Mathematics · Oct / Nov 2021 · Paper 41 · Question 1
YMA01
As printed in the paper
The question as text
1. The curve C has equation 2 + 3x 2 y = 4x 2 + 8 2x − 4y The point P (3, 2) lies on C. Find the equation of the normal to C at the point P, writing your answer in the form ax + by + c = 0 where a, b and c are integers to be found. (7)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
y
2
→
... d
y x
Attempt at chain rule
2 d
y
3
x y
2
→
...
x
+
...
xy
Attempt at product rule
d
x
d
y
d
y
2
−
8
y
+
6
xy
+
3
x
2
=
8
x
Correct differentiation
d
x
d
x
⇒ d
y
14
Substitutes
x
=
3,
y
=
2
x = −
d
11
11
Correct method for normal at (
3,2
)
2
(
3
)
⇒
y
−
=
x
−
14
11
x
−
14
y
−
5
=
0
d
y
y
2
... d
y x
M1: Attempts at chain rule
→
where ... is a constant
2 d
y
3
2
...
...
M1: Attempt at product rule
x y
→
x
+
xy
where ... are constants
d
x
d
y
d
y
2
2
8
6
3
2
8
A1: Correct differentiation
2
x
−
4
y
+
3
x y
2
=
4
x
2
+
8
→
−
y
+
xy
+
x
=
d
x
d
x
You may see
2
x
−
4
y
2
+
3
x y
2
=
4
x
2
+
8
→
2d
x
−
8 d
y y
+
6
xy x
d
+
3
x
2
d
y
=
8 d
x x
o.e. which is fine
d
y
d
y
d
y
=
2
−
8
y
+
6
xy
+
3
x
2
−
8
x
Note that
is A0 unless recovered
d
x
d
x
d
x
into a suitable equation and finds a value for d
y
x
=
3,
y
=
2
M1: Substitutes
x .
d
It is dependent upon having exactly two d
y
x terms, one from differentiating each of
d
A1: d
14
y
x = −
o.e.
d
11
dM1: Dependent upon the previous M. It is for a correct method of finding the equation of
11
y
2
''
''
(
x
3
)
the normal at (3,2). Look for
−
=
−
with the negative reciprocal of
14
their 14
−
being used.
11
If the form y
=
mx
+
c
is used they must proceed as far as c = ...
A1: 11
x
−
14
y
−
5
=
0
but accept any integer multiple of this. The ''= 0 '' must be seen
2
d
y
4
y
∫
1
∫
4
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