Mathematics · May / June 2024 · Paper 41 · Question 3
YMA01
As printed in the paper
The question as text
3. The curve C is defined by the equation 3 2 x y xy 8 3 2 9 Find an equation of the normal to C at the point (2, 5), giving your answer in the form ax + by + c = 0, where a, b and c are integers. (7)
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Show the answer and mark scheme
d
y
d
y
3
y
2
→
ky x
or 2
xy
→
α
y
+
β
x x
Either
d
d
d
y
d
y
3
y
2
→
ky x
and 2
xy
→
α
y
+
β
x x
Both
d
d
d
y
d
y
8
3
3
2
2
9
24
2
6
2
2
0
x
−
y
+
xy
=
→
x
−
y
+
y
+
x
=
d
x
d
x
d
y
d
y
d
y
53
(2, 5) : 96
−
30
+
10
+
4
=
0
=
...
=
d
x
d
x
d
x
1
3
d
y
53
d
x
13
x =
or
y =
o.e.
d
13
d
5
3
13
(
)
5
"
"
2
Normal:
y
−
= −
x
−
53
13
x
+
53
y
−
291
=
0
d
y
3
y
2
ky x
M1: Differentiates one term in y to a correct form. Award for either
−
→±
or
d
d
y
2
xy
→
α
y
+
β
x x
Accept equivalents if they differentiate with respect to y instead.
d
dM1: Differentiates both y terms to a correct form
d
y
d
y
Award for Both
−
3
y
2
→±
k
y
and 2
xy
→
α
y
+
β
x x
dx
d
'y for d
y
A1: Fully correct differentiation. (o.e. if with respect to y). Allow
x
d
You may see
24
x
2
d
x
−
6 d
y y
+
2 d
y x
+
2 d
x y
=
0
which is fine.
d
y
d
y
d
y
is A0 unless the extra d
y
Note that
=
24
x
2
−
6
y
+
2
y
+
2
x
=
0
x = is subsequently omitted
d
x
d
x
d
x
d
M1: For
y
x (or d
x
y ) in terms of both x and y and then substituting both (2, 5)
d
d
y
x (or d
x
y )
d
d
d
y
24
x
2
+
2
y
FYI
o.e.
=
d
x
6
y
−
2
x
It is dependent upon having exactly two terms in d
y
x , one each coming from differentiating the
d
and the 2xy .
A1: Correct value for either d
y
53
(accept awrt 4.08) or d
x
13
x =
y =
(awrt 0.245)
d
13
d
53
13
(
)
M1: Correct method for the equation of the normal at (2, 5). E.g.
y
−
5
= −
"
"
x
−
2
53
It is dependent upon them having attempted differentiation (may be only one term in d
d
x
which must be the negative reciprocal of their d
x
=
2,
y
=
5
to find the value of
−
d
y
d
If the form y
=
mx
+ is used then the method must proceed as far as c = ...
c
A1: Correct answer or any integer multiple of this. The = 0 must be seen
1 5
d
25
(
)
A
(
)
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