Mathematics · Oct / Nov 2024 · Paper 11 · Question 4
YMA01
As printed in the paper
The question as text
4. y C1 P C2 O 4 x Figure 1 Figure 1 shows a sketch of part of the curves C1 and C2 Given that C1 • has equation y = f(x) where f(x) is a quadratic function • cuts the x‑axis at the origin and at x = 4 • has a minimum turning point at (2, –4.8)
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Show the answer and mark scheme
4(a)[3]
Attempts to find k. E.g. 4.8
−
=
k
××
2 (2
−
4)
k
=
...
f( )
x
=
1.2 (
x x
−
4)
2
States or implies that
g( )
x
= λ
x x
(
−
4)
4(b)[3]
2
Attempts to find λ . E.g.
7.2
= λ××
6 (6
−
4)
λ=
...
2
g( )
x
=
0.3 (
x x
−
4)
2
4(c)[4]
Sets their
1.2 (
x x
−
4)
=
0.3 (
x x
−
4)
1.2
2
Valid attempt to solve 1.2x(
x −
4)
=
0.3x
(
x −
4)
x
=
4
+
0.3
x =
8
(
)
8, 38.4
(a)
M1: States or implies that f( )
x
=
kx x
(
−
4)
. Allow with k = 1. The f(x) need not be seen –
assume they are working with f(x) in part (a)
2
Alternatives include
f ( )
x
=
k x
(
−
2)
−
4.8
Allow with k = 1 and
f( )
x
=
ax
least one of 4
a
+
b
=
0
and/or 4
a
+
2
b
= −
4.8
dM1: Attempts to find all the constants in their equation for f( )x using (
)
2, 4.8
−
−
4.8
=
k
××
2 (2
−
4)
k
=
...
A1:
f( )
x
=
1.2 (
x x
−
4)
o.e. such as
f( )
x
=
1.2
x
2
−
4.8
x
or
f( )
x
=
1.2(
x
−
isw. Accept with (x − 0). The f(x) may be missing, it can be assumed (but use of g(x) is
A0). Allow if the full expression is given in latter parts if all the work was correct in (a)
to find the constants.
(b)
2
M1: States or implies that
g( )
(
4)
Allow with λ = 1. The g(x) need not be seen –
x
= λ
x x
−
assume they are working with g(x) in part (b).
g( )x
ax
3
bx
2
cx
with at least two of 64
16
Alternatives include
=
+
+
a
+
b
+
48
a
+
8
b
+
c
=
0,
and/or 216
a
+
36
b
+
6
c
=
7.2
(allow minor slips).
dM1: Attempts a full equation for g( )x using (
)
6,7.2
2
E.g.
7.2
= λ××
6 (6
−
4)
λ=
...
A1:
2
o.e. such as
3
2
and isw. Accept with (x
g( )
x
=
0.3 (
x x
−
4)
g( )
x
=
0.3
x
−
2.4
x
+
4.8
x
− 0). The g(x) may be missing, it can be assumed (but use of f(x) is A0). Allow if the
full expression in used in (c) if all the work was correct in (b) to find the constants.
(c)
2
B1ft: Sets their
1.2 (
x x
4)
0.3 (
x x
4)
. It is for a ''correct'' ft equation on their constants,
−
=
−
but the form (quadratic = cubic) must be correct.
M1: Valid attempt to solve their equations to find at least one nonzero solution as long as
they have a cubic and quadratic equated. E.g. 1.2x(
2
x −
4)
=
0.3x
(
x −
4)
Alternatively, they may expand, factor out the x and solve the quadratic (allowing for
slips), or even expand to a cubic (allowing slips) and solve via calculator (may need to
check, and you may ignore “incorrect factorisation” if the answers are correct for their
equation).
A1: x = 8 identified as the x coordinate of P. A0 if they give 0,4 and 8 and do not select the
correct answer.
(
)
A1:
8, 38.4 or exact equivalent for 38.4
Attempts
r
θ
= ×
5 1.2
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