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P75683A

F:1/1/1/1/1/



Please check the examination details below before entering your candidate information

Candidate surname Other names

Centre Number Candidate Number

Pearson Edexcel Level 3 GCE

Monday 3 June 2024

Afternoon (Time: 1 hour 30 minutes)

9FM0/02

Paper

reference

Total Marks

Further Mathematics

Advanced

PAPER 2:Core Pure Mathematics 2

You must have:

Mathematical Formulae and Statistical Tables (Green), calculator

Candidates may use any calculator permitted by Pearson regulations.

Calculators must not have the facility for algebraic manipulation,

differentiation and integration, or have retrievable mathematical formulae

stored in them.

Instructions

•

Use

black

ink or ball-point pen.

•

If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).

•

Fill in the boxes

at the top of this page with your name,

centre number and candidate number.

•

Answer

all

questions and ensure that your answers to parts of questions are

clearly labelled.

•

Answer the questions in the spaces provided

–

there may be more space than you need

.

•

You should show sufficient working to make your methods clear.

Answers without working may not gain full credit.

•

Inexact answers should be given to three significant figures unless otherwise stated.

Information

•

A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.

•

There are 9 questions in this question paper. The total mark for this paper is 75.

•

The marks for

each

question are shown in brackets

–

use this as a guide as to how much time to spend on each question

.

Advice

•

Read each question carefully before you start to answer it.

•

Try to answer every question.

•

Check your answers if you have time at the end.

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2





1.

 (a) Using the definition of sinh

x

in terms of exponentials, prove that

4 3 3

3

sinh sinh sinh

x x x

 

(2)

 (b) Hence solve the equation

sinh sinh3 19

x x

=

 giving your answers as simplified natural logarithms where appropriate.

(5)

_____________________________________________________________________________________

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Question 1 continued

_____________________________________________________________________________________

(Total for Question 1 is 7 marks)

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4





2.

 f(

x

) =

tanh

 







 





1 3

6

3

2

x

x x

 <

 (a) Show that

f′(

x

) = –

1

2 3

x

+

(4)

 (b) Hence determine f′′(

x

)

(1)

 (c) Hence show that the Maclaurin series for f(

x

), up to and including the term in

x

2, is

ln

p qx rx

+ +

 2

 where

p

,

q

and

r

are constants to be determined.

(3)

_____________________________________________________________________________________

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Question 2 continued

_____________________________________________________________________________________

(Total for Question 2 is 8 marks)

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3.

 (a) Explain why

1

9 16

2

4

3

x x







 d

 is an improper integral.

(1)

 (b) Show that

2

4

3

1d

9 16

x kπ

x









 where

k

is a constant to be determined.

(4)

_____________________________________________________________________________________

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7

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Question 3 continued

_____________________________________________________________________________________

(Total for Question 3 is 5 marks)

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4.

 Use the method of differences to show that

2

4 6 30 5 6

1

r r

n an b

n n

r

n



 

 

 

  

 



 

 

 





 where

a

and

b

are integers to be determined.

(6)

_____________________________________________________________________________________

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9

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Question 4 continued

_____________________________________________________________________________________

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A

10





Question 4 continued

_____________________________________________________________________________________

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Turn over

11



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 4 continued

_____________________________________________________________________________________

(Total for Question 4 is 6 marks)

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D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

12





5.

 The locus

C

is given by

z

  

4 4

 The locus

D

is given by

arg 3



π

z

 (a) Sketch, on the same Argand diagram, the locus

C

and the locus

D

(4)

 The set of points

A

is defined by

 

: 4 4 : 0 arg 3

 

    

 

 

 

 π

A z z z z

  

 (b) Show, by shading on your Argand diagram, the set of points

A

(1)

 (c) Find the area of the region defined by

A

, giving your answer in the form

3



pπ q



where

p

and

q

are constants to be determined.

(4)

_____________________________________________________________________________________

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Turn over

13



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 5 continued

_____________________________________________________________________________________

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D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

14





Question 5 continued

_____________________________________________________________________________________

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Turn over

15



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 5 continued

_____________________________________________________________________________________

(Total for Question 5 is 9 marks)

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󰀪󰁐󰀷󰀵󰀶󰀸󰀳󰁁󰀰󰀱󰀶󰀳󰀲󰀪

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

16





6.

 The motion of a particle

P

along the

x

-axis is modelled by the differential equation

2 5 2 4 12

2

d

x

t

x

t

x t

   

 where

P

is

x

metres from the origin

O

at time

t

seconds,

t



0

 (a) Determine the general solution of the differential equation.

(6)

 (b) Hence determine the particular solution for which

x

= 3 and

d

x

t

= –2 when

t

= 0

(3)

 (c) (i) Show that, according to the model, the minimum distance between

O

and

P



is (2 + ln 2) metres.

 (ii) Justify that this distance is a minimum.

(4)

 For large values of

t

the particle is expected to move with constant speed.

 (d) Comment on the suitability of the model in light of this information.

(1)

_____________________________________________________________________________________

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Turn over

17



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 6 continued

_____________________________________________________________________________________

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

󰀪󰁐󰀷󰀵󰀶󰀸󰀳󰁁󰀰󰀱󰀸󰀳󰀲󰀪

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

18





Question 6 continued

_____________________________________________________________________________________

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Turn over

19



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 6 continued

_____________________________________________________________________________________

(Total for Question 6 is 14 marks)

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

󰀪󰁐󰀷󰀵󰀶󰀸󰀳󰁁󰀰󰀲󰀰󰀳󰀲󰀪

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

20





7.

 (a) Determine the roots of the equation

z

6= 1

 giving your answers in the form ei

θ

where 0





θ



<

2

π

(2)

 (b) Show the roots of the equation in part (a) on a single Argand diagram.

(2)

 (c) Show that

3 64

6



 

  

i

(2)

 (d) Hence, or otherwise, solve the equation

z

6+ 64 = 0

 giving your answers in the form

r

ei

θ

where 0





θ



<

2

π

(3)

_____________________________________________________________________________________

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Turn over

21



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 7 continued

_____________________________________________________________________________________

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

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D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

22





Question 7 continued

_____________________________________________________________________________________

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Turn over

23



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 7 continued

_____________________________________________________________________________________

(Total for Question 7 is 9 marks)

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D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

24





8.



A



















3 1 1

1 1 1

3 6

0

k



(a) Find, in terms of

k

,

A

–1

(4)

 (b) Determine, in simplest form in terms of

k

, the coordinates of the point where the

following planes intersect.

3

x

+

y

–

z

= 3

 x

+

y

+

z

= 1

kx

+ 3

y

+ 6

z

= 6

(3)

_____________________________________________________________________________________

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Turn over

25



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 8 continued

_____________________________________________________________________________________

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󰀪󰁐󰀷󰀵󰀶󰀸󰀳󰁁󰀰󰀲󰀶󰀳󰀲󰀪

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

26





Question 8 continued

_____________________________________________________________________________________

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Turn over

27



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

Question 8 continued

_____________________________________________________________________________________

(Total for Question 8 is 7 marks)

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D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

28





9.



A

4 cm

4.5 cm

DE

F

B

C

DO

C

x

y

7 cm



Figure 1 Figure 2

 Figure 1 shows the central vertical cross-section

ABCDEFA

of a vase together with

measurements that have been taken from the vase.

 The horizontal cross-section between

AB

and

FC

is a circle with diameter 4 cm.

 The base of the vase

ED

is horizontal and the point

E

is vertically below

F

and the point



D

is vertically below

C

.

 Using these measurements, the curve

CD

is modelled by the parametric equations

3sin 2 cos 0 2

  

π

x a t y b t t

 

 where

a

and

b

are constants and

O

is the fixed origin, as shown in Figure 2.

 (a) Determine the value of

a

and the value of

b

according to the model.

(2)

 (b) Using algebraic integration and showing all your working, determine, according to

the model, the volume of the vase, giving your answer to the nearest cm3

(7)

 (c) State a limitation of the model.

(1)

_____________________________________________________________________________________

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Turn over

29



D

O

N

O

T

W

R

I

T

E

I

N

T

H

I

S

A

R

E

A

D

O

N

O

T

W

R

I

T

E

I

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Question 9 continued

_____________________________________________________________________________________

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Question 9 continued

_____________________________________________________________________________________

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Turn over

31

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Question 9 continued

_____________________________________________________________________________________

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32

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Question 9 continued

_____________________________________________________________________________________

(Total for Question 9 is 10 marks)

TOTAL FOR PAPER IS 75 MARKS

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s

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4

.

5

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935

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