Mathematics · May / June 2022 · Paper 21 · Question 2
YMA01
As printed in the paper
The question as text
2. y O x Figure 1 Figure 1 shows the graph of y = 1 − log10 (sin x) 0 < x < π where x is in radians. The table below shows some values of x and y for this graph, with values of y given to 3 decimal places. x 0.5 1 1.5 2 2.5 3 y 1.319 1.001 1.223 1.850
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
2(a)[2]
y
1.319
1.075
1.001
1.041
1.223
1.850
h =
0.5
2(b)[3]
seen or implied
0.5
✓
(
1.319
2("1.075" 1.001 "1.041" 1.223)
1.850
)
Area
≈
×
+
+
+
+
+
2
=
awrt 2.96
(
)
2(c)[3]
(
)
(
)
3
+
log
sin( )
x
=
4
−
1 log
−
sin( )
x
10
10
3
[
]
3
3
+
log (sin( ))
x
d
x
=
4
x
−
"2.96"
=
12
−
2
−
"2.96"
1
0
0.5
0
.
5
= 7.04 follow through 10
−
their 2.96
Alt (c)
3
3
log (sin( ))
x
d
x =
1
dx
−
"2.96"
=
2.5
−
"2.96"
10
0
.5
0.5
3
[
]
3
(
)
3
+
log (sin( ))
x
d
x
=
3
x
+
2.5
−
"2.96"
=
9 1.5
−
+
2.5
−
"2.96"
1
0
0.5
0
.5
= 7.04 follow through 10
−
their 2.96
Notes:
(a)
B1: Either of awrt 1.075 or awrt.1.041 correct
B1: Both of awrt 1.075 and awrt.1.041 correct
Remember to check the body of the script for the values if the table has not been completed.
Note that the incorrect values of 2.758 and 2.457 come from the use of degrees.
(Which gives 4.51 in (b))
(b)
B1: Correct interval width seen or implied by working.
M1: Correct application of the trapezium rule. Must all be y values, and should be first + last +
2(sum of middle values), but if one middle value is omitted allow as a slip. A repeated value is
M0. Follow through their values from (a). Allow obvious copying slips.
Note that "0.5"
✓×
1.319
+
2("1.075" 1.001 "1.041" 1.223)
+
+
+
+
1.850
scores M0 unless the
2
missing brackets are implied by subsequent work.
Allow this mark if they add the areas of individual trapezia e.g.
"0.5"
"0.5"
✓
✓
(
)
(
)
×
1.319
+
"1.075"
+
×
"1.075" 1.001
+
+
...
2
2
A1: For awrt 2.96. Allow to come from answers in (a) incorrectly rounded or wrong accuracy.
FYI: Use of rounded figure gives 2.96225, use of calculated figures gives 2.9625726… Actual value is 2.89
to 2 decimal places.
(
)
(c) Condone all work in (c) written without the “10” in
log
sin( )
x
10
(
)
(
)
B1: Writes
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