
following is correct?

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And we're given 4 statements, ABC and D,
which relate to whether P is necessary

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for QP is sufficient for Q,
or P is necessary and sufficient for Q.

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Let's start by looking at the statements
P&Q.

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So statement P is essentially saying that
F of X has three real roots.

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So I'm going to write that on so that I
can keep that in mind as I go through

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this question.

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Statement Q is essentially saying that F
of X has two stationary points.

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Now,
when thinking about whether a statement P,

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say,
is necessary or sufficient for another

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statement Q, say,
then I like to think about it in terms of

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the direction of implication.

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So you've probably seen these arrows
before.

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But if we write this,
then this is saying that P implies Q.

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And that is another way of saying that P
is sufficient for Q.

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So we're going to investigate,
is this true here?

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Is P sufficient for Q?

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Or does P imply Q?

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If we write P with a backwards arrow with
the other direction of implication,

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then that is the same as saying P is
necessary for Q.

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So we're also going to investigate
whether Q implies P,

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and that will tell us whether P is
necessary for Q.

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Let's think about sufficiency first,
and really when we think about polynomial

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functions,
there is a theorem that states that you

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can draw as many points on a coordinate
plane as you want as long as No2 would

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form a vertical line,
and you will be able to find a polynomial

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function that passes through each of
those points.

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Really what that's saying is that we can
construct any polynomial function that

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goes up and down as much as we want and
as high and low as we want.

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I've got an example to show you that
satisfies P that is a useful example to

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use in this context.

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So I'm imagining a polynomial of degree 4.

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Doesn't really matter where my where my Y
axis is.

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But let's imagine a polynomial of degree
4 like this.

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We can see that this polynomial satisfies
the statement P because it has exactly 3

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real roots.

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Now let's look at statement Q.

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Does this polynomial have two stationary
points?

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Well, no, it has three stationary points.

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And so this is a example to show that P
is not sufficient for Q.

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I've got another example to show you
that's helpful in this context.

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Now I'm going to think of a cubic
polynomial.

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So here I've drawn a cubic polynomial.

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Now if I draw it such that the X axis is
here,

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that means that it only has one real
route.

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So here's an example of a polynomial
function that satisfies statement Q but

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does not satisfy statement P That means
that in these examples for Q and PQ does

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not imply P,
which means P is not necessary for Q.

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So the answer to this question is going
to be option DP is not necessary and not

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sufficient for Q.