Mathematics · Oct / Nov 2024 · Paper 41 · Question 10

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The question as text

10. y R O x Figure 5 Figure 5 shows a sketch of the curve with parametric equations π 2 x 3 t y sin sin 2 t t 0 t 2 The region R, shown shaded in Figure 5, is bounded by the curve and the x-axis.

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10(a)[3]
d
x
t =
6
t
o.e.
d
Area
=
sin sin 2
t
t
×
6 d
t t
=
sin
t
×
2sin cos
t
t
×
6 d
t t
=
π
2
12
t
sin
2
t
cos d
t t
0
10(b)[5]
π
π
t
1
2
2
sin
3
sin
3
d
=
t
−
t t
3
3
0
0
π
π
t
12
(
)
2
2
=
sin
3
t
−
sin
t
1 cos
−
2
t
d
t
3
3
0
0
π
4
2
=
4 sin
t
3
t
+
4cos
t
−
cos
3
t
3
0
4
=
2
π
−
4
−
3
8
=
2
π
−
3
(a)
On EPEN this is M1M1A1 but we are marking this as B1M1A1
It is acceptable to work in a different variable such as θ instead of t provided the final
is in terms of t but withhold the final mark if there is a mixture of variables in the
same line
of working.
d
x
B1:
t =
6
t
seen or implied.
d
t implies B1 but
6 sin
t
2
t
cos
t does not imply B1.
6 sin sin2
t
t
M1: Uses sin2
t
=
2sin cos
t
t
within the expression for y i.e.
or may
y
=
sin sin 2
t
t
=
2sin
2
t
cos
t
(
)
(
be seen in their integral. May be implied by e.g.
pt
sin sin 2
t
t
d
t
→
2
pt
sin
2
t
cos
t
d
Condone the omission of t. i.e.
y =
2sin cos
2
A1: Correct integral including dt with k = 12 with at least one intermediate stage of working
(which may only be e.g. d
x
t =
6
t
) The integral sign, dt, limits only need to be present in the
d
π.
final answer. Ignore any attempts to verify the limits of 0 and 2
Do not condone poor notation e.g.
sint or the absence of variables in the main body of the
2
solution.
12 sin
t
2
t
cos
t with or without integration notation and with no other working implies
B1M1A0
(b) Condone working in mixed variables provided the intention is clear or they recover in further
work. Condone the absence of variables in trigonometric functions e.g. sin for sint or cos for
cost
M1: Attempts integration by parts in the correct direction to achieve an expression of the form
... sin
t
3
t
±
... sin
3
t
d
t
Condone the omission of their 12
(
)
... sin
3
... sin (1 cos
2
) d
M1: Uses
sin
3
t
=
sin
t
1 cos
−
2
t
to reach the integrable form
t
t
±
t
−
t
t
o.e.
May be seen as
... sin
t
3
t
±
... sin
t
−
sin cos
t
2
t
d
t
.
Alternatively uses compound angle formule to substitute
sin
3
t
=
...sin
t
±
...sin3
t
to reach an
(
)
(
)
integrable form.
... sin
t
3
t
±
...
...sin
t
±
...sin3 d
t
t
4 sin
t
3
t
−
(3sin
t
−
sin3 )d
t
t
In either approach it may be implied by their integrated expression. Condone the omission of
their 12
A1: Fully correct integration of
t
sin
2
t
cos d
t
t
1
or alternatively
4 sin
t
3
t
−
(3sin
t
−
sin3 )d
t
t
=
4 sin
t
3
t
+
3cos
t
−
cos3
t
.
3
4
Condone any multiple and condone missing variables e.g.
4 sin
t
3
+
4cos
−
cos
3
3
π to an expression of the form
sin
3
cos
cos
3
M1: Applies the limits 0 and
At
t
+
B
t
+
C
t
where
2
A B C ≠
, ,
0
α
cos
t
+
β
cos3
t
αβ≠
,
0
or in the alternative
where
A1: Correct answer in the required form. 8
3 does not need to be in simplest form. isw once a
correct answer is seen.
Alternative substitution method using
u
=
sin
t
to proceed to (
)
M1: Attempts to use the substitution
sin
12
u
2
sin
−
1
u u
d
and attempts
u
=
t
integration by parts in the correct direction to achieve an expression of the form
u
3
1
u
3
sin
−
1
u
−
d
u
but condone coefficient slips.
3
3
1
−
u
2
Condone the omission of their 12
M1: Proceeds to an integrable form by using the substitution
v
=
1
−
u
2
1
1
(
)
e.g.
u
3
sin
−
1
u
+
1
−
v
2
d
v
Condone the omission of their 12
3
3
(
)
1
1
1
3
A1: Fully correct integration
u
3
sin
−
1
u
+
1
−
u
2
−
1
−
u
2
(may not have reverted back to
3
3
9
being in terms of u). Condone any multiple and condone missing variables
(
M1: Applies the limits 0 and 1 to an expression of the form
Du
3
sin
−
1
u
+
E
1
−
u
2
+
F
1
−
u
where D, E and F
≠
0
A1: Correct answer in the required form. 8
3 does not need to be in simplest form. isw once a
correct answer is seen.
Alt (b) – Integration by parts twice
M1: Attempts integration by parts in the correct direction to achieve an expression of the form
... sin
t
3
t
±
... sin
3
t
d
t
Condone the omission of their 12
M1: Proceeds to an integrable form by attempting integration by parts again achieving the form
... sin
t
3
t
±
...cos sin
t
2
t
±
... sin cos
t
2
t t
d
Condone the omission of their 12
t
1
2
A1:
sin
3
t
+
cos sin
t
2
t
+
cos
3
t
o.e. Condone any multiple and condone missing
3
3
9
variables
π to an expression of the form
M1: Applies the limits 0 and
pt
sin
3
t
+
q
sin
2
t
cos
t
+
r
cos
3
t
2
where p, q and r
≠
0
A1: As above in main scheme notes
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