Mathematics · May / June 2021 · Paper 21 · Question 5
YMA01
As printed in the paper
The question as text
5. y R C2 C1 x O 2.5 4 Figure 2 Figure 2 shows a sketch of part of the graph of the curves C1 and C2 The curves intersect when x = 2.5 and when x = 4 A table of values for some points on the curve C1 is shown below, with y values given to 3 decimal places as appropriate. 2.5 2.75 3 3.25 3.5 3.75 4 x 5.453 7.764 9.375 9.964 9.367 7.626 5 y Using the trapezium rule with all the values of y in the table,
Read automatically from the paper; the printed page above is exact.
Show the answer and mark scheme
5(a)[4]
h =
4
1
1
{
}
A
5.453
5
2 7.764
(
...
7.62
6
)
≈
×
+
+
+
+
2
4
1
1
{
}
(
)
A
≈
×
5.453
+
5
+
2 7.764
+
9.375
+
9.964
+
9.367
+
7.626
2
4
=
awrt 12.33
∫
3
5
5(b)[3]
x
−
3
x
+
9
d
x
=
k
x
−
lx
2
+
m
x
(at least two powers correct)
2
2
∫
3
2
5
3
x
−
3
x
+
9
d
x
=
x
−
x
2
+
9 (
x
+
c
)
2
2
5
2
5(c)[3]
5
4
2
5
3
2
5
3
2 5
3 5
2
5
2
x
2
−
x
2
+
9
x
=
×
4
2
−
×
4
2
+
9 4
×
−
−
+
9
=
...
5
2
5
2
5 2
2 2
2
5
2
(
=
24.8 17.077...
−
=
7.722...
)
Area R = “12.33” – “7.722” = …
= awrt 4.6
Notes:
(a)
B1: For using h = 0.25 or equivalent. Score for sight of h = 0.25 or 1
1
or just 1
×
8 outside the
2
4
brackets.
M1: Correct bracket structure for the trapezium rule. Look for “h/2”(first + last y values +
2(attempt at sum of remaining values)). If one value is missing, allow the M as a slip, but M0 if 1st
or last value repeated. Must be y values used. Incorrect/invisible external bracketing may be
recovered by implication.
A1: Correct bracket “h/2”{…} with all 7 y values used, though allow a minor miscopy of a term if it
is clear the correct value is meant. May be implied by the correct answer.
A1: awrt 12.33
(b)
M1: Attempts to integrate. Power to increase by 1 in at least two terms.
A1: Any two terms correct.
A1: Fully correct. No need for constant of integration. Need not be simplified.
(c)
M1: Applies limits of 2.5 and 4 to their answer to (b) to get the area under C2. Limits may be either
way around for this mark. May be implied by awrt 7.72 and allow if scored from a calculator.
M1: Subtracts their area under C2 from the answer to (a). Must be subtracting a positive number
from “12.33”, so if the limits were the wrong way round (or a negative answer was found in error),
the answer must be made positive before subtracting.
A1: for awrt 4.6 following a correct answer to (a).
Way 1: Eqn is (
)
2
(
)
2
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