johortrial stpm 2012-mathst paper 2(qna)

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  • 7/31/2019 Johortrial Stpm 2012-Mathst Paper 2(Qna)

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    2

    1 . 1 . 1 . 1 . T h e r e l a t i o n b e t w e e n t h e v a r i a b l e s m a n d x i s r e p r e s e n t e d b y t h e d i f f e r e n e q u a t i o n ,

    I f m = 1 w h e n x = 3 , s h o w t h a t m x = A e m - x , w h e r e A i s a c o n s t a n t . [ 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 . 2 . 2 . 2 . P Q R i s a t r i a n g l e w i t h P Q R = 9 0 0 , P Q = d a n d Q R = d . I f S i s a p o i n t o n P R

    s u c h t h a t P Q S = 3 0 0 , f i n d t h e r a t i o P S : S R .

    H e n c e f i n d t h e l e n g t h o f P S a n d S R i n t e r m s o f d . [ 7 [ 7 [ 7 [ 7 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    3 . 3 . 3 . 3 .

    I n t h e d i a g r a m a b o v e , A D i s a d i a m e t e r o f a c i r c l e c e n t r e O a n d C i s a p o i n t o n t h e

    c i r c u m f e r e n c e n e a r e r t o A t h a n D . I f D C i s p r o d u c e d t o B s o t h a t B C = C D , p r o v e t h a t

    ( a ) t h e t r i a n g l e s D C A a n d B C A a r e c o n g r u e n 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 ] ] ] ]

    ( b ) O C i s p a r a l l e l t o A B . [ 3 [ 3 [ 3 [ 3 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    G i v e n t h a t A D = 1 0 c m a n d C D A = 3 5 0 , c a l c u l a t e t h e a r e a o f t h e t r i a n g l e A B D . [ 2 [ 2 [ 2 [ 2 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    4 . 4 . 4 . 4 . U s i n g t h e i d e n t i t y f o r c o s ( A + B ) , p r o v e t h a t c o s = l - 2 s i n 2 ( 2 1 ) . [ 3 [ 3 [ 3 [ 3 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    P r o v e t h a t 1 + s i n - c o s = 2 s i n ( 2

    1 ) [ c o s (

    2

    1 ) + s i n (

    2

    1 ) ] . [ 3 [ 3 [ 3 [ 3 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    H e n c e o r o t h e r w i s e , s o l v e t h e e q u a t i o n 1 + s i n - c o s = 0 , w h e r e 0 2 . [ 3 [ 3 [ 3 [ 3 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    5 . 5 . 5 . 5 . I n a c h e m i c a l r e a c t i o n , c o m p o u n d A c o n t i n u o u s l y d e c o m p o s e t o c oB . T h r o u g h o u t t h e d e c o m p o s i t i o n t h e t o t a l a m o u n t o f A a n d B i s c o n s t a n t . A t a n y t i m r a t e t h a t B i s p r o d u c e d i s d i r e c t l y p r o p o r t i o n a l t o t h e p r o d u cA a n d t h e

    a m o u n t o f B a t t h a t t i m e . A t t h e b e g i n n i n g o f t h e p r o c e s s , t h e A a n d t h e a m o u n t o f B a r e a g m o l - 1 a n d b g m o l - 1 r e s p e c t i v e l y . I f x g m o l - 1 d e n o t e s t h e a m o u n t o f B t h a t h a s b e e n p r o d u c e d a t t i m e t ( i n s e c o n d ) a f t e r t h e r e a c t i o n h a s b e g u n , s h

    , k i s a p o s i t i v e c o n s t a n t . [ 2 [ 2 [ 2 [ 2 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

    H e n c e , e x p r e s s x i n t e r m s o f t . [ [ [ [ 7 7 7 7 m a r k s ] m a r k s ] m a r k s ] m a r k s ] F i n d t h e t i m e w h e n 5 0 % o f A i s d e c o m p o s e d ? [ [ [ [ 2 2 2 2 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

    6 6 6 6 . . . . T h e p o s i t i o n v e c t o r s o f P , Q a n d R r e l a t i v e t o t h e o r i g i n O a r e , , , ,

    a n d r e s p e c t i v e l y . M i s m i d - p o i n t o f P Q .

    ( a ) E x p r e s s a n d i n t e r m s o f t . [ 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 O

    C

    D

    A

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    3

    ( b ) G i v e n t h a t t o n l y t a k e s p o s i t i v e v a l u e s , c a l c u l a t e t w h e n ( i ) =

    ( i i ) O , P , Q a n d M c o l l i n e a r . [ 5 [ 5 [ 5 [ 5 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

    ( c ) G i v e n t h a t

    i s t h e a n g l e b e t w e e n a n d , d e t e r m i n e t h e

    w h e n t = 1 . [ [ [ [ 3 3 3 3 m a r k s ] m a r k s ] m a r k s ] m a r k s ]

    7 . 7 . 7 . 7 . F i v e p e o p l e , o f w h o m t h r e e a r e w o m e n a n d t w o a r e m

    F i n d t h e p r o b a b i l i t y t h a t

    ( a ) n o t w o p e o p l e o f t h e s a m e s e x a r e t o s t a n d n e x 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 ] ] ] ]

    ( b ) t h e f i r s t a n d l a s t p e o p l e i n t h e q u e u e a r e b o t h w [ 3 [ 3 [ 3 [ 3 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    8 . 8 . 8 . 8 . T h e n u m b e r o f c a r b r e a k d o w n s , a l o n g a s t r e t c h o f h P o i s s o n d i s t r i b u t i o n w i t h a m e a n o f 2 . 7 .

    F i n d t h e p r o b a b i l i t y t h a t ( a ) a t m o s t 4 c a r s b r e a k d o w n s o c c u r r e d a l o n g t h e h d a y .

    [ 3 m a r k s ] ( b ) t h e s m a l l e s t i n t e g e r n s u c h t h a t t h e p r o b a b i l i t y t h a t m o r e t h an c a r

    b r e a k d o w n s o c c u r r e d a l o n g t h e h i g h w a y o n a p a r t i c u l a m a r k s ]

    9 . 9 . 9 . 9 . T h e c o n t i n u o u s r a n d o m v a r i a b l e X h a s a c u m u l a t i v e d i s t r i b u t i o n f u F ( x ) , g i v e n b y

    >

    <

    =

    5 , 1

    5 1 , 1

    1 4 5

    1 , 0

    ) (

    x

    x x

    x

    x F

    ( a ) F i n d t h e m e d i a n o f X [ 2 [ 2 [ 2 [ 2 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 p r o b a b i l i t y d e n s i t y f u n c t i o n o f X , f ( x ) a n d s k e t c h i t s g r a p h .

    [ 3 [ 3 [ 3 [ 3 m a r k s m a r k s m a r k s m a r k s ] ] ] ] ( c ) F i n d E ( X ) . [ 3 [ 3 [ 3 [ 3 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    1 0 . 1 0 . 1 0 . 1 0 . M a s s , X k g , o f s t u d e n t s f r o m a c e r t a i n c o l e d g e i s n o r m a5 4 k g a n d s t a n d a r d d e v i a t i o n 5 k g .

    ( a ) I f a s t u d e n t i s s e l e c t e d r a n d o m l y , f i n d t h e p r o b b e t w e e n 5 3 a n d 5 9 k g . [ 2 [ 2 [ 2 [ 2 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    ( b ) T w o s t u d e n t s a r e s e l e c t e d r a n d o m l y . C a l c u l a t t o t a l m a s s e x c e e d s 1 0 3 k 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 ] ] ] ]

    ( c ) I f a s a m p l e o f e i g h t s t u d e n t s i s s e l e c t e d r a n d o mt w o o f t h e s t u d e n t s h a v e m a s s e s t h a t e x c e e d 5 9 k g[ 4 [ 4 [ 4 [ 4 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    4

    1 1 . 1 1 . 1 1 . 1 1 . A n e l e c t r i c a l c o m p a n y s e l l s X t e l e v i s i o n s e t s a d a y w i t h t h e p r o b a bd i s t r i b u t i o n f u n c t i o n

    =

    = = o t h e r w i s e

    x x a x X P

    , 0

    4 , 3 , 2 , 1 , 0 ) , 5 ( ) (

    ( a ) D e t e r m i n e t h e v a l u e o f a . [ 2 [ 2 [ 2 [ 2 m a r k s m a r k s m a r k s m a r k s ] ] ] ] ( b ) W r i t e d o w n t h e c u m u l a t i v e d i s t r i b u t i o n f u n c t i X . [ [ [ [ 3 3 3 3 m a r k s m a r k s m a r k s m a r k s ] ] ] ] ( c ) F i n d P ( | X E ( X ) | < 1 ) . [ 4 [ 4 [ 4 [ 4 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    1 2 . T h e m a r k s s c o r e d b y 2 5 s t u d e n t s i n a M a t h e m a t i c

    5 2 5 2 4 8 6 5 4 0 7 6 5 4 7 8 3 2 6 4 7 0 3 0

    4 5 4 2 6 2 3 8 2 3 5 5 5 7 8 9 6 5 8 3 2 4 4 8

    ( a ) C o n t r u c t a s t e m p l o t f o r t h e a b o v e d a t a . [ 2 [ 2 [ 2 [ 2 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 m e d i a n a n d i n t e r q u a r t i l e r a n g e . [ 4 [ 4 [ 4 [ 4 m a r k s m a r k s m a r k s m a r k s ] ] ] ]

    ( c ) D r a w a b o x p l o t t o r e p r e s e n t t h e d a t a . [ 2 [ 2 [ 2 [ 2 m a r k s

    m a r k s m a r k s m a r k s ]

    ] ] ] ( d ) S t a t e t h e s h a p e o f t h e d i s t r i b u t i o n . [ 1 [ 1 [ 1 [ 1 m a r k m a r k m a r k m a r k ] ] ] ]

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    6

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    7

    R

    Q P

    S 6 0 0

    d

    3 0 0

    d 3

    Q S t e p s M a r k s N o t e s

    1 .

    m l n m = x + l n x + C

    1 l n 1 = 3 + l n 3 + C

    C = - 2 l n 3

    m l n m = x + l n x - 2 l n 3

    l n = m x + 2

    m x = 3 e m x + 2

    o r m x = 3 e 2

    e m x

    o r m x = 2 2 . 2 e m x

    1

    1

    1

    1

    1 5

    C o r r e c t s e p a r a b l e f o r m

    I n t e g r a t e e i t h e r s i d eo r e e c t l y

    S u b s t i t u t e i n t o h i s e q n

    R e w r i t e e q n w i t h h i c

    C o r r e c t a n s o n l y

    2

    9 0 0 -

    P R =

    = 2 d

    s i n = o r c o s =

    R S =

    P S : S R = 1 : 3

    P S =

    P S

    1

    1

    1

    1

    1

    1

    1

    7

    C o r r e c t a n s w e r

    E i t h e r o n e c o r r e c t

    U s i n g s i n e r u l e t o f i n d R S o r P S

    E i t h e r o n e c o r r e c t

    T h i s a n s w e r o n l y

    E i t h e r o n e ( h i s r a t i o )

    B o t h c o r r e c t

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    9

    = 0 , . 2 =

    = 0 , , 2 .

    1

    1

    9

    E i t h e r o n e c o r r e c t

    C o r r e c t a n s w e r

    5 A m o u n t o f A a f t e r t s e c o n d = a x A m o u n t o f B a f t e r t s e c o n d = b + x

    = k t + c

    W h e n t = 0 , x = 0

    = k ( 0 ) + c

    c =

    = k t +

    = k t

    = e ( a + b ) k t

    x = o r x =

    w h e n 5 0 % A d e c o m p o s e - - - - - - x =

    t =

    =

    1 *

    1

    1

    1

    1

    1

    1

    1

    1

    1

    1

    1 1

    s e e n

    C o r r e c t a n s w i t h * a c q u i r e d

    C o r r e c t s e p a r a b l e f o r m

    H i s P a r t i a l f r a c t i o n ( w o r k i n g s h o w n )

    s e e n

    S u b s t i t u t e i n i t i a l v a l u e s

    R e w r i t e e q n w i t h h i c

    E l i m i n a t e l o g ( m a y b e i n e a r l i e r s t e p )

    C o r r e c t a n s w e r

    S u b s t i t u t e x = i n t o

    h i s t .

    C o r r e c t a n s

    1 = A ( b + x ) + B ( a x ) A = B =

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    1 0

    6

    ( a )

    ( b ) ( i )

    9 t 2 = 4 t 2 4 0 t + 1 0 0 t 2 + 8 t 2 0 = 0

    ( t + 1 0 ) ( t 2 ) = 0 t = 2

    ( i i ) O , P , M , Q o n t h e s a m e l i n e e q u a l g r a d i e n t

    t = 3 ( c ) ,

    = - 0 . 1 7 8 9 o r - - - - 0 . 1 7 9

    1

    1

    1

    1

    1

    1 1

    1

    1

    1

    1

    1 1 2

    U s e r a t i o t h e o r e m o e q u i v a l e n t m e t h o d

    C o r r e c t a n s w e r o n l y

    A l l o w h i s

    C o r r e c t a n s w e r o n l y

    E q u a t i n g m a g n i t u d ei n t e r m s o f t .

    Q u a d r a t i c e q u a t i o n i n f a c t o r f o r m . C o r r e c t a n s w e r o n l y

    E q u a t i n g g r a d i e n t s

    C o r r e c t a n s w e r o n l y

    B o t h c o r r e c t ( h i s t )

    U s e s c a l a r p r o d u c t

    C o r r e c t a n s w e r o n l y

    7 W 1 W 2 W 3 M 1 M 2

    ( a ) P ( n o t w o p e o p l e o f t h e s a m e s e x a r e t o s t a n d n e x t t o

    o t h e r ) =

    = 0 . 1

    P ( t h e f i r s t a n d l a s t p e o p l e i n t h e q u e u e a r e b o t h w o

    =

    = 0 . 3

    1 1

    1

    1 1

    1

    6

    3 ! 2 ! s e e n

    D i v i d e d b y 5 !

    C o r r e c t a n s w e r

    D i v i d e d b y 5 !

    C o r r e c t a n s w e r

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    1 1

    4 5

    2 0 1

    1 5

    f ( x )

    x

    8 L e t X ~ P ( 2 . 7 ) ( a ) P ( X 4 )

    = e - 2 . 7 ( 1 + 2 . 7 + + + )

    = 0 . 8 6 2 9 / 0 . 8 6 3

    ( b ) P ( X > n ) < 0 . 0 1

    P ( X n ) 0 . 9 9

    C h e c k i n g :

    P ( X 5 ) = 0 . 8 6 2 9 + e - 2 . 7 ( = 0 . 9 4 3 3

    P ( X 6 ) = 0 . 9 4 3 3 + e - 2 . 7 ( = 0 . 9 7 9 4

    P ( X 7 ) = 0 . 9 7 9 4 + e - 2 . 7 ( = 0 . 9 9 3 4 > > > > 0 . 9 9 0 . 9 9 0 . 9 9 0 . 9 9

    n = 7

    1

    1 1

    1

    1

    1

    1

    7

    P ( X 4 ) s e e n

    A t l e a s t 3 c o r r e c t t e r m s C o r r e c t a n s w e r

    E i t h e r o n e s e e n

    0 . 9 4 3 3 o r 0 . 9 7 9 4 s e e n

    0 . 9 9 3 4 s e e n

    C o r r e c t a n s w e r

    9 ( a ) F ( m ) =

    m = 1

    ( b )

    ( c ) E ( X ) =

    =

    =

    1

    1

    1

    1

    1

    1

    1

    1

    8

    U s i n g F ( m ) =

    C o r r e c t a n s w e r

    A l l c o r r e c t

    C u r v e ( g e n e r a l s h a p

    L a b e l s : f ( x ) , x , 1 , 5 , a n d t w o o t h e r v a l u e b a s e d o n h i s f ( x ) .

    U s e E ( X ) f o r m u l a

    C o r r e c t i n t e g r a t i o n

    C o r r e c t a n s w e r o n l y

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

    9 5 4 / 2 * 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 C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *C O N F I D E N T I A L *

    1 2

    1 0 X ~ N ( 5 4 , 5 2 )

    ( a ) P ( 5 3 < X < 5 9 )

    = 0 . 4 2 0 6

    ( b ) X 1 + X 2 ~ N ( 1 0 8 , 5 0 )

    P ( X 1 + X 2 > 1 0 3 )

    = 0 . 7 6 0 3

    ( c ) P ( X > 5 9 ) = 0 . 1 5 8 6 6

    Y ~ B ( 8 , 0 . 1 5 8 6 6 )

    P ( Y = 2 ) = 8 C 2 ( 0 . 1 5 8 6 6 ) 2 ( 0 . 8 4 1 3 4 ) 6

    = 0 . 2 5 0

    1

    1

    1

    1

    1

    1

    1

    1

    1

    9

    S t a n d a r d i s a t i o n ( e x a c t v a l u e s )

    C o r r e c t a n s w e r

    1 0 8 o r 5 0 s e e n

    S t a n d a r d i s a t i o n ( h i sv a l u e s )

    C o r r e c t a n s w e r

    0 . 1 5 8 6 6 o r 0 . 1 5 8 7 s e e n

    B i n o m i a l d i s t r i b u t i os e e n ( c a n b e i m p l i e d

    U s e b i n o m i a l f o r m u l a

    C o r r e c t a n s w e r

    1 1 ( a )

    ( b )

    ( c ) E ( X ) =

    =

    P ( | X E ( X ) | < 1 ) = P ( | X | < 1 )

    = P (

    =

    1

    1

    1

    1

    1

    1

    1

    1

    1

    9

    H i s P ( X ) = 1

    C o r r e c t a n s w e r

    A t l e a s t 1 p a i r [ F ( x ) a n d i t s c o r r e s p o n d i nl i m i t ] c o r r e c t ( o t h e r t h a n 0 a n d 1 )

    A t l e a s t 3 p a i r s c o r r e c t ( o t h e r t h a n o a n d 1 )

    A l l c o r r e c t

    U s e E ( X ) f o r m u l a

    C o r r e c t a n s w e r

    C o r r e c t r a n g e b a s e do n h i s E ( X )

    C o r r e c t a n s w e r

  • 7/31/2019 Johortrial Stpm 2012-Mathst Paper 2(Qna)

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