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1. 82

422

2

−++

=xx

xxy

xxyxyyx 4282 22 +=−+ 08)42()2( 2 =−−+− yxyxy B1

For no real values of x; 042 <− acb 0)8)(2(4)42( 2 <−−−− yyy M1 0643216164 22 <−++− yyyy 04209 2 <+− yy 0)2)(29( <−− yy M1

292

<< y A1

Therefore for all real values of x, the function of 82

422

2

−++

=xx

xxy does not have values between

92 and 2. A1 [5]

2. x

x−110 = 3

x

x−110 = 9 M1

x10 = x99 − x9 = x109 − x81 = 210018081 xx +− M1 81261100 2 +− xx = 0 A1 )925)(94( −− xx = 0 M1

259

49 orx = A1

From the equation 31

10=

− xx ,

49

≠x259

=x A1 [6]

3. ')( BA∪ ⇒ })(:{ BAxx ∪∉ ⇒ }:{ BxandAxx ∉∉ B1 ⇒ }:{ '' BxandAxx ∈∈ B1 ⇒ })(:{ '' BAxx ∩∈ B1

∴ ''')( BABA ∩=∪ B1 )()( ' BABA −∪∪ = )()( ''' BABA ∩∪∩ B1 = '' )( BAA ∩∪ B1 = 'B∩ξ = 'B B1 [7]

9

2 2

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4. dxdyyx 22 secsec = M1

yx

dxdy

2

2

secsec

= A1

0tan,41

=π= yx B1

yδ ≈ xdxdy

δ⋅

xyx

δ⋅= 2

2

secsec M1

xyx

δ⋅

++

= 2

2

tan1tan1 M1

xx δ=δ⋅

++

= 20111 A1

yyy δ+≈ 0 xyy δ+≈∴ 20 A1 [7] 5. θ= sinx

θθ= ddx cos

dxx

x∫

−

21

0 2

2

1= θθ∫

θ−

θπ

dcossin1

sin4

0 2

2

M1 B1 ( correct limits)

= θθ

π

d24

0

sin∫ A1

= θθπ

d∫−4

02

2cos1 M1

= 4

022sin

21

πθθ

− A1

41

8−=

π M1 (substituting) A1 [7]

6. The distance from )6,5(−A to line 217 =+ yx -----(1)

d = 22 17

216)5(7

+

−+− M1

= 5050−

= 50−

25= unit A1

The equation of the circle is 222 )50()6()5( =−++ yx M1

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5036122510 22 =+−+++ yyxx 011121022 =+−++ yxyx A1

Gradient of line 7x + y =21 is -7

Therefore, gradient of line perpendicular to this line is 71

Equation of line perpendicular to this line is

)5(716 +=− xy M1

5427 +=− xy 477 += xy 477 −= yx ------(2)

Substituting (2) into (1) ,

21)477(7 =+− yy M1

2132949 =+− yy 35050 =y

7=y From (2), 247)7(7 =−=x

Therefore, the point of contact is (2 , 7 ) A1 [7]

7. (i) Inverse function 1−f exists because f is one to one function. B1

(ii) 0,ln2)( >= xxxf Let xy ln2=

xy ln2=

xey

=2 M1

ℜ∈− xexfx

;: 21 A1

1

1 2 3 x

y 2

e

0,ln2)( >= xxxf D1 ( shape ) D1 ( all correct )

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(iii) )(1 xgf − 2x

e= M1

ℜ∈− xexgfx

;: 41 A1

Range { }+ℜ∈yy : A1 [8] 8. 0))(( =β−α− xx M1 0)(2 =αβ+β+α− xx … ( 1 ) A1

02 =++ cbxax

02 =++acx

abx … ( 2 ) M1

Compare ( 1 ) & ( 2 ) ; ab

−=β+α , ac

=αβ A1

0)2( 2222 =+−− cyacbya

0)2(2

2

2

22 =+

−−

acy

aacby … ( 3 ) B1

From above ab

−=β+α , ac

=αβ

From ( 3 ) , sum of roots : 2

2 )2(a

acb − =

−

ac

ab 2

2

M1

= αβ−β+α− 2)]([ 2 = αβ−β+αβ+α 22 22 = 22 β+α A1

Product of roots : ( ) 2222

2

βα=αβ=ac M1

The roots are 2α and 2β . A1 [9]

9. 01

1366)(' =

−−=

xxxf M1

01

36)1(6=

−−−

xxx

03666 2 =−− xx 062 =−− xx A1

0)2)(3( =+− xx M1 3,2−=x 32 =∴≥ xx A1 [4]

13666)('

2

−−−

=x

xxxf B1

2

2

)1()1)(3666()612)(1()(''

−−−−−−

=x

xxxxxf M1

2

22

)1(366661812

−++−+−

=x

xxxx

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2

2

)1(42126

−+−

=x

xx A1

2

2

)1()72(6

−+−

=x

xx

2

2

)1(]6)1[(6

−+−

=x

x , 1≠x M1

2)1(366−

+=x

A1

Range of ]42,6(,)('' xf B1 [6] 10. a) X is singular ; 0=X ( ) ( ) ( ) 000100 22 =−−+− ba M1 022 =− ba ( )( ) 0=−+ baba M1 ba ±=∴ . A1 [3]

b) AB=

−

−

541213

421

−−−

−

761110917

8613 B1

=

−−

−

300030003

A1

−

−−=−=−

541213

421

31

311 AB B1

138613 −=−+ rqp

2010917 =+−− rqp 147611 −=−+ rqp

−

−=

−−−

−

142013

761110917

8613

rqp

B1

−

−

−

−−=

∴

142013

541213

421

31

rqp

M1

−

−−=

393

31

=

−13

1 A1

∴ The solutions are 1,3,1 =−== rqp . A1 [7]

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11. ( )( )2

2

212141710

xxxx

−++−

( )221212 xC

xB

xA

−+

−+

+≡

( ) ( )( ) ( )xCxxBxAxx ++−++−≡+− 221221141710 22 Substituting 42 =−= Ax M1

Substituting 221

== Cx

Substituting 10 == Bx A1 (any 2 correct)

( )( )2

2

212141710

xxxx

−+

+−

( )2212

211

24

xxx −+

−+

+≡ A1

= ( ) 124 −+ x

1

21

214

−

+

=

x

( ) ( )( )

+

−−

+

−+= ...

2!221

2112

2xx M1

...2

22

++−=xx A1

expansion is valid for 2<x

( ) ( )( ) ( )( ) ( ) ...2!2

2121121 21 +−−−

+−−+=− − xxx M1

...421 2 +++= xx A1

( ) ( )( ) ( )( ) ( )

+−

−−+−−+=− − ...2

!2322212212 22 xxx

...2482 2 +++= xx A1

Both expansions are valid for 21

<x

( )( )2

2

212141710

xxxx

−+

+− = ++−2

22xx 2421 xx ++ ...2482 2 ++++ xx

2

25795 xx ++= A1

Overall, expansion is valid for

−∈

21,

21x B1 [10]

x+24

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12. Asymptotes 1,2 =−= yx B1 [3]

i) 41

=h B1

B1 (all correct values)

Area

++++

≈

113

51

912

310

41

21 M1

3960743

=

1876.0= A1

ii) Area = ∫1

0

ydx

dxx

x∫ +

=1

02

dxx∫

+−=

1

0 221 M1

( )[ ]102ln2 xx +−= A1 ( ) ( )[ ]2ln203ln21 −−−=

32ln21+=

= 0.1891 A1

x y 0 0

41

91

21

51

43

113

1 31

1

1 -2

D1 (shape) D1 (all correct)

y

x

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% error = 1001891.0

1876.01891.0×

−

= 0.793 % B1 [8]

Volume dxy∫=1

0

2π

dxx

x 21

0 2∫

+π=

dxx

21

0 221∫

+−π= M1

( )

dxxx∫

++

+−=

1

022

42

41π

( ) ( )

1

0242ln4

+

−+−π=x

xx A1

= 0.0448π or 0.141 A1 [3]