maths stpm trial melaka_2012_paper1(q&a).pdf

7/31/2019 Maths STPM Trial MELAKA_2012_paper1(Q&A).pdf

1/10

2 0 1 2 2 0 1 2 2 0 1 2 2 0 1 2 M E L A K A M E L A K A M E L A K A M E L A K A S T P M S T P M S T P M S T P M T r i a l T r i a l T r i a l T r i a l M A T H E M A T I C S M A T H E M A T I C S M A T H E M A T I C S M A T H E M A T I C S P a p e r P a p e r P a p e r P a p e r 1 111 9 5 0 / 1 , 9 5 0 / 1 , 9 5 0 / 1 , 9 5 0 / 1 , 9 5 4 / 1 9 5 4 / 1 9 5 4 / 1 9 5 4 / 1

1111

1111 .... ( a ) B y u s i n g 'BABA = , p r o v e t h a t [ ] [ ] )()()()( BABBAABABA = [ 3[ 3[ 3[ 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

( b ) W i t h o u t u s i n g t a b l e s o r c a l c u l a t o r , s h o w t h a t

65l o g6l o g

1 2 5l o g8l o g2 7l o g=

+[ 3[ 3[ 3[ 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

2222 .... ( a ) E x p r e s s i n p a r t i a l f r a c t i o n s . [ 3[ 3[ 3[ 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

( b ) H e n c e , s h o w t h a t [ 3[ 3[ 3[ 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

3333 .... ( a ) D e t e r m i n e t h e v a l u e o f x i f 3231

3+

+

i

x ii s a r e a l n u m b e r a n d f i n d t h i s n u m b e r .

[ 4[ 4[ 4[ 4 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

( b ) F i n d t h e v a l u e o f 2 0 1 2 )1

1(

i

i

+[ 3[ 3[ 3[ 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

4444 .... E x p a n d i n t h e a s c e n d i n g p o w e r s o f u p t o t h e t e r m i n . H e n c e , b y s u b s t i t u t i n g x = 3,

e v a l u a t e c o r r e c t t o t h r e e d e c i m a l p l a c e s . [ 6[ 6[ 6[ 6 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

5555 .... D e t e r m i n e a l l t h e p o s s i b l e v a l u e s o f s u c h t h a t t h e r o o t s o f t h e q u a d r a t i c e q u a t i o n

052)3( 22 =+++ xx a r e r e a l .

I f a n d a r e t h e r e a l r o o t s o f t h i s e q u a t i o n , s h o w t h a t 2 4 )5( 222 ++=+ .

H e n c e f i n d t h e m a x i m u m v a l u e o f 22 + [ 7[ 7[ 7[ 7 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

6666 .... ( a ) A n a r i t h m e t i c p r o g r e s s i o n h a s f i r s t t e r m a a n d c o m m o n d i f f e r e n t 8 , t h e s u m o f t h e f i r s t n

t e r m s o f t h e p r o g r e s s i o n i s 1 0 0 0 . E x p r e s s a i n t e r m s o f n, a n d s h o w t h a t t h e n t h t e r m o f

t h e p r o g r e s s i o n i s [ 4[ 4[ 4[ 4 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

( b ) G i v e n t h a t t h e nt h t e r m i s l e s s t h a n 4 0 0 , s h o w t h a t , a n d h e n c e

f i n d t h e l a r g e s t p o s s i b l e v a l u e o f n . [ 3[ 3[ 3[ 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

7777 .... ( a ) F i n d t h e e q u a t i o n o f t h e t a n g e n t t o t h e c u r v e a t t h e p o i n t

w i t h p a r a m e t e r t = 2. [ 4[ 4[ 4[ 4 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

( b ) F i n d t h e c o o r d i n a t e s o f t h e p o i n t w h e r e t h i s t a n g e n t m e e t s t h e c u r v e a g a i n . [ 3[ 3[ 3[ 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

8888 .... F i n d t h e s o l u t i o n s e t o f t h e i n e q u a l i t y

( a )3

1

4

1

+>

xx[ 3[ 3[ 3[ 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

( b ) ) 75) (2(4 22 ++ xxxx [ 5[ 5[ 5[ 5 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

9999 .... ( a ) F i n d d x [ 4[ 4[ 4[ 4 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

( b ) F i n d t h e a r e a o f t h e b o u n d e d r e g i o n i n t h e f i r s t q u a d r a n t b e t w e e n

[ 5[ 5[ 5[ 5 m a r k s ] m a r k s ] m a r k s ] m a r k s ]


2/10


2222

1 01 01 01 0 .... ( a ) T h e f u n c t i o n )( xf i s d e f i n e d b y


3/10


3333

N o S C H E M E M a r k s

1 a

[ ] [ ]

[ ] [ ] )()(

)()(

)()(

)()(

''

'

BABBAA

BABBAA

BABA

BABA

=

=

=

M 1

B 1 ( d i s t r i b u t i v e l a w )

A 1 T o t a l 3

1 b

5

6l o g

)1 2 5

82 7l o g ( x

2

1

3

)5

6l o g (

)5

6l o g (

= 6

M 1

M 1 A 1 T o t a l 3

2 ( a )

L e t

L e t

B 1

M 1

A 1 T o t a l 3

2 ( b )

=

=

= (

=

=

M 1

M 1

A 1 T o t a l 3


4/10


4444

3 a

i

ix

i

x i

31

31

31

3

+

+

+

=4

)3(33 ixx ++

3

03

=

=+

x

x

33324

)3(3332

31

3=+

=+

+

i

x i

M 1 , x c o n j u g a t e

M 1 , e q u a t e h i s i p a r t = 0

A 1

A 1 T o t a l 4

3 b

i

i

ix

i

i

=

+

+

+

1

1

1

1

2 0 1 2 )1

1(

i

i

+= 1)()( 45 0 32 0 1 2 == xii

M 1 , x c o n j u g a t e

A 1

A 1 T o t a l 3

4

S u b s t i t u t e x = 3 ,

( 3 d . p . )

M 1

A 1 1s t 3 t e r m s c o r r e c t

A 1 1t h t e r m c o r r e c t

M 1

M 1

A 1 T o t a l 6


5/10


6/10


6666

7 b T h i s t a n g e n t m e e t s t h e c u r v e w h e n

r e p r e s e n t s t h e p o i n t w h e r e t h e t a n g e n t t o u c h e s t h e c u r v e .

S o t h e t a n g e n t m e e t s t h e c u r v e a g a i n w h e n a t t h e p o i n t

A 1

M 1

M 1A 1 T o t a l 4

8 a

8 b

03

1

4

1

+

xx

0)3) (4(

)4(3

+

+

xx

xx

0)3) (4(

7

+ xx0)3) (4( >+ xx

{ }3,4: xxx

)75) (2(4 22 ++ xxxx

)75) (2()2) (2( 2 +++ xxxxx

0)752) (2( 2 ++ xxxx

0)96) (2( 2 ++ xxx

0)3) (2( 2 + xx

Rxxc ex + 0)3(s i n02 2

{ }2: xx

M 1 , s i n g l e f r a c t i o n

M 1

A 1 ( s e t f o r m ) T o t a l 3

B 1

M 1 ( c o m p l e t e s q u a r e )

A 1

M 1 ( h i s )

A 1 T o t a l 5

9 a ) L e t x = 2 u d x = 2 d u

d x = 2

= 2

= 2 ( t a n u u ) + c

= 2 ( t a n ) + c = 2 t a n + c

M 1

A 1

M 1

A 1 T o t a l 4

9 b ) P t s o f i n t e r s e c t i o n s : s u b y = x - 2 i n t o M 1


7/10


7777

L e t f ( x ) =

f ( 1 ) = 3 - 7 + 4 = 0

D i v i d e f ( x ) b y ( x - 1 ) M 1 ( f a c t o r i s e h i s )

A 1

M 1 ( m u s t h a v e )

A 1 T o t a l 5

1 0 a 3)(l i m2

=

xfx

33 2 = ae

2=a

)(l i m2

xfx

e x i s t

32 =b

2

9=b

)(l i m xfcx

e x i s t

M 1

A 1

M 1 ( h i s

) )(l i m)(l i m22

xfxfxx +

=

A 1

M 1 ( h i s

) )(l i m)(l i m xfxfcxcx +

=

y = x- 2

4 y + 3 x = 7

R


8/10


8888

cc

2

9

4=

2,7 2 >= cc

A 1 T o t a l 6

1 0 b ( i )

1 0 b ( i i )

{ }3: = xxRf , { }1: = xxDg

gfDR fg d o e s n o t e x i s t

L e t yx =+ 32

3= yx

3,3:1 > xxxf

{ }3:1 >= xxDf

B 1 ( a n y o n e )

M 1 A 1

M 1

A 1

B 1 T o t a l 6

1 1

22 )4(3

843

++ yx =

22 )0(1

2

+y

25

843=

+y

yxo r 2

5

843+=

+y

yx

i e l o c u s o f P a r e 063 = xy a n d 023 =+ xy

e q u a t i o n o f l i n e 1l i s 1 2

3

4+= xy

s o l v i n g 1 23

4+= xy a n d 063 = xy

M = ( 6 , 4 )

2222 46:)1 24()06(: ++=O MA M

1 3:55 2:1 0 0 ==

)3

1) (

3

4(1

3

1

3

4

t a n

+

=

3t a n 1= , 3t a n 1 =

B 1

M 1 ( a n y o n e )

A 1 ( b o t h ) T o t a l 3

A 1

M 1

A 1

M 1 ( h i s c o o r d i n a t e M )

A 1 T o t a l 5

M 1 ( h i s )


9/10


9999

4 3.1 0 8,5 6.7 1=A 1 T o t a l 2

1 2 ( a ) B 1

B 1 ( t r a n s f o r m t o r e q u i r e d f o r m ) T o t a l 2

1 2 ( b )

S o l v e e q u a t i o n s ( 1 ) , ( 2 ) a n d ( 3 ) s i m u l t a n e o u s l y ,

M 1 ( a n y o n e e q u a t i o n

c o r r e c t )

M 1 ( a l l e q u a t i o n s c o r r e c t )

M 1 ( s o l v i n g f o r p , q a n d r ) A 1 ( a l l c o r r e c t )

T o t a l 4

( c )

M 1

J 1

M 1


10/10


1 01 01 01 0

C o f a c t o r o f m a t r i x A i s

A X = B B

B 1

B 1

M 1

A 1

M 1 ( m u l t i p l y h i s i n v e r s e )

A 1 T o t a l 9

maths stpm trial melaka_2012_paper1(q&a).pdf

Documents