WEBVTT

484e2621-5877-46e8-9500-8da46f94a02a-0
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Hi,
I'm Fiona and let's have a look at Tamura

484e2621-5877-46e8-9500-8da46f94a02a-1
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2021 paper one and question 8.

49e79209-851c-482a-bebc-b6f2be1bb3b3-0
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We're told that the line y = 2 X plus
three meets the curve y = X ^2 plus BX

49e79209-851c-482a-bebc-b6f2be1bb3b3-1
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plus C at exactly one point,
and that the line y = 4 X -2 also meets

49e79209-851c-482a-bebc-b6f2be1bb3b3-2
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the curve at exactly one point.

fa90b938-d65b-4c72-93ed-03fe466bbc6f-0
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And we're asked what is the value of b -,
C And then we're given options A to F to

fa90b938-d65b-4c72-93ed-03fe466bbc6f-1
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choose from once we get an answer.

cda2eff6-b7e1-4fe2-a035-d91faa8f0be1-0
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Now in order for this line to meet this
curve,

cda2eff6-b7e1-4fe2-a035-d91faa8f0be1-1
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I must have X ^2 plus BX plus C equal to
two X + 3.

a3f40080-605e-49e4-b043-8d889c08d72f-0
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So let's start there and see where we get.

f4d6073a-9f3b-4b85-a065-880aab9067a4-0
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Now I'm going to just rearrange this so
that I can get it in the form of a

f4d6073a-9f3b-4b85-a065-880aab9067a4-1
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quadratic that I can see and clearly the
coefficients on each term.

5400fe49-2513-4df5-8148-b3996c6f4723-0
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So I have X ^2 and this will be abx -2 X.

162dc68a-e109-452b-a733-a602b4c13e7f-0
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So that's going to give me plus b -,
2 * X and then my constant term will be C

162dc68a-e109-452b-a733-a602b4c13e7f-1
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- 3 and that will all be equal to 0.

44d31757-f58d-4d7b-bb18-6fd38294da67-0
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Now when we have a quadratic meeting a
line, there are different scenarios.

53bb63b8-49f0-410a-a84a-58509aaeafb7-0
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There are three different scenarios that
can happen.

229d0d35-f15e-4145-af27-1b982028bcbc-0
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This quadratic has a coefficient of X ^2,
has one as a coefficient of X ^2.

23bc7d54-d922-44df-85dc-1587236ec5ee-0
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So it will be U-shaped,
and this line has a positive gradient.

7fcf3248-b732-4135-8eaf-03853d01a2df-0
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And so for this scenario,
we would expect that either the line

7fcf3248-b732-4135-8eaf-03853d01a2df-1
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doesn't meet the curve or the line meets
the curve at 2 points, or in this case,

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what we're told is that the line meets
the curve at exactly one point.

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And the thing that will distinguish these
three scenarios is the discriminant.

be686014-0778-4c28-b2da-372f743755a2-0
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And so if the discriminant of this
particular quadratic is less than 0,

be686014-0778-4c28-b2da-372f743755a2-1
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that means that they don't meet.

31160ee0-c8ce-4cbe-902d-fcdf3764a931-0
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And we would have this scenario if the
discriminant is greater than 0,

31160ee0-c8ce-4cbe-902d-fcdf3764a931-1
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then they meet twice.

d94b557e-d256-4e64-ae62-479ca0116f3e-0
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And if the discriminant equals 0,
then they meet exactly once.

01ee5fe9-8f4f-4040-9679-e901a456d554-0
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And that's what we're looking for.

1c6ffb9b-ced2-4938-839c-f34e91ce2c7a-0
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So that's what we're going to use to get
into this question.

aac1b047-cb95-46f3-ab8b-9503a6bc58af-0
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So the discriminant of this quadratic,
well,

aac1b047-cb95-46f3-ab8b-9503a6bc58af-1
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any discriminant would be b ^2 - 4 AC.

c83a2625-650a-45e5-a603-e193f076dcb3-0
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I'm looking at the coefficients,
and for this 1B squared is going to be b

c83a2625-650a-45e5-a603-e193f076dcb3-1
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- 2 ^2 - 4 * a, which is 1 * C,
which is C - 3 here.

93b8de62-5d27-48ee-b395-8dc0b5891b20-0
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And we went back to equal 0.

311e70d7-52cf-4a00-91ae-7a3a12a09063-0
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OK, now let's just tidy this up a bit.

46527c5d-6238-46ab-b17e-468c2962581b-0
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I'm going to get b ^2 - 4 B plus 4 - 4 C
plus 12 = 0.

86e8a10a-a043-4bf2-917d-428306a33cf3-0
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And then just tidying that up one more
time, I'm going to have b ^2 -,

86e8a10a-a043-4bf2-917d-428306a33cf3-1
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4 B minus four, C + 16 = 0.

164ee150-9bca-4561-9edd-e342982bc155-0
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And everything that I've just done here
between this line and the curve,

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I'm now going to do for this line here,
y = 4 X -2 and the curve, OK,

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now I have got 2 equations,
and I've got 2 unknowns B&C.

8c2a2661-5477-4e30-86c4-63acf373c4d5-0
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And so I'm going to solve them
simultaneously.

f0da2ea3-0893-4d53-9b16-70fd72e0cd2f-0
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So in order to solve them simultaneously,
I'm going to put the left hand side of

f0da2ea3-0893-4d53-9b16-70fd72e0cd2f-1
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this one equal to the left hand side of
this one.

4d549638-bbd1-43da-959e-63faf30e990c-0
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So let's just bring that down here.

e7496801-d53b-4a51-8819-ef38e2eb13a6-0
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We can say that b ^2 - 4 B plus minus
four C + 16 = b ^2 - 8 B -4 C +8 and I

e7496801-d53b-4a51-8819-ef38e2eb13a6-1
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can see here I've got b ^2 on both sides.

6ae657a1-ae9e-451c-a030-722e1732ddad-0
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I've got A -,
4 C on both sides and so I'm going to get

6ae657a1-ae9e-451c-a030-722e1732ddad-1
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-4 B and moving this one over would be
plus 8B.

e60442c0-9f7f-4a71-849b-db54c0fe48ee-0
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I'm going to get a positive 4B on this
side and I'm going to have 8:00.

af42a8f3-328a-4e47-b7a0-a0d988271a07-0
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And then moving this 16 gives -16 so I'm
going to get four b = -,

af42a8f3-328a-4e47-b7a0-a0d988271a07-1
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8 and that will give me a value of -2 for
B.

eb4ef666-bb5b-4bd1-bd28-0f624704b10b-0
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Now I just need to go and find one of the
equations to use to get C so I'm going to

eb4ef666-bb5b-4bd1-bd28-0f624704b10b-1
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choose this one.

ca895d78-16c2-4a42-9c41-7e5b94d49965-0
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With B being -2 I will get -2 - 2,
which is -4 ^2 -4 C minus by minus plus

ca895d78-16c2-4a42-9c41-7e5b94d49965-1
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12 = 0.

76252bec-02aa-412b-99e8-2800e288172a-0
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So I'm going to have 16 - 4 C plus 12 = 0.

06188fdc-dba1-4ef2-a0d0-7e473f66718a-0
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That will give me that minus four C = -,
28 and then that will give me a value of

06188fdc-dba1-4ef2-a0d0-7e473f66718a-1
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seven for C Now I just need to calculate
b -, C which will be -2 -,

06188fdc-dba1-4ef2-a0d0-7e473f66718a-2
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7 and that will give me a value of -9.

d6944846-7a56-465d-b070-c4bd81a88bf1-0
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So looking at my options,
I can see that option A gives me the

d6944846-7a56-465d-b070-c4bd81a88bf1-1
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value of -9 and that's the answer to this
question.