AQA MFP2 p QP JUN15

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 P/Jun15/MFP2 Turn overs 03 QUESTIO

Centre Number Candidate Number

Surname

Other Names

Candidate Signature

General Certificate of Education Advanced Level Examination June 2015

Unit Further Pure 2

Tuesday 16 June 2015 1.30 pm to 3.00 pm

For this paper you must have:

*

the blue AQA booklet of formulae and statistical tables.

You may use a graphics calculator.

Time allowed

Instructions

drawing

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Write the question part reference (eg (a), (b)(i) etc) in the left-hand

margin

question If you require extra space, use an AQA supplementary

answer book; do not use the space provided for a different question

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Do all rough work in this book Cross through any work that you do

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Information

Advice

the booklet

For Examiner’s Use

Examiner’s Initials

1 2 3 4 5 6 7 8 TOTAL

(JUN15MFP201)

Answer all questions.

Answer each question in the space provided for that question

1 (a) Express 1

ðr þ 2Þr! in the form

A

ðr þ 1Þ! þ

B

ðr þ 2Þ!, whereA andB are integers.

[3 marks]

(b) Hence find Xn

r¼1

1

ðr þ 2Þr!.

[2 marks]

Answer space for question 1

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3 Answer space for question 1

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~

y

x O

2 (a) Sketch the graph of y ¼ tanh x and state the equations of its asymptotes

[3 marks]

(b) Use the definitions of sinh xand cosh x in terms ofex and ex to show that

sech2x þ tanh2x ¼ 1

[3 marks]

(c) Solve the equation 6 sech2x ¼ 4 þ tanh x, giving your answers in terms of natural

logarithms

[5 marks]

Answer space for question 2

(a)

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5 Answer space for question 2

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3 A curveC is defined parametrically by

x ¼ t

2þ 1

t , y ¼ 2 ln t

(a) Show that dx

dt

 2

þ dy dt

 2

¼ 1 þ 1

t2

[4 marks]

(b) The arc ofC from t ¼ 1 to t ¼ 2 is rotated through2p radians about thex-axis

Find the area of the surface generated, giving your answer in the form pðm ln 2 þ nÞ,

wherem and nare integers

[5 marks]

Answer space for question 3

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7 Answer space for question 3

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4 The expressionf ðnÞis given by f ðnÞ ¼ 24nþ3þ 33nþ1.

(a) Show that f ðk þ 1Þ  16f ðkÞ can be expressed in the form A  33k, whereAis an

integer

[3 marks]

(b) Prove by induction thatf ðnÞ is a multiple of11 for all integers n 5 1

[4 marks]

Answer space for question 4

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9 Answer space for question 4

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Im(z)

O

– – – – –

– – – – –

Re(z)

5

5

5 The locus of points,L, satisfies the equation

jz  2 þ 4ij ¼ jzj (a) SketchL on the Argand diagram below

[3 marks]

(b) The locusLcuts the real axis at Aand the imaginary axis at B

(i) Show that the complex number represented byC, the midpoint of AB, is

5

2  5

4 i

[4 marks]

(ii) The pointO is the origin of the Argand diagram Find the equation of the circle that

passes through the pointsO,A andB, giving your answer in the form jz  aj ¼ k

[2 marks]

Answer space for question 5

(a)

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11 Answer space for question 5

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6 (a) Given that y ¼ ðx  2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

5 þ 4x  x2

p

þ 9 sin1 x  2

3

, show that

dy

dx ¼ k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

5 þ 4x  x2

p

wherek is an integer

[5 marks]

(b) Hence show that

ð7 2

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

5 þ 4x  x2

p

dx ¼ p ffiffiffi

3

p

þ qp

wherep and qare rational numbers

[3 marks]

Answer space for question 6

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13 Answer space for question 6

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7 The cubic equation 27z3þ kz2þ 4 ¼ 0 has roots a,b and g.

(a) Write down the values of ab þ bg þ ga and abg

[2 marks]

(b) (i) In the case where b ¼ g, find the roots of the equation

[5 marks]

(ii) Find the value ofk in this case

[1 mark]

(c) (i) In the case where a ¼ 1  i, find a2 anda3

[2 marks]

(ii) Hence find the value ofk in this case

[2 marks]

(d) In the case where k ¼ 12, find a cubic equation with integer coefficients which has

roots 1

a þ 1, 1

b þ 1 and 1

g þ 1

[5 marks]

Answer space for question 7

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15 Answer space for question 7

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17 Answer space for question 7

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8 The complex numbero is given by o ¼ cos 2p

5 þ i sin 2p

5 . (a) (i) Verify thato is a root of the equation z5¼ 1

[1 mark]

(ii) Write down the three other non-real roots of z5¼ 1, in terms of o

[1 mark]

(b) (i) Show that 1 þ o þ o2þ o3þ o4¼ 0

[1 mark]

(ii) Hence show that o þ 1

o

þ o þ 1

o

 1 ¼ 0

[2 marks]

(c) Hence show that cos 2p

5 ¼

ffiffiffi 5

p

 1

[4 marks]

Answer space for question 8

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19 Answer space for question 8

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(19)

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Answer space for question 8

END OF QUESTIONS

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