Mathematics · May / June 2025 · Paper 21 · Question 5
YMA01
As printed in the paper
The question as text
5. 7 x –2 –0.5 1 2.5 4 5.5 y 12 4.243 1.5 0.530 0.188 0.066 0.023 The table above shows corresponding values of x and y for x y = 3 1 2 The values of y are given to 3 decimal places as appropriate.
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
5(a)[3]
Identifies h = 1.5
1.5 12
Area =
0.023
2 4.243 1.5
(
0.530
0.188
0.066
)
+
+
+
+
+
+
2
= 18.8
7
x
+
2
7
x
1
1
1
1
(b) (i)
3
d
=
3
d
=
×
''18.8''
=
4.7
x
x
2
4
2
4
−
2
−
2
7
1
(
)
7
(ii)
2
2
d
=
−
x
+
x
x
×
''18.8''
+
x
2
=
51.3
3
−
2
−
2
(a)
Note that awrt 18.8 with no working is 0 marks
B1: Identifies h = 1.5 or may be implied by e.g. 0.75 oe outside of the bracket in the expression for the area
M1: Correct method of finding the area. Requires 1
×
×correct bracket structure but condone the
their h
2
omission of a trailing bracket. Allow
h = if stated but if only an expression is seen then it must be
1
correct. Allow equivalent expressions e.g. they may add together individual trapezia.
Condone invisible brackets to be implied by further work or awrt 18.8 but do not allow just awrt 18.8
with no working seen to score this mark. Condone slips on the digits of the heights, e.g 4.234 for 4.243
A1: 18.8 or better (18.80775) following M1. i.e. if they have carried out the calculation and their 18.8 does
not come from 18.0775 (allow truncating or rounding) then A0. Isw
Note that the calculator answer is 17.2785...
(b)
Note in part (b) they may have rounded their value in part (a) as their final answer (e.g. 18.8)
but then used an earlier unrounded value in part (b). They must give an answer in part (b) that is
at least 1dp for the B1ft in (i) and/or A1ft (ii) to be awarded. You will need to check their follow
through answers if (a) is incorrect.
Do not accept solutions in (b) which involve redoing the table of values and attempting the
trapezium rule again.
(b)(i) Note if they have awrt 17.3 in (a) then this mark cannot be scored unless we see 1
4 ×
B1ft: Calculates 1
4 ×
''18.8''
=
"4.7"
Note that the calculator answer is 4.3196...
(b)(ii) Note if they have awrt 17.3 in (a) then A1ft can be scored following M1.
7
7
x
(
)
1
1
M1: Recognises that
2
−
x
d
=
×
3
d
AND integrates 2x to x2. We must see the x2 or allow to
x
x
3
2
−
2
−
2
(
)
2
be implied by limits substituted in and subtracting e.g.
7
2
−−
2
(or
7
2
−
2
2
or 49
−) but not just 45
4
A1ft: 51.3 or ft on their answer to part (a) by adding 45 to 1
. Allow fractions e.g. 769
15
3×
''18.8''
Note that the calculator answer is 50.7595...
x
+
11
(
)
2log
4
−
=
3
+
log
5(b)[3]
Identifies h = 1.5
1.5 12
Area =
0.023
2 4.243 1.5
(
0.530
0.188
0.066
)
+
+
+
+
+
+
2
= 18.8
7
x
+
2
7
x
1
1
1
1
(b) (i)
3
d
=
3
d
=
×
''18.8''
=
4.7
x
x
2
4
2
4
−
2
−
2
7
1
(
)
7
(ii)
2
2
d
=
−
x
+
x
x
×
''18.8''
+
x
2
=
51.3
3
−
2
−
2
(a)
Note that awrt 18.8 with no working is 0 marks
B1: Identifies h = 1.5 or may be implied by e.g. 0.75 oe outside of the bracket in the expression for the area
M1: Correct method of finding the area. Requires 1
×
×correct bracket structure but condone the
their h
2
omission of a trailing bracket. Allow
h = if stated but if only an expression is seen then it must be
1
correct. Allow equivalent expressions e.g. they may add together individual trapezia.
Condone invisible brackets to be implied by further work or awrt 18.8 but do not allow just awrt 18.8
with no working seen to score this mark. Condone slips on the digits of the heights, e.g 4.234 for 4.243
A1: 18.8 or better (18.80775) following M1. i.e. if they have carried out the calculation and their 18.8 does
not come from 18.0775 (allow truncating or rounding) then A0. Isw
Note that the calculator answer is 17.2785...
(b)
Note in part (b) they may have rounded their value in part (a) as their final answer (e.g. 18.8)
but then used an earlier unrounded value in part (b). They must give an answer in part (b) that is
at least 1dp for the B1ft in (i) and/or A1ft (ii) to be awarded. You will need to check their follow
through answers if (a) is incorrect.
Do not accept solutions in (b) which involve redoing the table of values and attempting the
trapezium rule again.
(b)(i) Note if they have awrt 17.3 in (a) then this mark cannot be scored unless we see 1
4 ×
B1ft: Calculates 1
4 ×
''18.8''
=
"4.7"
Note that the calculator answer is 4.3196...
(b)(ii) Note if they have awrt 17.3 in (a) then A1ft can be scored following M1.
7
7
x
(
)
1
1
M1: Recognises that
2
−
x
d
=
×
3
d
AND integrates 2x to x2. We must see the x2 or allow to
x
x
3
2
−
2
−
2
(
)
2
be implied by limits substituted in and subtracting e.g.
7
2
−−
2
(or
7
2
−
2
2
or 49
−) but not just 45
4
A1ft: 51.3 or ft on their answer to part (a) by adding 45 to 1
. Allow fractions e.g. 769
15
3×
''18.8''
Note that the calculator answer is 50.7595...
x
+
11
(
)
2log
4
−
=
3
+
log
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