AQA MFP3 p QP JUN15

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(JUN15MFP301)

Answer all questions.

Answer each question in the space provided for that question

1 It is given thatyðxÞsatisfies the differential equation

dy

dx ¼ f ðx, yÞ

where f ðx, yÞ ¼ x þ y

2

x

(a) Use the Euler formula

yrþ1 ¼ yrþ hf ðxr, yrÞ with h ¼ 0:05, to obtain an approximation toyð2:05Þ

[2 marks]

(b) Use the formula

yrþ1 ¼ yr1þ 2hf ðxr, yrÞ with your answer to part (a), to obtain an approximation toyð2:1Þ, giving your answer

to three significant figures

[3 marks]

Answer space for question 1

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3 Answer space for question 1

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2 By using an integrating factor, find the solution of the differential equation

dy

dx þ ðtan xÞy ¼ tan3x sec x

given that y ¼ 2 when x ¼ p

Answer space for question 2

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5 Answer space for question 2

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3 (a) (i) Write down the expansion of lnð1 þ 2xÞ in ascending powers ofx up to and including

the term inx4

[1 mark]

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

ln½ð1 þ 2xÞð1  2xÞ

in ascending powers ofxand state the range of values of xfor which the expansion is

valid

[3 marks]

(b) Find lim

x fi 0

3x  x ffiffiffiffiffiffiffiffiffiffiffi

9 þ x p ln½ð1 þ 2xÞð1  2xÞ

[4 marks]

Answer space for question 3

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4 (a) Explain why

ð1

2

x  2Þe2xdx



is an improper integral

[1 mark]

(b) Evaluate

ð1

2

x  2Þe2xdx



, showing the limiting process used

[6 marks]

Answer space for question 4

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5 (a) Find the general solution of the differential equation

d2y

dx2þ 6 dy

dx þ 9y ¼ 36 sin 3x

[7 marks]

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

d2y

dx2þ 6 dy

dx þ 9y ¼ 36 sin 3x such that f ð0Þ ¼ 0 and f0ð0Þ ¼ 0

(i) Show that f0 0ð0Þ ¼ 0

[1 mark]

(ii) Find the first two non-zero terms in the expansion, in ascending powers ofx, off ðxÞ

[3 marks]

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6 A differential equation is given by

4 ffiffiffiffiffi

x5

p d2y

dx2þ ð2 ffiffiffi

x

p

Þy ¼ ffiffiffi

x

p ðln xÞ2þ 5, x > 0

(a) Show that the substitution x ¼ e2t transforms this differential equation into

d2y

dt2 2 dy

dt þ 2y ¼ 4t2þ 5et

[7 marks]

(b) Hence find the general solution of the differential equation

4 ffiffiffiffiffi

x5

p d2y

dx2þ ð2 ffiffiffi

x

p

Þy ¼ ffiffiffi

x

p ðln xÞ2þ 5, x > 0

[10 marks]

Answer space for question 6

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15 Answer space for question 6

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7 The diagram shows the sketch of a curveC1.

The polar equation of the curveC1 is

r ¼ 1 þ cos 2y,  p

2 4 y 4 p

2 (a) Find the area of the region bounded by the curveC1

[5 marks]

(b) The curveC2 whose polar equation is

r ¼ 1 þ sin y,  p

2 4 y 4 p

2 intersects the curveC1 at the poleO and at the pointA The straight line drawn

throughA parallel to the initial line intersectsC1 again at the point B

(i) Find the polar coordinates ofA

[4 marks]

(ii) Show that the length ofOB is 1

4

 ffiffiffiffiffi 13

p

þ 1



[6 marks]

(iii) Find the length ofAB, giving your answer to three significant figures

[3 marks]

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O

Initial line

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21 Answer space for question 7

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