EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) – JUNE 2015 FINAL MARK SCHEME

Question Number Scheme Marks

1. (a) M1

B1,A1,A1

dM1A1

(6)

(b) B1 (1)

[7]

2. (a) B1

M1A1 (3)

(b) M1

A1 cso (2)

(c)

B1

or 120o( )2 3 2arg arctan arctan 3

2 3z π⎛ ⎞−= = − =⎜ ⎟⎜ ⎟

⎝ ⎠

CVs: ! 3, 6, 1− −

!

33 3 344 4 42 24 cos i sin 4 cos i sin

3 3 2 2z π π π π⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

! !( )( )23 5 6 0x x x+ + − = ( )( )( )3 6 1 0x x x+ + − =

! 1x >

or 21264096 or 4=

! OR !( )( ) ( )22 3 12 3 0x x x+ + − + =( )( )

( )3 2 12

03

x xx

+ + −>

+

OR: ! ( ) ( )6, 3 1,x∈ − − ∪ ∞

! 4z =

! oe or any other correct rooti2 2w =

! or !( )

66 62 2

4 cos i sin 4 cos 4 isin 43 3

z π ππ π

⎛ ⎞⎛ ⎞= + = +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

62i6 34ezπ⎛ ⎞

= ⎜ ⎟⎝ ⎠

! 6 3, 1x x− < < − >

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) – JUNE 2015 FINAL MARK SCHEME

M1

A1A1 (4)

[9]

3.

M1

M1A1

dM1A1

B1ft [6]

Question Number Scheme Marks

!sin 3cos 2 sin dy x x x x= ∫

! , IF !cot d ln sinx x x=∫ sin x=

! oe 3 2 2n w= = −

! oe !

33cos 2cos 'sin

x x cy

x− +

=3cos3 3cos '

2sinx x c

yx

− + +=

! oe1 2 2n w= =

! !( )2sin 3 2cos 1 sin dy x x x x= −∫ ( )3sin sin 3 sin d2

y x x x x= −∫

!

33 44 2 24 cos 2 isin 2

3 3n nπ ππ π

⎛ ⎞⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

!d

sin cos 3cos 2 sindy

x y x x xx+ =

!d

3cos 2d tany y

xx x+ =

! oe2 i2 2n w= = −

! !( )32sin 3 cos cos

3y x x x c⎡ ⎤= − + +⎢ ⎥⎣ ⎦

( )3 1sin cos3 cos2 3

y x x x c⎡ ⎤= − + +⎢ ⎥⎣ ⎦

!( )0 see aboven =

! 2

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) – JUNE 2015 FINAL MARK SCHEME

4. (a) M1 A1

A1 (3)

(b) M1

A1

A1

A1cso (4) [7]

Question Number Scheme Marks

!( ) ( )( )2 22 21 1n n n n+ × + − − ×

! ( )22 1n n= +

!( )23 2

1

1 14

n

r n n= +∑

!( )

223

1 1

1 12

n n

r n n r⎛ ⎞⎛ ⎞∴ = + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∑ ∑

So ! *( ) ( )23 3 3 31 2 3 .... 1 2 3...n n+ + + + = + + +

! ( ) ( )2 2 2 22 1 2 1r r r r r r+ + − − +

! ( )4 3 2 4 3 2 2 2 22 2 or 2 1 2 1r r r r r r r r r r r≡ + + − + − + + − + −

! *34r≡

!( ) ( ) ( )3 2 2 2 2 2 2 2 2 2

1

4 1 2 0 2 3 1 2 3 4 2 3 ...n

r⎛ ⎞= × − + × − × + × − ×⎜ ⎟

⎝ ⎠∑

! 3

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) – JUNE 2015 FINAL MARK SCHEME

Question Number Scheme Marks

5. (a)

M1A1

dM1

ddM1A1

(i) dddM1

(ii) A1A1 (8)

(b) Circle drawn on an Argand diagram in correct position ft their centre and radius

B1ft

Region inside correct circle shaded no ftB1 (2)

[10]

!

224 36

5 25u v⎛ ⎞+ + =⎜ ⎟⎝ ⎠

So a circle, Centre ! Radius ! (oe fractions or decimals)

4 ,05

⎛ ⎞−⎜ ⎟⎝ ⎠

65

! 3izw

z=

+

!( ) 3i 3i3i or

1 1w ww z z zw w

−+ = =

− −

!

3i2 21wzw

= =−

! 3i 2 1w w= −

! !iw u v= + ( ) ( )( )22 2 29 4 1u v u v+ = − +

!

! ( )2 2 2 29 9 4 1 2u v u u v+ = − + +

! 2 25 5 8 4 0u v u+ + − =

! 4

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) – JUNE 2015 FINAL MARK SCHEME

Question Number Scheme Marks

6. (a) M1

dM1

A1

ddM1A1

A1 (6)

(b)

M1

M1

dM1A1

A1 (5) [11]

!3 932 2

r a a= × =

!( )( )2cos 1 cos 1 0θ θ− + =

!( )

29 1 33 02 3 4 2 6a π π⎡ ⎤

+ + × + −⎢ ⎥⎣ ⎦

( ) 2 2d sin3 cos 3 cos 3 sin

dr

a a aθ

θ θ θθ

= + −3 cos 3 cos 2a aθ θ+

!( )

230

9 11 2cos cos 2 1 d

2 2a π

θ θ θ⎛ ⎞= + + +⎜ ⎟⎝ ⎠∫

!( )22 23

0

1 1Area d 9 1 cos d

2 2r a

π

θ θ θ= = +∫ ∫

!

! !1

cos2 3

πθ θ= = ( ) need not be seenθ π=

!

229 9 3 9 81 3

2 2 8 4 16a aπ π⎡ ⎤ ⎛ ⎞

+ = +⎜ ⎟⎢ ⎥ ⎜ ⎟⎣ ⎦ ⎝ ⎠

! terms in any order22cos cos 1 0θ θ+ − =

!

2 3

0

9 1 12sin sin 22 2 2a

π

θ θ θ θ⎡ ⎤⎛ ⎞= + + +⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦

! OR !sin 3 sin 3 sin cosr a aθ θ θ θ= +33 sin sin 22

a aθ θ+

!( )

223

0

91 2cos cos d

2a π

θ θ θ= + +∫

! 5

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) – JUNE 2015 FINAL MARK SCHEME

Question Number Scheme Marks

7. (a) B1

M1 A1

A1cso (4)

(b) M1A1

A1cso (3)

(c) B1(both)

M1(attempt both)

M1A1 (4)[11]

!

2 32 32

2 33

3 3 3

d 1 d 1 dtan3 d 2! 3 d 3! 3 d

y y yx y x x x

x x xππ π π

π π π⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + − + − + −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

!

24 2

2

3

d 6 2 4 2 80dyx π

⎛ ⎞= × − × =⎜ ⎟

⎝ ⎠

! ( )2 28sec tan 3sec 1x x x= −

OR !( )2d 2 tan 1 tan

dy

x xx= +

!

33 2

3

d 24sec sec tan 8sec tandy

x x x x xx= −

!

2 31763 8 3 40 33 3 3 3

x x xπ π π⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + − + − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠

! *4 26sec 4secx x= −

!

( )3

23

3

d 8 4 3 3 2 1 352 3dyx π

⎛ ⎞= × × × − =⎜ ⎟

⎝ ⎠

! !( )4 2 22sec 4 sec 1 secx x x= + − ( )2 2 22sec 6 sec 1 secx x x= + −

! !( ) ( )

2

3

3 3yπ = = ( )2

3

d 22 3 8 3d 1yx π

⎛ ⎞ ⎛ ⎞= × =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

! !

24 2 2

2

d 2sec 4 tan secdy

x x xx

= +2

2 2 22

d 2sec 2 3tan secdy

x x xx

= + ×

!2d 2 tan sec

dy

x xx=

! 6

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) – JUNE 2015 FINAL MARK SCHEME

Question Number Scheme Marks

8. (a) B1

M1

M1A1

dM1

A1cso (6)

(b)

M1A1

A1

M1

dM1A1

!

2

2

d d 0d dy yau u= =

! *2

2

d d8 16 2d duy y y uu

− + =

PI: try ! ( or ! different derivatives, c = 0)y au b= + 2y cu au b= + +

!

22

2

d d7 16 2lnd dy y

x x y xx x− + =

(CF =)! ( ) 4e uA Bu+

!d d d ded d d d

uy y u yx u x u

−= × =

!( )

22 2

2

d d de e 7e e 16 2ln ed d d

u u u u uy y y yu u u

− −⎛ ⎞× − + − × + =⎜ ⎟

⎝ ⎠

! oe (decimals must be 0.125 and 0.0625)1 18 16

a b= =

!

2 2 22

2 2 2

d d d d d d de e ed d d d d d d

u u uy u y y u y yx x u u x u u

− − − ⎛ ⎞= − + = − +⎜ ⎟

⎝ ⎠

! !( )24 0m − = 4m =

! ( )0 8 16 2a au b u− + + =

!d d d d 1e e or e or or d d d d

u u ux u x ux xu x u x x

−= = = = =

!2 8 16 0m m− + =

! 7

EDEXCEL FURTHER PURE MATHEMATICS FP2 (6668) – JUNE 2015 FINAL MARK SCHEME

B1ft (7)

(c) B1 (1)

[14]

!( ) 4 1 1ln ln

8 16y A B x x x= + + +

!( ) 4 1 1e

8 16uy A Bu u∴ = + + +

! 8