AQA MFP3 p QP JUN15

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Question

General Certificate of Education
Advanced Level Examination
June 2015

Mark

1
2
3

Mathematics

MFP3

5

Unit Further Pure 3

Wednesday 13 May 2015

4

6

9.00 am to 10.30 am

7
For this paper you must have:
*
the blue AQA booklet of formulae and statistical tables.
You may use a graphics calculator.

TOTAL

Time allowed
*
1 hour 30 minutes
Instructions
*
Use black ink or black ball-point pen. Pencil should only be used for
drawing.
*
Fill in the boxes at the top of this page.
*
Answer all questions.
*
Write the question part reference (eg (a), (b)(i) etc) in the left-hand
margin.

*
You must answer each question in the space provided for that
question. If you require extra space, use an AQA supplementary
answer book; do not use the space provided for a different question.
*
Do not write outside the box around each page.
*
Show all necessary working; otherwise marks for method may be lost.
*
Do all rough work in this book. Cross through any work that you do
not want to be marked.
Information
*
The marks for questions are shown in brackets.
*
The maximum mark for this paper is 75.
Advice
*
Unless stated otherwise, you may quote formulae, without proof, from
the booklet.
*
You do not necessarily need to use all the space provided.

(JUN15MFP301)

P88829/Jun15/E3

MFP3

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2

Answer all questions.
Answer each question in the space provided for that question.

It is given that yðxÞ satisfies the differential equation

1

dy
¼ f ðx, yÞ
dx
where
and

f ðx, yÞ ¼

x þ y2
x

yð2Þ ¼ 5

Use the Euler formula

(a)

yrþ1 ¼ yr þ hf ðxr , yr Þ
with h ¼ 0:05 , to obtain an approximation to yð2:05Þ.
[2 marks]
Use the formula

(b)

yrþ1 ¼ yrÀ1 þ 2hf ðxr , yr Þ
with your answer to part (a), to obtain an approximation to yð2:1Þ, giving your answer
to three significant figures.
[3 marks]
QUESTION
PART
REFERENCE

Answer space for question 1

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3
QUESTION
PART
REFERENCE

Answer space for question 1

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dy
þ ðtan xÞy ¼ tan3 x sec x
dx
given that y ¼ 2 when x ¼

QUESTION
PART
REFERENCE

p
.
3

[9 marks]

Answer space for question 2

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6
3 (a) (i)

Write down the expansion of lnð1 þ 2xÞ in ascending powers of x up to and including
the term in x 4 .
[1 mark]

(ii) Hence, or otherwise, find the first two non-zero terms in the expansion of

ln½ð1 þ 2xÞð1 À 2xÞ
in ascending powers of x and state the range of values of x for which the expansion is
valid.
[3 marks]
Find xlim
ﬁ0

(b)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !
3x À x 9 þ x
.
ln½ð1 þ 2xÞð1 À 2xÞ
[4 marks]

QUESTION
PART
REFERENCE

Answer space for question 3

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QUESTION
PART
REFERENCE

Answer space for question 3

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x À 2ÞeÀ2x dx is an improper integral.

2

ð1
Evaluate

(b)

À

[1 mark]

x À 2ÞeÀ2x dx , showing the limiting process used.

2

[6 marks]
QUESTION
PART
REFERENCE

Answer space for question 4

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REFERENCE

Answer space for question 4

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10
Find the general solution of the differential equation

5 (a)

d2 y
dy
þ 9y ¼ 36 sin 3x
þ
6
dx 2
dx

[7 marks]

It is given that y ¼ f ðxÞ is the solution of the differential equation

(b)

d2 y
dy
þ 9y ¼ 36 sin 3x
þ
6
dx 2
dx
0

such that f ð0Þ ¼ 0 and f ð0Þ ¼ 0 .
(i)

00

Show that f ð0Þ ¼ 0 .
[1 mark]

(ii) Find the first two non-zero terms in the expansion, in ascending powers of x, of f ðxÞ .
[3 marks]
QUESTION
PART
REFERENCE

Answer space for question 5

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QUESTION
PART
REFERENCE

Answer space for question 5

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14
A differential equation is given by

6

pﬃﬃﬃﬃﬃ d2 y
pﬃﬃﬃ
pﬃﬃﬃ
4 x 5 2 þ ð2 xÞy ¼ xðln xÞ2 þ 5, x > 0
dx
Show that the substitution x ¼ e2t transforms this differential equation into

(a)

d2 y
dy
À 2 þ 2y ¼ 4t 2 þ 5eÀt

2
dt
dt
Hence find the general solution of the differential equation

(b)

pﬃﬃﬃﬃﬃ d2 y
pﬃﬃﬃ
pﬃﬃﬃ
4 x 5 2 þ ð2 xÞy ¼ xðln xÞ2 þ 5, x > 0
dx

QUESTION
PART
REFERENCE

[7 marks]

[10 marks]

Answer space for question 6

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16
QUESTION
PART
REFERENCE

Answer space for question 6

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18
The diagram shows the sketch of a curve C1 .

7

O

Initial line

The polar equation of the curve C1 is

r ¼ 1 þ cos 2y,

p
p
À 4y4
2
2

Find the area of the region bounded by the curve C1 .

(a)

[5 marks]
The curve C2 whose polar equation is

(b)

r ¼ 1 þ sin y,

p
p
À 4y4
2
2

intersects the curve C1 at the pole O and at the point A. The straight line drawn
through A parallel to the initial line intersects C1 again at the point B.
(i)

Find the polar coordinates of A.
[4 marks]

1 pﬃﬃﬃﬃﬃ
13 þ 1 .
(ii) Show that the length of OB is
4
[6 marks]
(iii) Find the length of AB , giving your answer to three significant figures.

[3 marks]
QUESTION
PART
REFERENCE

Answer space for question 7

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QUESTION
PART
REFERENCE

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QUESTION
PART
REFERENCE

Answer space for question 7

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REFERENCE

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END OF QUESTIONS

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