Mathematics · May / June 2024 · Paper 11 · Question 8
YMA01
As printed in the paper
The question as text
8. The curve C1 has equation 2) y = x(4 – x
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8(a)[3]
−2
2
O
8(b)[1]
A
(
)
x
4
−
x
2
=
4
x
2
−
x
4
=
A
x
x
4
−
4
x
2
+
A
=
0*
8(c)[3]
A > 0
b
2
=
4
ac
16
=
4
A
A
=
...
0 < A < 4
Notes:
(a)
B1: Negative cubic curve in any position. Must have one maximum and one minimum.
B1: Positive or negative cubic curve passing through (not touching) the origin.
The origin does not have to be labelled.
B1: Cuts the xaxis at x = 2 and x = −2. May be as shown or as (2, 0) and (−2, 0).
Must cut so extends beyond the xaxis from above or below but not a turning point.
Condone (0, 2) and (0, −2) as long as they are in the correct positions.
Allow away from the sketch but must be (2, 0) and (−2, 0).
If there is any ambiguity, the sketch takes precedence.
Examples:
100 110 010 010 001
110 011 010 000
B1*: Equates the curves and shows sufficient working to obtain the printed answer with no errors.
Note that (
)
x
4
x
2
may be expanded first before equating which is fine.
−
A
There must be at least one intermediate step between (
)
x
4
−
x
2
=
and the printed answer.
x
A
A
As a minimum accept (
)
x
4
−
x
2
=
4
x
−
x
3
=
x
4
−
4
x
2
+
A
=
0
x
x
(c)
B1: For A > 0 as one boundary. May be seen embedded in their final answer. x > 0 is B0.
M1: Attempts
b
2
=
4
ac
or equivalent e.g.
b
2
−
4
ac
=
0
with a = 1, b = −4 and c = A to obtain a value for A.
Condone use of an inequality rather than “=” to obtain a value or inequality for A e.g.
b
2
−
4
ac
b
2
−
4
ac
<
0
leading to A > … or A < …
May be implied by e.g. A = 4, A < 4, A > 4 etc.
A1: Correct range for A. Allow equivalents e.g. “A > 0 and A < 4”, “(0, 4)”
but not “A > 0 or A < 4”, “A > 0, A < 4”
Correct answer with no working scores 3/3
Partially correct answer with no working e.g. 0
<
A
4
can score B1 M1 (implied) A0
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