WEBVTT

1feb0389-c8a2-4ab1-b6bf-cd0cee90009f-0
00:00:05.200 --> 00:00:06.480
Hi, my name is Richard.

f06641c6-f959-4c78-b42c-b12202daec92-0
00:00:06.480 --> 00:00:10.080
We're looking at Tamura 2021 paper two,
question 16.

b346802f-a515-418a-8083-db949ac002a0-0
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Here it is.

98be5c8e-47f6-42ce-b1f7-51231a0f18af-0
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It says P&
Q are real numbers and the equation X

98be5c8e-47f6-42ce-b1f7-51231a0f18af-1
00:00:15.278 --> 00:00:22.269
times the modulus of X equals PX plus Q
has exactly K distinct real solutions for

98be5c8e-47f6-42ce-b1f7-51231a0f18af-2
00:00:22.269 --> 00:00:22.439
X.

c60b1e36-03b6-471d-b83f-817759001fb9-0
00:00:22.800 --> 00:00:26.840
Which of the following is the complete
list of possible values for K?

8c09e90c-a575-498a-9149-c3c3b5d87318-0
00:00:27.240 --> 00:00:31.636
And we've got various selections of
possible values of K through the answers

8c09e90c-a575-498a-9149-c3c3b5d87318-1
00:00:31.636 --> 00:00:34.434
A to F OK,
so how do we approach a question like

8c09e90c-a575-498a-9149-c3c3b5d87318-2
00:00:34.434 --> 00:00:34.720
this?

8825e3fe-a2d1-4d18-a49d-846ea6fe9391-0
00:00:35.520 --> 00:00:38.760
I think the best way to attack this is
visually.

b58c5c62-0d6a-4463-839b-4727bda5093c-0
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We'd like to draw a graph to represent
this equation.

2d701e4b-bf8a-432a-b1d0-430611893536-0
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The right hand side is very familiar to
us.

bcd85882-d745-4927-8b95-2f3cf4ab54c0-0
00:00:43.960 --> 00:00:45.280
PX plus Q.

f87dafd2-14de-4a25-885b-6b419ed9536e-0
00:00:45.440 --> 00:00:47.400
We can just represent that as a straight
line.

0a500de0-2639-402b-b0d9-c742d1c8d1e0-0
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It's got the same structure as MX plus C
Notice that it couldn't be a vertical

0a500de0-2639-402b-b0d9-c742d1c8d1e0-1
00:00:51.851 --> 00:00:52.640
straight line.

93e71d42-04e2-404a-8a65-f8fd73c7c346-0
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It couldn't be like X = 4.

bae1a7b6-9f90-474c-811e-61dce2e0cc77-0
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We can only get straight lines which are
not vertical on the right hand side.

db5b87bf-85df-4fac-b9c9-193c388f9c86-0
00:00:58.760 --> 00:00:59.040
Here.

a068e494-2282-41e0-936c-e35bd38a3b04-0
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The left hand side is a little bit more
difficult to think about because it

a068e494-2282-41e0-936c-e35bd38a3b04-1
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involves the modulus of X.

1c22a0a1-3d8b-41b2-8722-f38416a56eb5-0
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The modulus of X is defined in cases if X
is greater than or equal to 0,

1c22a0a1-3d8b-41b2-8722-f38416a56eb5-1
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then the modulus of X is X itself.

b66065e7-2aef-4168-a658-c6e87fd036f3-0
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Whereas if X is less than 0,
the modulus of X is actually minus X.

e545dd99-e7e0-449a-b73e-ca16c9551063-0
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So if we think about the consequences of
that for X times the modulus of X,

e545dd99-e7e0-449a-b73e-ca16c9551063-1
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we can say that it's in two cases.

c31223c3-a1ef-46d3-b314-f7474fc66548-0
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If X is greater than or equal to 0,
then the modulus of X = X and so this

c31223c3-a1ef-46d3-b314-f7474fc66548-1
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would just be X ^2.

0fed26e0-b350-4369-bd8c-52d81e28cada-0
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Whereas if X is less than 0,
then the modulus of X would be minus X

0fed26e0-b350-4369-bd8c-52d81e28cada-1
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and we would have X * -, X,
so we would have minus X ^2.

1fe83271-ec0e-491a-ac98-4a5d60dda655-0
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So this is what the function on the left
hand side looks like.

583109ea-f845-45f4-b345-706aa0e1eac3-0
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So for X greater than or equal to 0,
it's graph is part of y = X ^2 and for X

583109ea-f845-45f4-b345-706aa0e1eac3-1
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less than 0 it's graph is part of y = -,
X ^2.

26915c50-a985-45a9-922d-89f86827b69a-0
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So let's now get on and draw a graph
represent of the function on the left

26915c50-a985-45a9-922d-89f86827b69a-1
00:02:03.376 --> 00:02:04.040
hand side.

2f9a898c-251a-4c35-9942-950827882d8f-0
00:02:04.520 --> 00:02:09.730
So for X greater than 0,
we want part of y = X ^2 which will look

2f9a898c-251a-4c35-9942-950827882d8f-1
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like this.

88bdb196-a06d-4d71-a3d9-c12ceb0f1fab-0
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And for X less than 0,
we need to reflect what we would normally

88bdb196-a06d-4d71-a3d9-c12ceb0f1fab-1
00:02:15.778 --> 00:02:20.080
have for y = X ^2 here in the X axis so
that we get -X ^2.

7d920989-0322-4571-aba3-21c55d2be810-0
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So that will look something like this.

a04eb0de-0ee1-494f-8a37-e94440fd4c25-0
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So I'll just make a note that this is
part of the graph of y = -,

a04eb0de-0ee1-494f-8a37-e94440fd4c25-1
00:02:26.376 --> 00:02:30.237
X ^2 for this part,
and this is just part of the regular y =

a04eb0de-0ee1-494f-8a37-e94440fd4c25-2
00:02:30.237 --> 00:02:31.440
X ^2 for this part.

c2f320ed-4bcb-4851-8498-d96ecab24e2c-0
00:02:32.080 --> 00:02:36.300
So now the question becomes,
if I draw a straight line onto this

c2f320ed-4bcb-4851-8498-d96ecab24e2c-1
00:02:36.300 --> 00:02:41.040
diagram which is not vertical,
how many times might it cross this curve?

e651fee8-e456-466d-9051-be3e876594f4-0
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What are the possibilities for the number
of times it might cross this curve?

63189847-e365-47a4-a37c-91767189741a-0
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OK,
so let's first of all think about whether

63189847-e365-47a4-a37c-91767189741a-1
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we can ever get 0 crossing points.

96f1f5b1-516c-40ea-88cd-5e2f14fb9c61-0
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Hopefully you can convince yourself that
no matter where I put my straight line

96f1f5b1-516c-40ea-88cd-5e2f14fb9c61-1
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here,
it will cross this curve at least once.

826d0c43-1b0f-4165-b651-4fc1e79bf357-0
00:02:58.440 --> 00:03:01.920
So we cannot have zero solutions to this
equation.

15a5f144-450e-4148-9b8e-6a6dc37e71d7-0
00:03:01.920 --> 00:03:03.800
We cannot have zero crossing points.

24abe666-2130-4f92-ba66-8748ecbe6ec2-0
00:03:03.800 --> 00:03:05.360
So let's rule that out straight away.

7eff4062-2075-4637-b412-e8b2c0e4c502-0
00:03:06.720 --> 00:03:09.680
Let's now ask the question,
can we just get one crossing point?

2d787ce2-ed15-4d44-b1fb-7e3a1e89d872-0
00:03:10.040 --> 00:03:12.625
Well,
I think if you imagine a straight line

2d787ce2-ed15-4d44-b1fb-7e3a1e89d872-1
00:03:12.625 --> 00:03:17.278
whose gradient is negative like this one,
you can see that we would just get one

2d787ce2-ed15-4d44-b1fb-7e3a1e89d872-2
00:03:17.278 --> 00:03:20.840
point of intersection and hence one
solution to the equation.

33317695-d65c-4f2e-bcc2-5d65e1a93828-0
00:03:21.160 --> 00:03:24.840
So it looks like yes,
it will be possible to get KB one.

e3db8aa8-be59-41ba-8750-b44f6e8a2e34-0
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Now can we think of a situation where we
could get K being 2SO2 crossing points

e3db8aa8-be59-41ba-8750-b44f6e8a2e34-1
00:03:32.044 --> 00:03:32.720
for this?

1cb18de1-3e0a-482f-9303-6d161315d903-0
00:03:32.720 --> 00:03:38.714
If we imagine taking a point,
say on this positive part of y = X ^2 and

1cb18de1-3e0a-482f-9303-6d161315d903-1
00:03:38.714 --> 00:03:44.126
taking the tangent there,
you can imagine that the tangent would

1cb18de1-3e0a-482f-9303-6d161315d903-2
00:03:44.126 --> 00:03:48.040
cross this curve again once over on this
side.

450a1a4f-eeb8-445b-9965-ef412b7be843-0
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So we would have one crossing point and a
second crossing point here,

450a1a4f-eeb8-445b-9965-ef412b7be843-1
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and that shows us that K = 2 is possible.

f1ee90e5-b00a-48e3-9e6e-463845ebaf67-0
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Is K = 3 possible?

ae22e552-5c58-4cba-8980-ecd6da35dec4-0
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A nice way to think about this is to
think about the straight line which

ae22e552-5c58-4cba-8980-ecd6da35dec4-1
00:04:03.393 --> 00:04:05.840
passes through the origin with a positive
gradient.

7ac1be73-f8b5-4fe1-8b50-bbea06577662-0
00:04:06.200 --> 00:04:11.096
So if I draw in a line which is passing
through the origin with a positive

7ac1be73-f8b5-4fe1-8b50-bbea06577662-1
00:04:11.096 --> 00:04:14.425
gradient,
hopefully you can see that this straight

7ac1be73-f8b5-4fe1-8b50-bbea06577662-2
00:04:14.425 --> 00:04:18.800
line will cross our curve once again at
the origin and once again.

bad0d05b-a03a-4c23-ae9a-b28eb9578206-0
00:04:19.120 --> 00:04:22.080
And so we can see that K = 3 is possible.

41fe1635-524e-445a-9e89-e79f501a2c05-0
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It's possible that this equation has
three real solutions.

25f20134-f683-433f-9ad9-b53cd96748d4-0
00:04:25.360 --> 00:04:29.120
So is it possible for our equation to
have four solutions?

afc337ff-2bad-4ac8-a3f6-0e630288e646-0
00:04:29.360 --> 00:04:33.200
I'm going to redraw my diagram so that we
can think about that case separately.

80947ff0-5a1c-4d4d-a188-3f249a717392-0
00:04:40.000 --> 00:04:44.284
So we're now thinking about whether we
can draw a straight line on this diagram

80947ff0-5a1c-4d4d-a188-3f249a717392-1
00:04:44.284 --> 00:04:46.480
which crosses this curve exactly 4 times.

670d5409-60ff-4f95-a35b-b7ce757ec544-0
00:04:47.280 --> 00:04:50.144
The first thing to say is that if if this
was to be possible,

670d5409-60ff-4f95-a35b-b7ce757ec544-1
00:04:50.144 --> 00:04:52.640
this straight line would not pass through
the origin.

8ec13dfa-418e-439a-bb31-f6c3f81dae52-0
00:04:52.960 --> 00:04:55.607
If I draw in a straight line through the
origin,

8ec13dfa-418e-439a-bb31-f6c3f81dae52-1
00:04:55.607 --> 00:04:59.767
it will either cross once or it will
cross 3 three times like in the example

8ec13dfa-418e-439a-bb31-f6c3f81dae52-2
00:04:59.767 --> 00:05:00.200
earlier.

eea68c9e-d022-4372-8b1a-1023ec0899d5-0
00:05:00.560 --> 00:05:04.000
So if this straight line exists,
it does not go through the origin.

67d6f8de-f5ba-4be3-aa0f-64de4536610e-0
00:05:05.720 --> 00:05:09.480
So now let's think about how we might try
to create such a straight line.

64beafd0-40c9-4d40-97e2-aa63b2dad3ba-0
00:05:09.720 --> 00:05:13.058
Our straight line can cross y = X ^2 at
most twice,

64beafd0-40c9-4d40-97e2-aa63b2dad3ba-1
00:05:13.058 --> 00:05:17.040
because the underlying equation for that
will be a quadratic.

51763637-17d0-4f1b-8bec-68375357bb90-0
00:05:17.600 --> 00:05:21.907
So let's think about the possibility
where we have a straight line crossing y

51763637-17d0-4f1b-8bec-68375357bb90-1
00:05:21.907 --> 00:05:23.840
= X ^2 in this positive part twice.

689172c5-f136-446a-a6ae-e6b2c3f8a435-0
00:05:24.360 --> 00:05:27.313
Well,
that straight line would then continue

689172c5-f136-446a-a6ae-e6b2c3f8a435-1
00:05:27.313 --> 00:05:31.120
and it would pass through the Y axis at a
negative value.

9b7f8959-3389-4a30-97cc-02ec05133fa1-0
00:05:32.360 --> 00:05:37.044
If we were to aim to have two crossing
points over here with negative X values

9b7f8959-3389-4a30-97cc-02ec05133fa1-1
00:05:37.044 --> 00:05:40.246
like these two,
remember we could have at most two of

9b7f8959-3389-4a30-97cc-02ec05133fa1-2
00:05:40.246 --> 00:05:44.160
these because the underlying equation
again would be a quadratic.

923d90f6-cada-4270-9841-688b3fc4bc38-0
00:05:44.720 --> 00:05:48.678
You can see that if we were to continue
that straight line,

923d90f6-cada-4270-9841-688b3fc4bc38-1
00:05:48.678 --> 00:05:53.560
it would pass through the Y axis at a
point with a positive Y coordinate.

2cecd841-f69e-4cce-8a96-18f6f93c994c-0
00:05:54.520 --> 00:06:00.838
So we cannot have the origin if we would
need to have two points with a positive X

2cecd841-f69e-4cce-8a96-18f6f93c994c-1
00:06:00.838 --> 00:06:06.547
coordinate and two points with a a
negative X coordinate to get 4 crossing

2cecd841-f69e-4cce-8a96-18f6f93c994c-2
00:06:06.547 --> 00:06:07.080
points.

d39ba577-9dbb-43b0-ace3-49def5eaef11-0
00:06:07.400 --> 00:06:11.200
But you can see that these two situations
are not consistent with one another.

59f7f062-b09a-483b-9cf5-539e482acb12-0
00:06:11.200 --> 00:06:13.880
There can be no single straight line
which does this.

a5b9537d-4ba4-4835-998b-4cb95470971f-0
00:06:14.360 --> 00:06:19.132
A straight line which passes through this
part of the curve twice has a negative Y

a5b9537d-4ba4-4835-998b-4cb95470971f-1
00:06:19.132 --> 00:06:22.122
intercept,
and a straight line which passes through

a5b9537d-4ba4-4835-998b-4cb95470971f-2
00:06:22.122 --> 00:06:25.400
this part of the curve twice has a
positive Y intercept.

d02cdf47-633e-46b5-b71d-46fd48b34bb5-0
00:06:25.400 --> 00:06:29.506
So there can be no single straight line
which passes through this part of the

d02cdf47-633e-46b5-b71d-46fd48b34bb5-1
00:06:29.506 --> 00:06:32.560
curve twice and this part of the curve
another two times.

864cbced-7023-4bef-a39d-5a0a509c0ba7-0
00:06:32.840 --> 00:06:37.723
So it's not possible for us to get 4
crossing points and not possible to have

864cbced-7023-4bef-a39d-5a0a509c0ba7-1
00:06:37.723 --> 00:06:40.040
four real solutions to this equation.

e45eac24-94e9-4351-82a9-c949d8555755-0
00:06:40.680 --> 00:06:43.240
So we have seen we cannot get 0.

4b28ad1b-29a5-4c49-826e-4d54a299af2b-0
00:06:43.280 --> 00:06:47.760
We can get one, we can get 2,
we can get three, we cannot get 4.

11692e1a-0545-45ec-95e5-64142a9b3126-0
00:06:48.080 --> 00:06:52.600
And that means the correct answer is here,
E12 and three.

d1c0f8a3-1f67-4c2e-847c-8c865b98d7dc-0
00:06:53.200 --> 00:06:55.233
OK,
so two points to reflect on for this

d1c0f8a3-1f67-4c2e-847c-8c865b98d7dc-1
00:06:55.233 --> 00:06:55.680
question.

ee05e09b-a0e4-40ae-81e6-a3f98c7c41bc-0
00:06:55.880 --> 00:07:00.680
The first one is that you may have
thought that we cannot draw a graph of

ee05e09b-a0e4-40ae-81e6-a3f98c7c41bc-1
00:07:00.680 --> 00:07:03.080
the left hand side y = X times mod X.

0bd4f39c-40aa-4c73-9dca-25a3f8b5eff6-0
00:07:03.280 --> 00:07:05.240
So we've seen that we can deal with that.

7e4d5463-7a9d-4664-8c11-05cd8e980903-0
00:07:05.240 --> 00:07:08.200
We just need to do it in two cases,
positive X and negative X.

034cc075-3ca7-46c8-9988-9bf532b8d06a-0
00:07:08.720 --> 00:07:11.440
The other thing is this is towards the
end of the paper.

03035f47-367e-40aa-80d2-f5de46dfd160-0
00:07:11.760 --> 00:07:14.480
The K = 4 case is particularly tricky.

092f0178-72eb-481d-ae9e-70f2aa5fe951-0
00:07:14.480 --> 00:07:18.875
So you may have found yourself in a
situation where you knew that zero was

092f0178-72eb-481d-ae9e-70f2aa5fe951-1
00:07:18.875 --> 00:07:21.688
not possible,
which would rule out ABC and D as

092f0178-72eb-481d-ae9e-70f2aa5fe951-2
00:07:21.688 --> 00:07:25.204
possible answers,
and you were still thinking about whether

092f0178-72eb-481d-ae9e-70f2aa5fe951-3
00:07:25.204 --> 00:07:29.013
it the answer was your F,
which depends on whether you can get 4

092f0178-72eb-481d-ae9e-70f2aa5fe951-4
00:07:29.013 --> 00:07:29.600
solutions.

9100e143-4e67-4a46-9642-500b9fd019d2-0
00:07:29.880 --> 00:07:33.307
So this might be something that you come
back to at the end of the paper when

9100e143-4e67-4a46-9642-500b9fd019d2-1
00:07:33.307 --> 00:07:36.648
you're a little bit tight on time,
but you know that at that point you only

9100e143-4e67-4a46-9642-500b9fd019d2-2
00:07:36.648 --> 00:07:40.119
have to consider the possibility of
caving for whether that's possible or not.