Mathematics · May / June 2021 · Paper 31 · Question 1
YMA01
As printed in the paper
The question as text
1. The curve C has equation y = x 2 cos 1 2 x 0 < x π 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
1(a)[4]
d
y
1
1
1
=
2 cos
x
x
−
x
sin
x
2
d
x
2
2
2
Sets d
y
1
4
4
=
0
⇒
tan
x
=
⇒
x
=
2arctan
*
d
x
2
x
x
1(b)[3]
4
x
=
2arctan 2
2
awrt 2.214
x =
2.155
x =
cao
2
6
(a)
1
M1: Differentiates using the product rule to obtain an expression of the form
Ax
cos
x
+
Bx
2
If the product rule is quoted it must be correct.
A1: For a correct simplified or unsimplified derivative.
dM1: Sets their d
y
1
x =
0
and proceeds to an equation involving
tan 2 x
. This may be implied by their working.
d
The M1 must have been scored.
4
A1*: Correctly achieves the given answer of
x
=
2arctan
x
1
sin 2
x
1
=
4
2
x
tan 2 x
=
An intermediate line of
or
must be seen
x
1
1
x
2
cos
x
2
2
Note that some may work backwards having differentiated e.g.
( )
4
x
4
x
4
x
sin
x
x
x
x
=
2arctan
⇒
=
arctan
⇒
tan
=
⇒
2
=
4
⇒
4cos
=
x
sin
x
2
x
2
x
cos
( )
2
2
x
2
x
1
x
x
1
x
⇒
2 cos
x
=
x
2
sin
⇒
2 cos
x
−
x
2
sin
=
0
2
2
2
2
2
2
Score M1 for a complete method to reach d
y
and A1 if correct with (minimal) conclusion.
x =
0
d
(b)
4
M1: Attempts to use the given iteration formula. Award for
x
=
2arctan 2
or awrt 2.21
2
Note that this mark may also be implied by awrt 127 if they are using degrees and no working is shown.
2x = awrt 2.214
A1:
6x = 2.155 (Note that this is not awrt)
A1:
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