AQA MFP2 p QP JUN15

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Question

General Certificate of Education
Advanced Level Examination
June 2015

Mark

1
2
3

Mathematics

MFP2

5

Unit Further Pure 2

Tuesday 16 June 2015

4

6

1.30 pm to 3.00 pm

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

8
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.

(JUN15MFP201)

P88824/Jun15/E4

MFP2

Do not write
outside the
box

2

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

Express

1 (a)

1
A
B
þ
in the form
, where A and B are integers.
ðr þ 2Þr!
ðr þ 1Þ! ðr þ 2Þ!
[3 marks]

Hence find

(b)

QUESTION
PART

REFERENCE

n
X

1
.
ðr þ 2Þr!
r¼1

[2 marks]

Answer space for question 1

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4
Sketch the graph of y ¼ tanh x and state the equations of its asymptotes.

2 (a)

[3 marks]
Use the definitions of sinh x and cosh x in terms of e x and eÀx to show that

(b)

sech2 x þ tanh2 x ¼ 1
[3 marks]
Solve the equation 6 sech2 x ¼ 4 þ tanh x , giving your answers in terms of natural
logarithms.
[5 marks]

(c)

QUESTION
PART
REFERENCE

Answer space for question 2

(a)

y~

~

O

x

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(04)

P/Jun15/MFP2

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5
QUESTION
PART
REFERENCE

Answer space for question 2

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s

(05)

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P/Jun15/MFP2

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box

6
A curve C is defined parametrically by

3

t2 þ 1
, y ¼ 2 ln t
t

2 2
dx
dy
1 2
Show that
þ
¼ 1þ 2 .
t
dt
dt

x¼

(a)

[4 marks]
The arc of C from t ¼ 1 to t ¼ 2 is rotated through 2p radians about the x-axis.
Find the area of the surface generated, giving your answer in the form pðm ln 2 þ nÞ ,
where m and n are integers.
[5 marks]

(b)

QUESTION
PART
REFERENCE

Answer space for question 3

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8
The expression f ðnÞ is given by f ðnÞ ¼ 24nþ3 þ 33nþ1 .

4
(a)

Show that f ðk þ 1Þ À 16f ðkÞ can be expressed in the form A Â 33k , where A is an
integer.
[3 marks]

(b)

Prove by induction that f ðnÞ is a multiple of 11 for all integers n 5 1 .
[4 marks]

QUESTION
PART
REFERENCE

Answer space for question 4

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REFERENCE

Answer space for question 4

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s

(09)

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10
The locus of points, L , satisfies the equation

5

jz À 2 þ 4ij ¼ jzj
Sketch L on the Argand diagram below.

(a)

[3 marks]
The locus L cuts the real axis at A and the imaginary axis at B.

(b)
(i)

Show that the complex number represented by C , the midpoint of AB, is

5 5
À i
2 4

[4 marks]

(ii) The point O is the origin of the Argand diagram. Find the equation of the circle that
passes through the points O, A and B, giving your answer in the form jz À aj ¼ k .
[2 marks]
QUESTION
PART
REFERENCE

Answer space for question 5

(a)

Im(z)

~

5–
–
–
–
–

5

~

–

–

–

–

O

–

–

–

–

–

–
À5

Re(z)

–
–

–
–
À5 –

(10)

P/Jun15/MFP2

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11
QUESTION
PART
REFERENCE

Answer space for question 5

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12

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
x
À
2
À1
Given that y ¼ ðx À 2Þ 5 þ 4x À x 2 þ 9 sin
, show that
3

6 (a)

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
dy
¼ k 5 þ 4x À x 2
dx
where k is an integer.
[5 marks]
Hence show that

(b)

ð 7 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
pﬃﬃﬃ
2
5 þ 4x À x 2 dx ¼ p 3 þ qp
2

where p and q are rational numbers.
[3 marks]
QUESTION
PART
REFERENCE

Answer space for question 6

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(12)

P/Jun15/MFP2

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13
QUESTION
PART
REFERENCE

Answer space for question 6

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(a)

[2 marks]
(b) (i)

In the case where b ¼ g , find the roots of the equation.
[5 marks]

(ii) Find the value of k in this case.
[1 mark]
(c) (i)

In the case where a ¼ 1 À i , find a 2 and a 3 .
[2 marks]

(ii) Hence find the value of k in this case.
[2 marks]
In the case where k ¼ À12 , find a cubic equation with integer coefficients which has

(d)

roots

1
1
1
þ 1 , þ 1 and þ 1 .
a
b

g
[5 marks]

QUESTION
PART
REFERENCE

Answer space for question 7

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(14)

P/Jun15/MFP2

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15
QUESTION
PART
REFERENCE

Answer space for question 7

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QUESTION
PART
REFERENCE

Answer space for question 7

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2p
þ i sin .
5
5

Verify that o is a root of the equation z5 ¼ 1 .
[1 mark]

(ii) Write down the three other non-real roots of z5 ¼ 1 , in terms of o .
[1 mark]
(b) (i)

Show that 1 þ o þ o 2 þ o 3 þ o 4 ¼ 0 .
[1 mark]

(ii) Hence show that

1
oþ
o

2

1
À 1 ¼ 0.
þ oþ
o
[2 marks]

2p
¼
Hence show that cos
5

(c)

pﬃﬃﬃ
5À1
.
4
[4 marks]

QUESTION
PART
REFERENCE

Answer space for question 8

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(19)

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P/Jun15/MFP2

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20
QUESTION
PART
REFERENCE

Answer space for question 8

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AQA MFP2 p QP JUN15

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