Mathematics · May / June 2025 · Paper 41 · Question 3
YMA01
As printed in the paper
The question as text
3. Given that y = 4 at x = 6 d y π π 2 y cos 2 x x 3sin 2 x y 0 x d 4 4 giving your answer in the form y 2 = g(x) (6)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
d
y
sin 2
x
=
(
)
=
→
y
cos
2
2
x
3sin 2
x
y
d
y
3
d
x
d
x
cos 2
2
x
y
d
y
=
3sec(2 )tan(2 )d
x
x
x
2
y
3 sec2
(
)
=
x
+
c
2
2
16
3 sec 2
π
=
+
=
( 5)
=
c
c
...
2
2
6
y
2
=
3sec2
x
+
10
Main method
sin 2
x
M1: For an attempt to separate the variables OR use of
x sec2 tan2
x
x
cos 2
2
For the attempt at separation of variables it is not necessary to have the integral signs but the dx and dy
must be present and in the correct positions (on at least one line)
d
y
sin 2
x
d
y
sin 2
x
(
)
=
=
=
→
E.g.
y
cos
2
2
x
3sin 2
x
y
d
y
3
d
x
or
y
d
x
3
d
d
x
cos 2
2
x
d
x
cos 2
2
x
Condone a slip on the position (or omission) of the 3
sin 2
x
A1:
Correct separation of variables AND use of
x = sec2 tan2
x
x
cos 2
2
=
E.g.
y
d
y
3sec(2 )tan(2 )d
x
x
x
o.e. with or without the integral signs
1
3 y
d
y
=
sec(2 )tan(2 )d
x
x
x
Note that
is also correct
(
)
M1:
Correct attempted integration of the sec2 tan2
x
x to reach
f ( )
y
=
α
sec2
x
+
c
with or without the c
A1: Correct integration.
2
y
3 sec2
(
)
For example,
=
x
+
c
Accept with or without the c
2
2
y
2
(
)
(
)
=
+
=
+
Note that that
y
2
3sec2
x
c
and
sec2
x
c
are also examples of correct work.
3
π
dM1: Substitutes in
y =
4
and
x
=
into an integrated equation of the form f( )
y
=
α
sec2
6
find c. It is dependent on the previous method mark. You may need to check calculations if the
y
x
y
2
3 sec2
substitution isn’t shown. You may see this attempted via
=
π
x
2
2
4
6
3
2
=
+
y
2
=
+
10
A1:
y
3sec2
x
10
or equivalent such as
but ISW after a correct answer.
cos2
x
The answer is required to be in the form
2y =
A score of 1,0,1,1,1,1 is possible for candidates who never have a dx and/or a dy when separating the
variables, but do everything else correctly.
3 ALT
d
y
sin 2
x
(
)
=
→
=
→
=
y
cos
2
2
x
3sin 2
x
y
d
y
3
d
x
y
d
y
3(cos2 )
x
n
d
x
cos 2
2
x
where
n = − or 3
1
−
y
2
3 cos2
(
)
−
(
)
=
x
1
+
c
2
2
16
3
=
+
=
( 5)
=
c
c
...
π
2
2cos 3
3
y
2
=
+
10
cos2
x
sin 2
x
Alt versions will integrate
x using substitution or reverse chain rule
cos 2
2
The key here is realising that the integral of this is (
)
n
=
cos 2
x
or
u where
n
u
cos2
x
sin 2
x
(
)
n
M1: For an attempt to separate the variables OR use of
d
x
cos2
x
2
cos 2
x
d
y
=
(
)
=
→
For separation of variables look for example for
y
cos
2
2
x
3sin 2
x
y
d
y
d
x
condoning a slip on the position (or omission) of the 3 and omission of the integral signs
sin 2
x
(
)
n
A1:
Correct separation of variables AND use of
d
x
cos2
x
where
n = − or 3
1
cos 2
2
x
sin 2
x
=
n
n = − or 3
If the candidate sets
u
cos2
x
it is for integrating
d
x
to
u where
1
cos 2
2
x
α
sin 2
x
(
)
M1: Correct attempted integration of the
d
x
to reach
f( )
y
=
+
c
o.e with or without the c
cos 2
2
x
cos2
x
2
y
3
(
)
A1: Correct integration. For example,
=
+
c
o.e. Accept with or without the c
2
2cos2
x
3
y
2
(
)
(
)
(
)
=
+
=
−
1
+
Note that that
y
2
c
and
cos2
x
c
are also examples of correct work.
cos2
x
3
π
α
=
=
dM1: Substitutes in
y =
4
and
x
into an integrated equation of the form f( )
y
6
cos2
x
to find c. It is dependent on the previous method mark.
3
3
2
=
+
2
=
+
A1:
y
10
or equivalent such as
y
10
but ISW after a correct answer.
cos2
x
cos2
x
The answer is required to be in the form
2y =
…………………………………………………………………………………………………………………..
3
Also note that you may attempt using integration by parts. FYI
2
=
3sin 2 sec 2
x
x
d
x
sin 2 tan 2
x
2
1st A1: Correct separation of variables and
= ±
λ
±
,
3sin 2 sec 2
x
2
x
d
x
sin 2 tan 2
x
x
cos2
x
2nd M1:
2
=
λ
±
,
λ
>
0
3sin 2 sec 2
x
x
d
x
sin 2 tan 2
x
x
cos2
x
3
3
2nd A1:
3sin 2 sec 2
x
2
x
d
x
=
sin 2 tan 2
x
x
+
cos2
x
2
2
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