7362_01_rms_20110309

Mark Scheme (Results)

January 2011

O Level

GCE O Level Pure Mathematics (7362/01)

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January 2011

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GCE AO Level Pure Mathematics (7362) Paper 1 January 2011

1

Pure Mathematics 7362, Mark Scheme January 2011 Paper 1 Question Number Scheme Marks

1 2

2

cos3d 2 cos3 3 sin 3d

y x xy x x x xx

=

= −

M1 A1 A1

(3) 2 sin 52 sin

4.6 5.75.7sin 52sin

4.677.5

B

B

B

=

=

=

M1 A1 A1 (3)

3

( )( )

2 2

2 2

2 2

8 2 3 4 2( 8) 3( 8) 42 2 12 0 2 30 100 0

6 0 15 50 03 2 0 ( 5)( 10) 03 5

2 10

x x x or y y yx x or y y

x x or y yx x or y y

x yx y

+ = + − = − + − −

+ − = − + =

+ − = − + =

+ − = − − =

= − == =

M1 A1 M1A1 A1 (5)

4 (a) (i) 1y = − (ii) 3x = (b) (i) ( )0 2 3 5 (or 5,0 )y x x= = − =

(ii) ( )2 2 23 3 30 1 1 (or 0, 1 )x y −= = − = − −

x-3 -2 -1 1 2 3 4 5 6 7

y

-5

-4

-3

-2

-1

1

2

3

4

(5, 0)

(0, -5/3)

(c)

B1B1 B1 B1 B1 curve B1 asymptotes B1 intercepts

(7)


2

Question Number Scheme Marks

5 ( )( )( )( )

( )( )

2

2

e 2 ed(a) d 2

e 2 1 2 1

22*

kx kx

kx

k xyx x

kx k y kx kxx

+ −=

+

+ − + −= =

++

alternative

( )( ) ( )

( )( )( ) ( )

2 1

2

d 2 2d

( 2 1)1 2 22

*

kx kx

kx

y e x ke xx

e y kx kk xxx

− −= − + + +

+ −= − + + =

++

( )12

2 1d 1 5(b) 0 d 2 2 4

3 *

kyx yx

k

−= = = × =

=

( )

d 5 4d 4 5

1 42 5

(c) 0 grad normal 0 10 5 8 (oe)

yxx

y xy x

= = = −

− = − −

− = −

M1A1A1 M1A1 M1A1 A1 B1 M1 A1 (11)

6

2 0 6 2(a) gradient 16 2 4 0 4 0

2 oe 1 2

y x or

y x y x

− − −= = =

− − −− = = +

( ) ( )(b) is 0, 2 2 f 0 *A r⇒ = = 3 2(c) : 6 4 4 4 2

1 16 4 4 15 : 1 3 3 1 2 6 3 9 3 3

B p qp q p q

D x y p qp q

p p q

= − − += − − + == = − − = − − +

+ == = =

( ) ( ){ }( )

4 2 4 2

4 3 2

0

4 3 2

0

4 4 43 2 3 34 2 4 20 0 0

(d) 2 3 3 2 d

3 4 d

2 2 2

64 64 32 32 8 8 (64 64 24 8) 32

x x x x

x x x x x

x x x x

x x or x x x

or

+ − − − +

= − + +

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= − + + + − − − +⎣ ⎦ ⎣ ⎦ ⎣ ⎦= − + + = + − − − + =

∫∫

M1 A1 B1 M1 A1 M1A1 M1 M1A1 M1A1 (12)


3


7

( )( )

5 5

3

3

55

5

2 2 2 25

2

(a) 16807 16807 7(b) 7 1 6

6 1 317

log 5 12 12log5 3log(c) 12log 5 3 3log log 5 log log log5

4 log 5 1 12 3(log ) 12(log5) 3(log )

log 5 0.25 4 (lo

pp

p

p

m mn

n

por p orp p

or p or p

or

= = =

− =

+= =

= = =

= = =

= =

( ) ( ) ( )

1 12 2

2 2 25

5

2 2

125

4 4 4 4 4 4 4

4 4

g ) 4(log5) (log )

log 5 0.5 2 log 2log5 log

5 or 5 5 or 5 25 or (d) log 3 log 2 log 3 log 4 log 3 log 8 log 3 4log 3 log 16

p

p or p

or p or p

p p or p pp p

− −

=

= ± ± = ± =

= = = == =

+ + + + + +

= + +

( )

1 14 4 4 42 2

4 4 43 4

4 4 4 4 4

4 4 4 4 435184

4 4 481

1 log 3 log 3 1 log 3 1 log 3 4log 3 2log 4 1 3 4log 3

log (3 6 12 24) log (4 3 ) 3log 4 4log 3 3 4log 3 log 3 log 6 log 12 log 24 4log 3

log log 4 3log 4 3(e) l

*or

oror

+ + + + + +

= + + = +

× × × = × = + = ++ + + −

= = = =

4 4 4

4 4 4 44 4 4 4

4 4

4 4 44

4 4

3 44 4 4

og 3log 4log 3 4log 4log 3 log log 3 3

log log 3 3 3 log (3 6 12 24) 3 log 3log

log (5184) 3 log5184 5184 51 log 3 4

x xx x x

or x x xor x x

x

xx x

+ == = =

= = =× × × = + +

= +

= ⇒ = ⇒ =84 81 3

64x= ⇒ =

M1A1 M1 A1 M1 A1 M1 A1

M1 M1A1 M1

M1A1

(14)


4


8

( ) ( )( )

( )( )

2 2

2 2 2 12

2 2 2 2 21 12 2

2 2 2 12

1 12 2

(a) cos 2 cos sin

(i) cos 2 1 sin sin sin 1 cos 2

1 cos 2 ((cos sin ) (cos sin )) sin

(ii) cos 2 cos 1 cos cos cos 2 1

cos 2 1 ((co

*

*or

or

θ θ θ

θ θ θ θ θ

θ θ θ θ θ θ

θ θ θ θ θ

θ

= −

= − − = −

− = + − − =

= − + = +

+ =

( ) ( )( )

( )

2 2 2 2 2

4 2

21 12 2

2

12

2

s sin ) (cos sin )) cos

(b) 8sin 4sin 5 8 ( 1 cos 2 ) 4 1 cos 2 5

2 1 2cos 2 cos 2 2 2cos 2 5

1 6cos 2 2 cos 4 1 cos 4 6cos 2

cos4 6cos 2 (2cos 2 1)

*or

θ θ θ θ θ

θ θ

θ θ

θ θ θ

θ θ θ θ

θ θ θ

− + + =

+ −

= × − + × − −

= − + + − −

= − − + × + = −

− = −

( )

2

2 2 2

2 4 2

4 2

4 2

12

6(1 2sin )2(1 2sin ) 1 6 12sin2(1 4sin 4sin ) 7 12sin8sin 4sin 5

(c) 4sin 2sin 3cos 2 2.4 cos 4 6cos 2 5 3cos 2 2.4 cos 4 0.2 4 1.772, 4.511

*

θ

θ θ

θ θ θ

θ θ

θ θ θθ θ θ

θ θ

− −

= − − − +

= − + − +

= + −

+ + =

− + + =

= − =

( )4

8

4 2 2

4 2 2

14

0.443, 1.13 4sin 2sin 3(1 2sin ) 2.4

4sin 4sin 0.6 0, sin 0.184 or 0.816 sin 0.429 or 0.903 0.443, 1.13

(d) 4 cos 4 6cos 2 d 4 sin 4 3si

or

π

π

θ

θ θ θ

θ θ θθ θ

θ θ θ θ

=

+ + − =

− + = == ± ± =

− = −∫ [ ]

( )( )( )

4

8

1 14 2 4 2 4

214 2

n 2

4 sin 3sin sin 3sin

4 3 3

13 6 2 13, 6m n

π

π

π π π

θ

π= − − −

= − − + ×

= − + = − =

M1A1 A1 M1

M1 M1 A1

M1

M1

A1A1 M1A1 M1 B1 A1

(16)


5


9 ( )

( )( )

( )

1 11 2 2

55

1 12 2

2

42

12

12

1

(a) : through ,6

grad. grad perp 1 perp. bisector: 6

(b) : through 1,6 grad 2 grad perp perp. bisector: 6 1

(c) :

l

PQ

y x

lQR

y x

l

=

= −

− = − −

−

= =

= −

− = − +

( )( )

1

1 1 1 12 22 2 2 2

1 12 2

2

1

7 4 3 7 3, 4 lies on

: 5 4 1 5 3, 4 lies on 1 , 3, 4 eliminate , 4, 3(d) length 5 area 5 25(e) is perp. bisector

**

y x l

l y x lor x x y or x y x

PS

lπ π

+ = + = ∴

+ = + = ∴

= = = = =

=

= =

( )

2 22

1 13 2 2

of 3 ( 2) 5

is perp. bisector of 3 4 =5 radius circle passes through and .(f) passes through midpoint of ie 1 ,8

PQ or QS PS QS

l QR or RS RS QSSQ SR Q Rl PR

= − − = ∴ =

= + ∴ == = ∴

B1 M1 A1ft B1 B1 B1 B1 M1A1 A1 M1A1 M1 A1 M1A1 (16)


6


10

( )

( )( )

4 3 753 4 4

3 154 4

1212 2

43

(a) 8 5 3 4 19th term 18 18 25 18 4th term 3 4 5 3 19th term 5 4th term (b) 2 11

300 6 2 11 300

*

a d d

d d

a

a d a dd a

a d a a or da d a a a or d

S a d

a or

+ = +

== + = + × = + =

= + = + = + =

∴ = ×

= +

= + × ( )

( )( )

( )

34

43

2

2

6 2 11 300 12 88 3 (c) 4

(d) 2000 6 4 1

2 2000 0

1 1 16000 critical value(s) 31.4 31.94

greatest 31

* 300 9 66 4, 3 *

d

a

p

da a a

d

p

p p

p

or d d d a= × +

= + == =

> + −

+ − <

− ± += −

=

= + ⇒ = =

M1 A1 M1A1 M1A1 A1 B1 M1 A1 M1A1 A1ft (13)

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7362_01_rms_20110309