Mathematics · Oct / Nov 2021 · Paper 21 · Question 3
YMA01
As printed in the paper
The question as text
3. y R O x 2 14 Figure 1 Figure 1 shows a sketch of part of the curve with equation y = log10 x The region R, shown shaded in Figure 1, is bounded by the curve, the line with equation x = 2, the x-axis and the line with equation x = 14 Using the trapezium rule with four strips of equal width,
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
3(a)[3]
States or uses
h =
''3'' log 2
{
}
(
)
Attempts
+
log 14
+ ×
2
.....
2
10
10
3 log 2
{
(
)
}
=
+
log 14
+ ×
2
log 5
+
log 8
+
log 11
= 10.10 (2dp)*
2
10
10
10
10
10
3(b)[1]
Increase the number of strips
14
∫
1
(c) (i)
log
x
d
x =
×
10.10
=
5.05
2
10
2
log 100
x
3
=
2
+
3log
x
(ii)
10
10
14
∫
[
]
log 100
3
d
2
14
3 10.10
54.30
x
x
=
x
+ ×
=
10
2
2
(a)
B1:
States or uses
h =
3
. (If there is a conflict between what is stated and used award bod if
one is correct.)
M1: An attempt at the trapezium rule with their value of h. Look for
''3'' log 2
{
}
(
)
+
log 14
+ ×
2
.....
where the ….. is an attempt at at least two
2
10
10
intermediate terms and does not repeat the end points. The bracket structure must be
correct or implied. (Use of h = 4 is likely to have two terms in this inner bracket.)
Allow with values correct to 2 s.f. for the first and last terms.
A1*: Reaches 10.10 (or allow 10.1) following at least one correct intermediate line and no
incorrect lines that would give a significantly different answer.
3 log 2
{
(
)
}
Example are
+
log 14
+ ×
2
log 5
+
log 8
+
log 11
or possibly
2
10
10
10
10
10
3 log 28
3 0.301 1.146
{
}
{
}
(
)
+
2log 440
or
+
+ ×
2
0.699
+
0.903 1.041
+
with
2
2
10
10
values correct to at least 2d.p.
(Note that integration via the calculator gives 10.23)
(b)
B1:
States ''increase the number of strips'', ''decrease the width of the strips'', “use more
intervals” o.e.
(c)(i)
(awrt) 5.05 or allow 1
B1:
×
( )
a
if they make a slip miscopying the 10.10
2
(This is a B mark, so allow even if this answer comes from a repeat at using the trapezium rule.)
(c)(ii)
B1:
States or implies that
log 100
x
3
=
2
+
3log
x
10
10
∫
14
[
]
14
log 100
x
3
d
x
=
ax
+ ×
3 10.10
M1:
or equivalent work for finding the area of the
10
2
2
rectangle.
Use of the trapezium rule again here is M0. Must see use of (a).
A1:
54.30 Accept 54.3.
3(c)[4]
States or uses
h =
''3'' log 2
{
}
(
)
Attempts
+
log 14
+ ×
2
.....
2
10
10
3 log 2
{
(
)
}
=
+
log 14
+ ×
2
log 5
+
log 8
+
log 11
= 10.10 (2dp)*
2
10
10
10
10
10
Increase the number of strips
14
∫
1
(c) (i)
log
x
d
x =
×
10.10
=
5.05
2
10
2
log 100
x
3
=
2
+
3log
x
(ii)
10
10
14
∫
[
]
log 100
3
d
2
14
3 10.10
54.30
x
x
=
x
+ ×
=
10
2
2
(a)
B1:
States or uses
h =
3
. (If there is a conflict between what is stated and used award bod if
one is correct.)
M1: An attempt at the trapezium rule with their value of h. Look for
''3'' log 2
{
}
(
)
+
log 14
+ ×
2
.....
where the ….. is an attempt at at least two
2
10
10
intermediate terms and does not repeat the end points. The bracket structure must be
correct or implied. (Use of h = 4 is likely to have two terms in this inner bracket.)
Allow with values correct to 2 s.f. for the first and last terms.
A1*: Reaches 10.10 (or allow 10.1) following at least one correct intermediate line and no
incorrect lines that would give a significantly different answer.
3 log 2
{
(
)
}
Example are
+
log 14
+ ×
2
log 5
+
log 8
+
log 11
or possibly
2
10
10
10
10
10
3 log 28
3 0.301 1.146
{
}
{
}
(
)
+
2log 440
or
+
+ ×
2
0.699
+
0.903 1.041
+
with
2
2
10
10
values correct to at least 2d.p.
(Note that integration via the calculator gives 10.23)
(b)
B1:
States ''increase the number of strips'', ''decrease the width of the strips'', “use more
intervals” o.e.
(c)(i)
(awrt) 5.05 or allow 1
B1:
×
( )
a
if they make a slip miscopying the 10.10
2
(This is a B mark, so allow even if this answer comes from a repeat at using the trapezium rule.)
(c)(ii)
B1:
States or implies that
log 100
x
3
=
2
+
3log
x
10
10
∫
14
[
]
14
log 100
x
3
d
x
=
ax
+ ×
3 10.10
M1:
or equivalent work for finding the area of the
10
2
2
rectangle.
Use of the trapezium rule again here is M0. Must see use of (a).
A1:
54.30 Accept 54.3.
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