Mathematics · May / June 2022 · Paper 41 · Question 8
YMA01
As printed in the paper
The question as text
8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. 20 cm y R O 10 x Figure 2 Figure 3 Figure 2 shows the curve with equation −1 2 x y = 10xe 0 x 10 The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the line with equation x = 10 The region R is rotated through 2π radians about the x-axis to form a solid of revolution.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
8(a)[2]
2
10
1
−
x
10
V
=
π
10
xe
d
x
=
;
2
100
π
x
2
e
−
x
d
x
0
0
8(b)[3]
x
2
e
−
x
d
x
= −
x
2
e
−
x
+
2
x
e
−
x
d
x
= −
x
2
e
−
x
+
2
x
e
−
x
d
x
= −
x
2
e
−
x
+
2
−
x
e
−
x
+
e d
−
x
x
−
x
2
e
−
x
−
2 e
x
−
x
+
2 e d
−
x
x
= −
x
2
e
−
x
−
2 e
x
−
x
−
2e
−
x
(
+
c
)
8(c)[5]
10
Total volume
2 "100 "
x
2
e
−
x
d
x
=
×
π
0
10
10
(
) (
x
2
e
−
x
= −
x
2
e
−
x
−
2 e
x
−
x
−
2e
−
x
= −
(10) e
2
−
10
−×
2 10e
−
10
−
2e
−
10
−−
2
)
0
0
10
=
2 122e−
−
(1.9944…)
5000
Density
=
"200 " "1.9944..."
π
×
awrt 3.99 (g / cm3)
Notes
(a)
M1: Forms a correct unsimplified expression for the volume.
10
A1: Achieves
100
π
x
2
e
−
x
d
x
Condone a missing dx but limits must be present.
0
(b)
M1: Attempts integration by parts in the right direction to achieve an expression of the form
...
x
2
e
−
x
±
...
x
e
−
x
d
x
Condone missing dx
dM1: Dependent on the previous method mark. Attempts integration by parts a second time to achieve an
2
±
expression of the form
...
x
e
−
x
±
... e
x
−
x
... e
−
x
d
x
Condone missing dx
−
x
2e
−
x
−
2 e
x
−
x
−
2e
−
x
(
+
c
)
with or without the constant of integration
A1:
For attempts via the DI (tabular) method, look for first two rows of the table to have correct forms for M1, all
rows with correct forms and answer extracted for dM1 and A1 for correct answer.
(c)
M1: A correct strategy to find the total volume with their values of k.
M1: Substitutes the limits of 10 and 0 into their part (b) and subtracts. Alternatively allow M1 for limits 20 and 0
used (as a mistaken attempt to double).
2 122e−
−
10
or awrt 1.99
A1:
5000
dM1: Dependent on second M. Attempts to find the density using
their Volume . The attempt at the volume
need not be correct but an attempt at using (b) must have been made. E.g. if they forget k or forget to
double, allow for the attempt with their volume. Must be with 5000 in numerator, or with correct work to
reach correct units later.
5000
A1: awrt 3.99 (g / cm3) oe. Accept exact simplified answers such as
200 (2 122e
−
10
)
π
−
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