OCR MEI C3 — Question 5

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicIntegration by Substitution
TypeShow definite integral equals specific value (algebraic/exponential substitution)
DifficultyModerate -0.3 This is a straightforward integration by substitution question with clear guidance (the substitution is given). Students must express x in terms of u, change limits, and integrate a simple rational function that splits into polynomial minus logarithmic form. The algebraic manipulation is routine and the question is slightly easier than average due to the explicit substitution provided.
Spec1.08h Integration by substitution
Using the substitution \(u = 2x + 1\), show that \(\int_0^1 \frac{x}{2x + 1} dx = \frac{1}{4}(2 - \ln 3)\).
Question 5:
5
8 Fig. 8 shows part of the curve y (cid:1) f(x), where
f(x) (cid:1) (ex (cid:3) 1)2 for x (cid:4) 0.
y
O x
Fig. 8
(i) Find f(cid:5)(x), and hence calculate the gradient of the curve y (cid:1) f(x) at the origin and at the point
(ln 2, 1). [5]
The function g(x) is defined by g(x)=ln ( 1+ x ) for x (cid:4) 0.
(ii) Show that f(x) and g(x) are inverse functions. Hence sketch the graph of y (cid:1) g(x).
Write down the gradient of the curvey (cid:1) g(x) at the point (1, ln 2). [5]
(iii) Show that Û Ù ( e x -1 )2 dx = 1e 2x -2e x +x+c.
ı 2
ln2
Hence evaluate Û Ù ( e x -1 )2 dx, giving your answer in an exact form. [5]
ı
0
(iv) Using your answer to part (iii), calculate the area of the region enclosed by the curve
y (cid:1) g(x), the x-axis and the line x (cid:1) 1. [3]
PMT
4753/01
ADVANCED GCE UNIT
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
MONDAY 11 JUNE 2007 Afternoon
Time: 1 hour 30 minutes
Additional materials:
Answer booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions.
• You are permitted to use a graphical calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
ADVICE TO CANDIDATES
• Read each question carefully and make sure you know what you have to do before starting your
answer.
• You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
HN/5 © OCR 2007 [K/100/3632] OCR is an exempt Charity [Turn over
Question 5:
5
8 Fig. 8 shows part of the curve y (cid:1) f(x), where
f(x) (cid:1) (ex (cid:3) 1)2 for x (cid:4) 0.
y
O x
Fig. 8
(i) Find f(cid:5)(x), and hence calculate the gradient of the curve y (cid:1) f(x) at the origin and at the point
(ln 2, 1). [5]
The function g(x) is defined by g(x)=ln ( 1+ x ) for x (cid:4) 0.
(ii) Show that f(x) and g(x) are inverse functions. Hence sketch the graph of y (cid:1) g(x).
Write down the gradient of the curvey (cid:1) g(x) at the point (1, ln 2). [5]
(iii) Show that Û Ù ( e x -1 )2 dx = 1e 2x -2e x +x+c.
ı 2
ln2
Hence evaluate Û Ù ( e x -1 )2 dx, giving your answer in an exact form. [5]
ı
0
(iv) Using your answer to part (iii), calculate the area of the region enclosed by the curve
y (cid:1) g(x), the x-axis and the line x (cid:1) 1. [3]
PMT
4753/01
ADVANCED GCE UNIT
MATHEMATICS (MEI)
Methods for Advanced Mathematics (C3)
MONDAY 11 JUNE 2007 Afternoon
Time: 1 hour 30 minutes
Additional materials:
Answer booklet (8 pages)
Graph paper
MEI Examination Formulae and Tables (MF2)
INSTRUCTIONS TO CANDIDATES
• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions.
• You are permitted to use a graphical calculator in this paper.
• Final answers should be given to a degree of accuracy appropriate to the context.
INFORMATION FOR CANDIDATES
• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
ADVICE TO CANDIDATES
• Read each question carefully and make sure you know what you have to do before starting your
answer.
• You are advised that an answer may receive no marks unless you show sufficient detail of the
working to indicate that a correct method is being used.
This document consists of 4 printed pages.
HN/5 © OCR 2007 [K/100/3632] OCR is an exempt Charity [Turn over
Using the substitution $u = 2x + 1$, show that $\int_0^1 \frac{x}{2x + 1} dx = \frac{1}{4}(2 - \ln 3)$.

\hfill \mbox{\textit{OCR MEI C3  Q5}}
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