WEBVTT

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Hi I'm Fiona and this is Tamura 2021
paper one and question 14.

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We're first told that this question uses
radians and then we're asked to find the

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number of distinct values of X that
satisfy this equation X + 1 on 3 -,

b87aa076-1fd5-4a16-b945-f9c3c6318ea3-2
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X = 2 on one minus cosine of π X.

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And then we're given some options to
choose from,

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23456 or seven when we get to that point.

bb65c3a5-daa9-4152-a2de-d84b0f1c49fc-0
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Now we're not asked to solve this
equation.

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In fact,
that would be very difficult to do.

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We are asked to find the number of
distinct values of X that satisfy the

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equation, in other words,
the number of solutions.

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And when we're asked something like this,
then we should be first thinking about a

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graphical approach.

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So I'm going to have a think about the
right hand side,

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the left hand side of this equation.

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What's going on there?

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I'm going to have a think about the right
hand side of the equation.

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I'm going to graph them and then see how
many intersections and therefore how many

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solutions I'm going to find.

503dd242-85e6-4674-a891-5a994f32e1e8-0
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Looking at this left hand side,
I can see that this is a quadratic.

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Let's multiply out these brackets just to
get more of a feel of this function.

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So here's my quadratic -X ^2 + 2 X plus
three.

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Because the coefficient of X ^2 is
negative, this graph will have AU, sorry,

79175c9b-eb03-49d4-ba99-4f60c9e13e79-1
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an N shape.

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I can also see that the roots will be at
X = - 1 and X = 3,

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and so if the where crosses the X axis
will be at a value of X = - 1 and X = 3.

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The other thing that I think will be
valuable for us to investigate will be

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this maximum point.

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And to do that,
I'm going to think about completing the

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square.

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First I'm just going to take out a factor
of -1 and then completing the square I

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get X -, 1 in a bracket squared.

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This -1.

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The constant term in this bracket that's
generated will be a +1,

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so I need a -1 just to balance.

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Rebalance that out and a -, 3.

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And now my completed square is this.

86c7a8d2-50f1-4abd-9dde-11a0b7d690e5-0
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So that tells me that my function is
going to have a oh hang on,

86c7a8d2-50f1-4abd-9dde-11a0b7d690e5-1
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that should be a + 4.

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My function is going to have a maximum
value when this is 0,

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which will be when X = 1.

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So I will have a maximum at 14.

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So this coordinate point here will be 1/4.

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I think I've done enough investigation on
this quadratic.

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Now I'll move on to thinking about the
right hand side of this equation.

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And 1st,
I'm going to multiply out the brackets.

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So I have 2 -, 2 cosine π X.

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Now in order to draw the graph of this
function,

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I'm going to need to think about it in
stages.

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So let's go through in stages and build
up from what we do now to a graph of this

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function.

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Now let's first consider cosine π X.

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Now I'm familiar with the graph of Cos X
and it has a period of 2π when we're

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thinking about radians.

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So the graph of Cos of π X will be very
similar,

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but it will have a period of 2 instead of
having a period of 2π.

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So let's draw that.

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And this function will range in value
between y = - 1 and y = 1.

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Now let's think about this function when
it's multiplied by two.

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So now we're going to think about two Cos
π X and this will be a.

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Multiplying by two here will give a
stretch in the Y direction,

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so let's draw that.

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Now here's our graph of two Cos of π X.

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If we think about what happens when we
have -2 Cos of π X, Well,

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when we multiply a function by -1,
that's the same as a reflection in the X

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axis.

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So let's think about the function -2 Cos
of π X.

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And I'm going to draw that in orange and
just draw that on top of this graph so we

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can see the difference.

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It's going to be the same as this graph,
but reflected in the X axis.

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And now the final thing I need to do is
to take this orange function and add 2 to

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it.

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So that is going to shift it up by two
units in the Y in the positive Y

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direction.

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And now finally we've arrived at our
graph,

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the right hand side of this equation,
which is the graph of 2 -, 2 Cos π X.

704ed36f-5287-478e-969a-3da148825193-0
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And we can see that because we've shifted
this orange graph up two in the Y

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positive Y direction,
the function now ranges in value between

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y = 0 and y = 4.

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Now let's go back to our quadratic.

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So our quadratic had a maximum at 14.

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So that's this coordinate point here.

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I'm going to draw the quadratic or the
left hand side of this equation in purple,

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and it had roots at X = - 1 and X = 3.

a57dbf1e-8deb-459a-aad4-f8d0ae9f991e-0
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So that's this point here and this point
here.

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So if I overlay the graph of 2 -,
2 Cos π X with the graph of the quadratic

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function that forms the left hand side of
this equation, I have the following.

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And I can see that the graphs of these
two functions intersect at three points.

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So that gives me the number of distinct
values of X that satisfy this equation.

aed21996-5d8a-4794-aa57-550754cbfb1b-0
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And I can look and see that B is the
value I'm looking for, which is 3.

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So the answer to this question is B.