AO Jan 2010 Paper 2

Mark Scheme (Results) January 2010

GCE O Level

Pure Mathematics (7362/02) Paper 2

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January 2010

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AO Pure Mathematics 7362 Mark Scheme Paper 2

Q. Scheme Marks

1 2 2de sin 3 2e sin 3 3e cos3d

x xyy x x 2x xx

= = + M1A1A1

(3) 2 2 2 2

212

6.9 8.3 5(a) cos2 6.9 8.3

36.97... 37.0 (b) Area 8.3 6.9sin 17.2 cm

C

CABC C

+ −=

× ×= = °

∆ = × × =

M1A1 A1 (3)

M1A1A1 (3) (6) 3 (a) (i) (ii) 3y = 2x = B1B1 (2) (b) (i) 2 2

2 30 2 2xy x−= = = B1

(ii) 0 3 1 4x y= = + = B1 (2)

G1(2 branches) G1(asymptotes) G1(crossing pts)

(3) (7)

B2(all corr) (B1 for 2 or 3)

4 (a)

(

(

(c)

(c)

GCE AO Le

x 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

y 1 0.649 -1.28 -4.52 -8.61 -12.8 -15.9 -15.9 -9.40

b) Graph drawn (see diagram overleaf)

2

2

2 2

c) (i) 4 e 8 e 4 8, 1.9 (ii) e 4 8, e 4 3, 1.3

x

x

x x

xx xx x x

= +

− = − =

= − − = − =

G1 (pts plotted) G1 (smooth curve)

(2)

M1A1

M1M1A1 (5)

(9)

vel Pure Mathematics (7362) Paper 2 January 2010

4 (b)

Diagram of graph

GCE AO Level Pure Mathematics (7362) Paper 2 January 2010

5

M1 A1A1 (3)

G1 shape G1 3 and 4

(2)

M1 M1

M1A1 (5)

(9) 6 ( )

( )( )

( )

4141 1

4 4 4 1 116 2

12

32

44 12

11

(a) 80 75

75 80 1 80 5

(b) 80 1

100 80 120

(c) 30 120 30

4

a rar rS x S x

x x r r r r

a x

a x a x x

ar x

x

−

∞ − −= = = =

= − = = =

=

⇒ = × =

= − =

×

m

>

±

6 1 4x= =

M1A1

M1A1 (4) M1

M1A1 (3) M1A1ft

A1 (3) (10)

M1

A1 M1 M1A1

B1 (6)

M1A1 (2)

7

M1

A2,1,0 ft

A1 (4) (12)

(b)

GCE AO Level Pure Mathematics (7362) Paper 2 January 2010

8

2

2

2 3

2

2

3 2

2

2125 250

d 250d

3

25034

d 500d

dd

(a) Vol 125 cm 2 2 2 *(b) 4

250 4

2.709... 2.71

(c) 4

0 0 min.

(

r rAr r

Ar r

Ar

r hA rh rrh A r

r

r

r r

r

π

π

π π

π ππ

π

π

= =

= +

= = +

= − +

=

= = =

= +

⇒ ∴> >2250

2.71d) 2 2.709... 138.3... 138A π= + × = =

B1

B1

M1A1 (4)

M1 M1 A1 (3)

M1

M1A1 (3)

M1A1 (2) (12) 9 ( ) ( )

( ) ( )

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

2 2 3 2 2 2 2 3 3

2 2 3 2 2 2 2 3 3

3 3 2 2

2

(a)

(b) 7 3

3

7 49 9 280

(c)

3

3

α β α αβ β α α β αβ βα αβ β α β

α β α αβ β α α β αβ βα αβ β α β

α β αβ

α β α β α αβ β

α β α β αβ

α

+ − + ≡ − + + − + ≡ +

− + + ≡ + + − − + ≡ −

+ = − =

+ = + − +

= + + −

= − − = −

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

( )3 3

2 2 2 2

2 2

2 2

23 3 2 2

2809

1 13

2 280 19 3

2

4 49 12 37

(d)

37 49 3 46 37

(e)

0

9 280 3 0

x x

x x

β α βαβ α α β

βααββ α

β α β αβ

α β α β α αβ β α β α β αβ

+

− = + − = − =

− = − + + = − + −

= − =

+ = = −

× = =

+ + =

+ + =

M1A1 A1 (3)

B1

M1

A1 (3)

M1A1 (2) M1M1 A1 (3) M1

B1

M1

A1 (4) (15)

GCE AO Level Pure Mathematics (7362) Paper 2 January 2010

10 ( )

sin cos cos sincos cos cos cos

sin sincos cos

sin cos cos sin(a) tancos cos sin sin

1

tan tan *1 tan tan

(b) (i) 45 30

A B A BA B A B

A BA B

A B A BA BA B A B

A BA B

A B

++ =

−+

=−

+=

−= ° = °

13

13

2

2

2

tan 45 tan 30 tan 75 1 tan 45 tan 301 3 1 ,1 3 1

1 3 1 (ii) tan15tan 75 3 1

2 tan(c) tan21 tan

2 tan(d) 22.5 11 tan

tan 2 tan

θθθ

θθθ

θ θ

+=

−+ +

= =− −

−= =

+

=−

= ° =−

+ −

( )

( )

( )2 2

45 25 204

5 21 21225

20 20 202984121 20

1 0

2 4 4 tan 1 2

2 tan 22.5 0 tan 22.5 2 1

(e) tan 21

sin 2

θ

θ

θ+

=

− ± += = −

∴ =

= = × =−

= = =

>

±

−

M1

M1A1 (3) M1

A1A1

M1A1 (5)

B1 (1)

M1

M1

A1 A1 (4)

M1A1

M1A1 (4)

(17)

GCE AO Level Pure Mathematics (7362) Paper 2 January 2010

GCE AO Level Pure Mathematics (7362) Paper 2 January 2010

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