Mathematics · May / June 2025 · Paper 31 · Question 7
YMA01
As printed in the paper
The question as text
7. A continuous curve has equation π –x2 sin3x 0 x 3 y = e The curve has a stationary point at the point P.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
7(a)[4]
d
y
y
=
e
−
x
2
sin3
x
=
3e
−
x
2
cos3
x
−
2 e
x
−
x
2
sin3
x
d
x
2
2
3e
−
x
cos3
x
−
2 e
x
−
x
sin3
x
=
0
3cos3
x
−
2 sin3
x
x
=
0
3
3cos3
x
=
2 sin3
x
x
tan3
x
=
2
x
1
3
x
=
arctan
*
3
2
x
(b)(i)
1
3
x
=
0.4
x
=
arctan
1
2
3
2 0.4
×
(
)
x =
0.4367
2
(ii)
(
)
x =
0.4307
4
Notes:
(a)
2
2
M1: Differentiates using the product rule to obtain
α
e
−
x
cos3
x
+
β
x
e
−
x
sin3 ,
x
αβ
,
≠
0
A1:
Correct derivative in any form
dM1: Depends on the first method mark.
Sets their d
y
(which may be implied), cancels
e x
−
2
and rearranges to
±
tan3
x
=
...
x =
0
d
A1*: Correct proof with no errors seen.
1
3
Allow e.g.
tan − or e.g. artan for arctan and allow e.g.
1
x
=
arctan
3
2
x
Ignore any reference or attempts to solve
e
−
x
2
=
0
7(b)[3]
M1: Attempts to substitute x = 0.4 into the formula.
Allow if seen embedded in the formula or may be implied by
x =
awrt 0.44
2
A1:
Awrt 0.4367
A1:
0.4307 cao not awrt 0.4307
Both values correct as defined with no working scores M1A1A1
7(c)[2]
1
3
e.g. ( )
f
x
=
x
−
arctan
3
2
x
1
3
(
)
(
)
f 0.4305
=
0.4305
−
arctan
=
6.38 10
×
−
5
3
2 0.4305
×
1
3
(
)
(
)
f 0.4295
=
0.4295
−
arctan
= −
1.141... 10
×
−
3
3
2 0.4295
×
or e.g.
( )
2
2
f
x
=
3e
−
x
cos3
x
−
2 e
x
−
x
sin3
x
)(
)
(
)
(
)
2
(
)
(
)
(
)
2
(
f 0.4305
=
3e
−
0.4305
cos3 0.4305
−
2 0.4305 e
−
0.4305
sin3 0.4305
= −
4.968... 10
×
−
4
)(
)
(
)
(
0.4295
)
2
(
)
(
)
(
0.4295
)
2
(
f 0.4295
=
3e
−
cos3 0.4295
−
2 0.4295 e
−
sin3 0.4295
=
8.885... 10
×
−
3
Sign change therefore x is 0.430 to 3dp
(c)
M1: Chooses a suitable function and attempts its value at both 0.4305 and 0.4295
For the attempt we need to see embedded values as in scheme or one value correct to 1sf for their suitable
function.
The function they are using must be seen or implied by embedded values.
Allowable functions are:
1
3
x
−
arctan
or multiples of this
3
2
x
or multiples of this following through on their d
y
3e
−
x
2
cos3
x
−
2 e
x
−
x
2
sin3
x
x
d
or multiples of this following through on their d
y
3cos3
x
−
2 sin3
x
x
x
d
Unlikely, but it is also acceptable to pick a tighter interval e.g. 0.4305 and 0.4301
A1: Requires correct differentiation (if d
y
is used)
d
x
both values correct (rounded or truncated to 1sf)
a valid reason that includes a reference to the sign change e.g. < 0, > 0
a minimal conclusion which could be Π, QED, root
Note we condone the omission of a reference to continuity.
Attempts to use repeated iteration score no marks.
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