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Integration – Trapezium Rule
Mark Scheme 1
Level
International A Level
Subject
Maths
Exam Board
CIE
Topic
Integration
Sub Topic
Integration – Trapezium Rule
Booklet
Mark Scheme 1
Time Allowed:
57 minutes
Score:
/47
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
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1
2
(a
Integrate to obtain form k ln( 2x − 7)
Obtain correct 3ln( 2x − 7)
Substitute limits correctly (dependent on first M1)
Use law for logarithm of a quotient or power (dependent on first M1)
Confirm ln125 following correct work and sufficient detail (AG)
(b) Evaluate y at (1), 5, 9, 13, 17
Use correct formula, or equivalent, with h = 4 and five y-values
va
Obtain 13.5
3
(a) Integrate to obtain form k sin( 13 x + 2) where k ≠ 4
Obtain 12sin( 13 x + 2)
( + c)
20,
68,
148
(b) State or imply correct y-values 2,
Use correct formula, or equivalent, with h = 4 and four y-values
va
Obtain 79.2
M1
A1
DM1
DM1
A1
[5]
M1
A1
[3]
M1
A1
[2]
B1
A1
[3]
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4
(a
(i) Attempt to divide by e2x and attempt to integrate 2 terms
Integrate a term of form ke−2x correctly
Fully correct integral x – 3e−2x (+ c)
cos 2 x + 12 or equivalent
Integrate an expression of the form a + b cos 2x, where ab ≠ 0, correctly
3sin 2 x 3 x
State correct integral
+ (+c)
4
2
(ii) State correct expression
1
2
(b) State or imply correct ordinates 5.46143…, 4.78941…, 4.32808…
Use correct formula, or equivalent, correctly with h = 0.5 and three ordinates
Obtain answer 4.84 with no errors seen
5
(a
b)
(b)
M1
A1
A
B1
M1
A1
[3
B1
M1
A1
[3]
Expand brackets and use sin2 x + cos2 x = 1
Obtain 1 – sin 2x
Integrate and obtain term of form ±k cos 2x, where k = ½, 1 or 2
cos 2x
State correct integral x +
(+c)
2
M1
A1
M1
State or imply correct ordinates 1.4142…, 1.0823…, 1
B1
Use correct formula, or equivalent, correctly with h =
π
and three ordinates
8
Obtain answer 0.899 with no errors seen
(ii) Make a recognisable sketch of y = cosec x for 0 < x Y 12 π
Justify statement that the trapezium rule gives an over-estimate
[3]
A1
[4]
M1
A1
[3]
B1
B1
[2]
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6
(a) State or imply correct ordinates 0.125, 0.08743…, 0.21511…
Use correct formula, or equivalent, correctly with h = 0.5 and three ordinates
Obtain answer 0.11 with no errors seen
B1
M1
A1
[3]
M1
A1
A1
A1
[4]
B1
B1
[2]
(ii) State or imply correct ordinates 0.61370..., 0.80277..., 1.22741..., 1.78112...
Use correct formula, or equivalent, correctly with h = 1 and four ordinates
Obtain answer 3.23 with no errors seen
B1
Ml
Al
[3]
(iii) Justify statement that the trapezium rule gives an over-estimate
B1
[1]
(b) Attempt to expand brackets and divide by e2x
Integrate a term of form ke−x or ke−2x correctly
Obtain 2 correct terms
Fully correct integral x + 4e−x - 2e−2x + c
7
(i
Obtain correct derivative
Obtain x = 2 only