Mathematics · Oct / Nov 2022 · Paper 41 · Question 11
YMA01
As printed in the paper
The question as text
11. North y P East O x Q Figure 4 Figure 4 shows a sketch of the closed curve with equation (x + y) 3 + 10y 2 = 108x
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
11(a)[5]
x
+
y
+
10
y
2
=
108
x
y
d
Differentiates
10y to
2
20
y x
d
d
y
Differentiates (
)
to (
)
x
+
y
3
x
+
y
2
+
3
1
d
x
y
y
d
d
(
x
y
)
2
y
3
+
1
+
+
20
=
108
x
x
d
d
d
y
d
y
(
)
(
)
x
+
y
2
+
y
=
−
x
+
y
2
3
20
108
3
d
x
d
x
(
)
2
y
108
−
3
x
+
y
d
=
*
x
d
(
)
2
20
y
+
3
x
+
y
(
)
2
(
)
2
11(b)[4]
Deduces that P and Q are where
−
x
+
y
=
x
+
y
=
108 3
0
36
Attempts to substitute (
)
= ± into (
)
3
x
+
y
x
+
y
+
10
y
2
=
108
x
6
to form
an equation in y (or x)
y
(
y
)
Solves
216 10
+
2
=
108 6
−
and finds the negative root
Awrt 13900 metres
(a) Allow use of y' for d
y
x . This is a proof. Look carefully for evidence of candidates working
d
backwards.
y
d
B1: Differentiates
10y to
2
20
y x
d
y
M1: Attempts to differentiate (
)
d
x
+
y
3
Via the chain rule look for (
)
2
k x
+
y
1
+
See **
x
d
Alternatively you will see attempts where (
)
x
+
y
3
is multiplied out. Terms may not be
collected but
d
y
d
y
look for 3 of
x
→
x
x y
→
xy
+
x
xy
→
y
+
xy
y
→
3
...
2
,
...
2
2
2
,
...
2
2
2
and
3
d
x
d
x
differentiated to the correct form.
d
y
d
y
A1: Correct differentiation (
)
x
+
y
2
+
+
y
=
3
1
20
108
d
x
d
x
d
y
d
y
d
y
d
y
x
+
xy
+
x
+
y
+
xy
+
y
+
y
=
or
3
2
3 2
2
3
2
2
3
2
20
108
d
x
d
x
d
x
d
x
dM1: Collects the terms in d
y
x .
d
y
Look for either d
x being factorised out of the relevant terms
d
y
or the terms in d
x being set on one side of the equation and the other terms on the other side
d
It is dependent upon BOTH the B and M marks.
A1*: Proceeds to the given answer.
y
y
y
y
d
d
d
d
In the approach
3
x
2
+
3 2
xy
+
x
2
+
3
y
2
+
2
xy
+
3
y
2
+
20
y
=
108
it is
x
x
x
x
d
d
d
d
acceptable
to go from (
)
d
y
x
2
+
xy
+
y
2
+
y
=
−
x
2
−
xy
−
y
2
3
6
3
20
108
3
6
3
straight to the given answer
d
x
................................................................................................................................................................
.................
Watch for candidates who use the given answer and work backwards **.
(
)
2
y
108
−
3
x
+
y
y
y
d
d
d
(
)
2
(
)
2
=
3
x
+
y
+
20
y
=
108
−
3
x
+
y
d
x
(
)
2
d
x
d
x
y
+
x
+
y
20
3
Therefore candidates who start with
d
y
d
y
d
y
d
y
(
)
(
)
or indeed (
)
(
)
x
+
y
2
+
y
=
−
x
+
y
2
x
+
y
2
+
x
+
y
2
+
y
=
3
20
108
3
3
3
20
x
x
x
x
d
d
d
d
without any other working or evidence can only score B1 M0 A0 M0 A0
................................................................................................................................................................
..............
(b)
(
)
(
)
−
x
+
y
2
=
x
+
y
2
=
M1: Deduces that P and Q are where
108 3
0
36
This is implied if there is a statement that (
)
x
+
y
= ±
6
dM1: Attempts to solve (
)
= ± and (
)
x
+
y
x
+
y
3
+
y
2
=
x
6
10
108
simultaneously to form
a quadratic equation in y or x.
(
)
ddM1: Solves
216 10
+
y
2
=
108 6
−
y
and finds the negative root. Allow if both roots are found
Note that (
x
y
)
+
=
6
must have be used.
If an equation was set up in x, then look for an attempt to find the larger root of
(
)
2
216 10 6
+
−
x
=
108
x
which should then be used to find a negative y value
A1: Awrt 13900 metres or awrt 13.9 km with the correct units. Condone answers like 13.9
−
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