Mathematics · Oct / Nov 2022 · Paper 41 · Question 5
YMA01
As printed in the paper
The question as text
5. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. y R O x 1 k 2 Figure 2 Figure 2 shows a sketch of part of the curve with equation x 12 y = ) ( 1 5 . 2 x + 2 3 The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = 1 2 , the x-axis and the line with equation x = k. This region is rotated through 360° about the x-axis to form a solid of revolution. Given that the volume of this solid is 713 648 π , use algebraic integration to find the exact value of the constant k. (6)
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
x x
,
or
x
A1: Fully correct
2
k
12
x
5.
States or implies Volume =
π
d
x
1.5
(2
x
2
+
3)
1
2
2
x
x
12
144
−
2
x
=
x
= −
x
2
+
Attempts
d
d
18(2
3)
1.5
3
x
2
+
x
2
+
(2
3)
(2
3)
18
π
18
π
713
Sets
−
+
=
π
k
2
2
1
648
(2
+
3)
2
(2
×
+
3)
2
18
9
713
−
+
=
k
2
+
2
=
k
2
=
(2
3)
729
...
8
648
k
2
2
(2
+
3)
k =
2 3
2
k
x
12
B1: States or implies Volume =
π
d
x
o.e. including limits and π.
1.5
(2
x
2
+
3)
1
2
You can condone a missing dx and the limits the wrong way around.
You may not see the limits until the integration has been attempted which is fine.
2
x
x
2
−
2
M1: Attempts
d
to achieve
A
(2
x
+
3)
o.e.
x
2
1.5
(2
+
3)
−
2
With substitution of
u
=
2
x
2
+ look for
3
Au
2
12
x
x
2
−
2
−
2
A1: Correct integration of
d
to
−
18(2
x
+
3)
or
−
18u
which may be left
2
1.5
(2
x
+
3)
unsimplified.
Note that this can be scored with or without the π and may even have an incorrect
coefficient of 2π
y NOT y
2
M1: Uses the limits (subtracts either way around) within their attempted integration of
equal to 713
648π. Condone poor attempts at integrating the
y here.
2
and sets
Note that if they change limits to u where
u
=
x
2
+ look for limits of
k +
2
and 4
2
3
2
3
ddM1: Dependent upon both previous M's. It is for proceeding to
2
±
2
B >
k or k
2
either
(2
k
+
3)
=
B
where
0
leading to a value for
k leading to a value for
2
k or k
2
or setting up a quadratic equation in
.
FYI
4
2
2
2
2
k
+
3
k
−
180
=
0
k
−
12
k
+
15
=
0
k
=
12
k =
k = ±
A1:
2 3
or exact equivalent such as 12 Condone if
2 3
o.e.
Make a free account to save your answers and see your weak topics and ask the AI why a wrong answer is wrong.