johortrial stpm 2012-mathst paper 1(qna)

7/31/2019 Johortrial Stpm 2012-Mathst Paper 1(Qna)

1/10

C O N F I D E N T I A L * T r i a l S T P M J O H O R 2 0 1 2 P a p e r 1

9 5 4 / 1

* T h i s q u e s t i o n p a p e r i s C O N F I D E N T I A L u n t i l t h e e x a m i n a t i o n i s o v e r .

C O N F I D E N T I A L *

2

1 . I f ,1

2

2

xe

xy

+= s h o w t h a t 0242

2

22 =+

+ x

d x

d y

d x

yde x . [ 4 m a r k s ]

2 . D e t e r m i n e t h e s e t o f r e a l v a l u e s o f k s u c h t h a t t h e m a t r i x

k

k

210413

12

i s n o t

s i n g u l a r . [ 4 m a r k s ]

3 . I t i s k n o w n t h a t i f ,xxy = t h e n t h e f i r s t d e r i v a t i v e w i l l b e ) .l n1( xyd x

d y+=

B y u s i n g a s u i t a b l e s u b s t i t u t i o n , s h o w t h a t .)l n1(2 22 Cxd xxy x +=+[ 4 m a r k s ]

4 . G i v e n .12 51 3 ++= iZ F i n d t h e m o d u l u s a n d a r g u m e n t o f i t s c o n j u g a t e .

[ 6 m a r k s ]

5 . T h e f u n c t i o n f i s d e f i n e d b y . ,3)1(:f2

I Rxxx ++ S k e t c h t h e g r a p h o f

f u n c t i o n f . [ 2 m a r k s ]

H e n c e ,

( a ) s h o w t h a t f i s n o t a o n e - t o - o n e f u n c t i o n a n d s t a t e i t s r a n g e . [ 2 m a r k s ]

( b ) s t a t e t h e p o s s i b l e r e s t r i c t e d d o m a i n s f o r w h i c h t h e c o r r e s p o n d i n g r e s t r i c t e d

f u n c t i o n o f f h a s t h e r a n g e ]2,( a n d i s o n e - t o - o n e . [ 2 m a r k s ]

6 . E x p r e s s 61 23 0 a s 32 qp + w h e r e p a n d q a r e i n t e g e r s . [ 7 m a r k s ]

7 . E x p r e s s x

x

21

1

+a s a s e r i e s o f a s c e n d i n g p o w e r s o f x u p t o t h e t e r m i n .3x

[ 6 m a r k s ]

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

8 . S h o w t h a t t h e e q u a t i o n 0 3 62 43 649 22 =++ yxyx i s a n e q u a t i o n o f a n

e l l i p s e . [ 3 m a r k s ]

S k e t c h i t s g r a p h . [ 2 m a r k s ]

B y t r a n s l a t i o n , m o v e t h e e l l i p s e t o a s u i t a b l e p o s i t i o n . H e n c e , f i n d t h e a r e a o f t h e

e l l i p s e . [ 5 m a r k s ]

9 . AAAA a n d BBBB a r e t w o m a t r i c e s s u c h t h a t

=

421

134

432

AAAA a n d

=

615

1 441 5

941 0

BBBB ....

F i n d A BA BA BA B a n d d e d u c e 1

AAAA . [ 5 m a r k s ]

H e n c e , s o l v e t h e s y s t e m o f l i n e a r e q u a t i o n

834

142

8432

=++

=++

=++

zyx

zyx

zyx

[ 5 m a r k s ]


2/10

S U L I T * S U L I T * S U L I T * S U L I T *

9 5 4 / 1 [ L i h a t s e b e l a h * K e r t a s s o a l a n i n i S U L I T s e h i n g g a p e p e r i k s a a n k e r t a s i n i t a m a t . S U L I T * S U L I T * S U L I T * S U L I T *

3

1 0 . T h e f u n c t i o n s f a n d g a r e d e f i n e d a s f o l l o w s :

Rxxx

ex x

+

>

,1:g

1 x ,:f

2

1

( a ) S h o w t h a t f h a s a n i n v e r s e f u n c t i o n 1

f

. D e f i n e 1

f

a n d s k e t c h o n t h e s a m e c o o r d i n a t e a x e s , t h e g r a p h o f f u n c t i o n s f a n d 1f . [ 6 m a r k s ]

( b ) D e t e r m i n e w h e t h e r t h e c o m p o s i t e f u n c t i o n s gf a n d fg e x i s t . G i v e y o u r

r e a s o n s . [ 4 m a r k s ]

1 1 . G i v e n t h a t ,)(f 23 qxp xxx +++= w h e r e p a n d q a r e c o n s t a n t s , i s d i v i s i b l e

b y ) .1( x W h e n d i v i d e d b y ) .2( x t h e r e m a i n d e r i s 1 7 . F i n d t h e v a l u e s o f a n d .

[ 4 m a r k s ]

( a ) S h o w t h a t 0)(f =x h a s o n l y o n e r e a l r o o t . F i n d t h e s e t o f v a l u e s o f s u c h

t h a t 0)(f >x . [ 6 m a r k s ]

( b ) E x p r e s s )(f

91 1

x

x +i n p a r t i a l f r a c t i o n s . [ 5 m a r k s ]

1 2 . T h e e q u a t i o n o f t h e c u r v e i s g i v e n b y .)2) (3(

1

+

+=

xx

xy

( a ) F i n d t h e e q u a t i o n s o f a l l t h e a s y m p t o t e s o f t h e c u r v e . [ 2 m a r k s ]

( b ) F i n d t h e p o i n t s w h e r e t h e c u r v e i n t e r s e c t s t h e a x i s -x a n d t h e

a x i s . -y D e t e r m i n e t h e t u r n i n g p o i n t s i f t h e y e x i s t . S k e t c h t h e c u r v e . [ 7

m a r k s ]

( c ) F i n d t h e s e t o f v a l u e s o f k s u c h t h a t t h e e q u a t i o n )2) (3(1 += xxkx h a s

t w o r e a l r o o t s . [ 1 m a r k ]

( d ) W i t h o u t u s i n g t h e g r a p h a n d 7 , s h o w t h a t t h e r e i s a r o o t p f o r

1)2) (3(

1=

+

+

xx

x, s u c h t h a t .7.26.2


3/10


9 5 4 / 1



4


4/10



5

M A R K I N G M A R K I N G M A R K I N G M A R K I N G S C H E M E S C H E M E S C H E M E S C H E M E ( 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 T TTT P A P E R P A P E R P A P E R P A P E R 1 ) 1 )1 )1 )

1

[ 4 ]

0242

224

222

2)2(

1

1

2

22

2

2

22

22

22

22

2

2

=+

+

=++

=++

=+

+=

+=

x

d x

d y

d x

yde

d x

d ye

d x

ydex

xd x

d yex

xd x

d yeey

xy e

e

xy

x

xx

x

xx

x

x

1( p r o d u c t r u l e ) 1 ( C A O )

1( e l i m i n a t e y )

1( a l l c o r r e c t )

2

[ 4 ]

F o r t h e m a t r i x t o b e n o n - s i n g u l a r ,

T h e s e t o f v a l u e o f k

i s ,

i . e .

i . e .

1 ( C A O )

1 ( t o u s e d e t )

1(

1

3

[ 4 ]

y2 y

) ]l n1(2 y [ y )l n1(2 y

2

2

2

Cx

Cy

d

d xxd xx

x +=

+=

=

+=+

1 ( t o u s e )l n1(

yxy

d x

d+= )

1

1 ( i n t e g r a t i o n )

1

4

[ 6 ]

12 51 3 ++= iZ

112 5)(62 ++= iiZ ,

iZ 61 +=

iZ 61* =

1 ( 12 =i a n d

i= 1 )

11


5/10


9 5 4 / 1



6

3 7)6(1 22* =+=Z

4 7 7.01

6t a na r g 1 == Z

1 ( m e t h o d )

1

1

5

[ 6 ]

1l

( a ) T h e h o r i z o n t a l l i n e 1

l c u t s t h e g r a p h o f f u n c t i o n f a t 2

p o i n t s , t h e r e f o r e f i s n o t a o n e - o n e f u n c t i o n .

R a n g e o f ( ]3,=f

( b ) T h e p o s s i b l e r e s t r i c t e d d o m a i n s w h i c h h a s t h e r a n g e

]2,( a r e ]2,( a n d [ ),0 .

D 1 ( c u r v e )

D 1 ( a l l c o r r e c t )

1

1

1 , 1

6

[ 7 ]

3261 23 0 qp+=

623261 23 0 22 p qqp ++=

)1(323 0 22 qp +=

( 2 ) 21 2 p q=

F r o m ( 2 ) , ( 3 )6

pq =

S u b s t i t u t e ( 3 ) i n t o ( 1 )

3i . e . ,9,

9,6

0)9) (1 22(

01 0 83 02

3 01 0 8

2

2

2

22

24

2

2

==

=

=

=+

=+

ppZp

p

pp

pp

pp

1

1 ( b o t h )

1 ( m e t h o d )

1

1

1 ( m e t h o d )

3

2

0 x- 1

y


6/10



7

2,3;2,3 ==== qpqp

.2,3

,03261 23 0

==

>+=

qp

qp

1

7

[ 9 ]

( ) ( ) 21

21

21121

1 +=

+xx

x

x

( )( ) ( ) ( )

+++=

+

+

++=+

32

323

21

21

221

21

1 6

1

8

1

2

11

!3!22

111 2

1

xxx

xxxx

( ) ( ) ( )( )

( )

( )( )( ) ( )

++++=

++

+

+=

32

3252321

223

21

2

5

2

31

2!3

2!2

22

1121 2

1

xxx

x

xxx

( ) ( )

++

+

++++

++++=+

3

32

32

32

1 6

1

8

1

8

1

4

3

2

1

2

1

2

5

2

31211 2

121

x

xx

xxx

xxxxx

++++=

+ 321 6

5 1

8

1 5

2

31

21

1xxx

x

x

F o r e x p a n s i o n t o b e v a l i d ,

t h e s e t o f v a l u e s o f i s

1

1 ( m e t h o d )

1

1

1 ( m e t h o d )

1

1

1 ( i n t e r s e c t i o n )

1

8

[ 1 0 ]

19

)3(

4

)2(

3 6)3(4)2(9

03 6)3(4)2(9

03 6)6(4)4(9

03 62 43 649

22

22

22

22

22

=

+

=+

=+

=++

=++

yx

yx

yx

yyxx

yxyx

S i n c e t h e e q u a t i o n i s i n t h e f o r m 1)()(

2

2

2

2

=

+

b

ny

a

mx,

1 ( c o m p l e t i n g t e s q u a r e )

1


7/10


9 5 4 / 1



8

t h e r e f o r e t h e e q u a t i o n i s t h e e q u a t i o n o f a n e l l i p s e .

19

)3(

4

)2(e l l i p s e o fA r e a

22

=

+ yx

2

2

2

2

2

2

2

2

2

22

u n i t 6

2

2s i n6

)2c o s1(6

c o s26

c o s43

43

2

93 62

194

e l l i p s e o fA r e a

2

2

2

2

2

2

2

2

=

+=

+=

=

=

=

=

=+=

d

d

d

d xx

d xx

yx

1

1 ( S h a p e + p o s i t i o n )

1 ( a l l c o r r e c t )

1 ( t o u s e

194e l l i p s e

22

=+

yx

1

1 ( t o u s e

s i n2=x )

1 ( t o u s e

2c o s1c o s2 2 +=)

1

9

[ 1 0 ]

H e n c e ,

1 ( m e t h o d )

1

1

1 ( m e t h o d )

1

1

y

x

( 2 , 6 )

( 2 , 0 )

( 0 , 3 ) ( 4 , 3 )

1

9

)3(

4

)2( 22=

+

yx


8/10



9

A 1

1 ( m e t h o d )

1

2 ( a n y 2 c o r r e c t , 1 )

1 0

[ 1 0 ]

L e t

i s a o n e - t o - o n e f u n c t i o n . H e n c e , h a s a n i n v e r s e f u n c t i o n .

L e t

1 ( m e t h o d )

1

1 ( m e t h o d )

1

D 1 ( f o r f - 1)

D 1 ( a l l c o r r e c t )

1 ( a l l f o u r c o r r e c t )

1 ( m e t h o d ) 1

1

1 1

[ 1 5 ] ( 2 ) - ( 1 ) ,

1 ( m e t h o d ) 1 ( b o t h c o r r e c t )

1 ( s o l v i n g )

y

( 1 , 1 )

x

y = x

O


9/10


9 5 4 / 1



1 0

( o n l y r e a l r o o t )

B u t

T h e s o l u t i o n s e t i s .

( b ) L e t

:

C f : :

1 ( c o r r e c t p a n d q )

1

1 ( m e t h o d ) 1

1 ( m e t h o d )

11

1

1 ( m e t h o d )

2 ( a n y 2 c o r r e c t , 1 )

1

1 2

[ 1 5 ]

( a ) T h e a s y m p t o t e s a r e 0 .y,2,3 === xx

( b ) T h e p o i n t o n t h e )0,1(a x i s - =x

T h e p o i n t o n t h e )6

1,0(a x i s - =y

)6(

72

)6(

)12) (1()6(

6

1

.)2) (3(

1

22

2

22

2

2

+

=

+

+++=

+

+=

+

+=

xx

xx

xx

xxxx

d x

d y

xx

xy

xx

xy

1 ( 2,3 == xx )

1 0 )y( =

1 ( b o t h )

1 ( m e t h o d )


10/10



1 1

{ }

p o i n t s . t u r n i n g n o ,0

) ,2,306)1(72(0 22

johortrial stpm 2012-mathst paper 1(qna)

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