melaka trial stpm 2012-mathst paper 2(q&a)

7/31/2019 Melaka Trial Stpm 2012-Mathst Paper 2(Q&A)

1/9

T r i a l T r i a l T r i a l T r i a l S T P M S T P M S T P M S T P M M e l a k a M e l a k a M e l a k a M e l a k a 2 0 1 2 2 0 1 2 2 0 1 2 2 0 1 2 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 2 222 9 5 4 / 2 9 5 4 / 2 9 5 4 / 2 9 5 4 / 2

---- 1111 ----

---- 1111 ----

1 . S o l v e t h e e q u a t i o n c o s + 2 c o s 2 + c o s 3 = 0 f o r . ( 4 m a r k s )

2 . T h r e e f o r c e s o f m a g n i t u d e 2 N , 3 N a n d 5 N a c t o n a p a r t i c l e i n t h e d i r e c t i o n s a n d

r e s p e c t i v e l y . F i n d t h e r e s u l t a n t o f t h e t h r e e f o r c e s i n t e r m s o f a n d .

( 5 m a r k s ) 3 . R e l a t i v e t o a n o r i g i n O , t h e p o s i t i o n v e c t o r s o f t h e v e r t i c e s A , B a n d C o f t h e t r i a n g l e A B C

a r e , a n d r e s p e c t i v e l y . I f A D , B E a n d C F a r e t h e a l t i t u d e s o f t h e t r i a n g l e a n d R i s t h e

i n t e r s e c t i o n p o i n t o f A D a n d B E , s h o w t h a t a n d

. ( 4 m a r k s )

H e n c e , p r o v e t h a t t h e a l t i t u d e s o f a t r i a n g l e a r e c o n g r u e n t . ( 3 m a r k s )

4 . ( a ) A m a n w a n t s t o m e a s u r e t h e h e i g h t o f a t o w e r . H e t a k e s t w o s i g h t i n g o f t h e p e a k o f

t h e t o w e r f r o m p o i n t s A a n d B . P o i n t s A a n d B a r e a t a d i s t a n c e d m e t r e s a p a r t a n d o n a

h o r i z o n t a l l i n e t o t h e f o o t o f t h e t o w e r . T h e t w o a n g l e s o f e l e v a t i o n h e o b t a i n e d a r e a n d

. S h o w t h a t t h e h e i g h t o f t h e t o w e r i s . ( 4 m a r k s )

( b ) A n o t h e r m a n m e a s u r e s t h e h e i g h t o f t h e h i l l b y t a k i n g t h r e e s i g h t i n g o f t h e p e a k o f t h e

h i l l f r o m p o i n t s C, D a n d E. P o i n t s C, D a n d Ea r e o n a s t r a i g h t l i n e a n d f o r m a h o r i z o n t a l

p l a n e w i t h t h e f o o t o f t h e h i l l . I f C D = a, D E= 2a a n d t h e a n g l e o f e l e v a t i o n s f r o m p o i n t s

C, D a n d Ea r e a n d r e s p e c t i v e l y , s h o w t h a t ( 5 m a r k s )

5 . F i g u r e b e l o w s h o w s p o i n t s A , P a n d B o n t h e c i r c u m f e r e n c e o f a c i r c l e . M P N i s a t a n g e n t , A Q B i s a c h o r d o f t h e c i r c l e , a n d

( a ) S h o w t h a t t r i a n g l e P A Mi s s i m i l a r t o t r i a n g l e B P Q . ( 3 m a r k s )

( b ) F i n d a n o t h e r p a i r o f s i m i l a r t r i a n g l e s . ( 3 m a r k s )

( c ) S h o w t h a t P Q2 = A M B N. ( 3 m a r k s )

P

B

A

Q

NM


2/9


---- 2222 ----

---- 2222 ----

6 . T h e p o p u l a t i o n o f a s p e c i e s o f b i r d i n t h e f o r e s t s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n

w h e r e P i s t h e p o p u l a t i o n o f t h e s p e c i e s o f b i r d a n d ti s t h e t i m e i n

y e a r s . A t t h e b e g i n n i n g , t h e r e a r e 1 0 0 0 o f t h a t s p e c i e s o f b i r d s .

( a ) F i n d a p a r t i c u l a r s o l u t i o n o f t h e d i f f e r e n t i a l e q u a t i o n . ( 7 m a r k s )

( b ) W h a t i s t h e p o p u l a t i o n a f t e r 1 0 y e a r s ? ( 2 m a r k s )

( c ) A t w h a t t i m e t h e p o p u l a t i o n i n c r e a s e m o s t r a p i d l y ? ( 5 m a r k s )

( d ) S k e t c h t h e g r a p h o f P a g a i n s t t. ( 2 m a r k s )

7 . A n u m b e r o f d i f f e r e n t t y p e s o f f u n g i a r e d i s t r i b u t e d a t r a n d o m i n a f i e l d . F o r t y p e r c e n t o f

t h e s e f u n g i a r e m u c o r s , a n d t h e r e m a i n d e r s a r e m u s h r o o m s . F i v e p e r c e n t o f t h e

m u s h r o o m s a r e p o i s o n o u s . A w o m a n , w h o c a n n o t d i s t i n g u i s h i n b e t w e e n m u c o r s a n d m u s h r o o m s , w o n d e r s a c r o s s t h e

f i e l d a n d p i c k s a t o t a l o f 2 0 0 f u n g i . D e t e r m i n e , c o r r e c t t o 3 s i g n i f i c a n t f i g u r e s , t h e p r o b a b i l i t y t h a t t h e w o m a n h a s p i c k e d( a ) e x a c t l y s e v e n p o i s o n o u s m u s h r o o m s . ( 2 m a r k s )

( b ) a t l e a s t 1 1 5 m u s h r o o m s , b y u s i n g s u i t a b l e a p p r o x i m a t i o n . ( 4 m a r k s )

8 . T h e d i s c r e t e r a n d o m v a r i a b l e Xc a n o n l y t a k e t h e v a l u e s 1 , 3 , 5 a n d 7 w i t h t h e p r o b a b i l i t i e s

P (X= 1 ) = k, P (X= 3 ) = 0 . 3 , P ( X= 5 ) = 2 k, P (X= 7 ) = 0 . 4

( a ) F i n d t h e v a l u e o f k. ( 2 m a r k s ) ( b ) F i n d E ( 4 X 3 ) a n d V a r ( 4 X 3 ) . ( 5 m a r k s )

9 . I f A a n d B a r e e v e n t s t h a t P ( B) = , P ( ) = a n d P ( B | A) = , c a l c u l a t e P ( A) a n d P (A | B ) w h e r e B i s t h e e v e n t B d o e s n o t o c c u r . ( 5 m a r k s )

S t a t e , w i t h r e a s o n , i f A a n d B a r e( i ) i n d e p e n d e n t o f e a c h o t h e r , ( 1 m a r k )

( i i ) m u t u a l l y e x c l u s i v e . ( 1 m a r k )

1 0 . I n a s t u d y o f w o m e n a b o v e 5 0 y e a r s o l d , i t w a s f o u n d t h a t 2 % o f t h e m s u f f e r f r o m

o s t e o p o r o s i s .

( a ) A s a m p l e o f 1 0 w o m e n i s s e l e c t e d r a n d o m l y . D e t e r m i n e t h e p r o b a b i l i t y t h a t

( i ) e x a c t l y t w o w o m e n ( 2 m a r k s ) ( i i ) a t l e a s t t w o w o m e n ( 3 m a r k s )

s u f f e r f r o m o s t e o p o r o s i s . ( b ) A r a n d o m s a m p l e o f 2 0 0 w o m e n w a s o b t a i n e d . U s i n g a s u i t a b l e a p p r o x i m a t i o n ,

d e t e r m i n e t h e p r o b a b i l i t y t h a t a t m o s t t w o w o m e n s u f f e r f r o m o s t e o p o r o s i s .

( 4 m a r k s )


3/9


---- 3333 ----

---- 3333 ----

1 1 . A c o n t i n u o u s r a n d o m v a r i a b l e Xh a s 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 o n

( a ) S h o w t h a t a n d f i n d t h e v a l u e o f . ( 3 m a r k s )

( b ) F i n d P ( X> ) . ( 2 m a r k s )

( c ) F i n d t h e v a l u e o f m s u c h t h a t P ( X


4/9


---- 4444 ----

---- 4444 ----

N o S c h e m e M a r k s

1 c o s + 2 c o s 2 + c o s 3 = 0

( c o s + c o s 3

o r

.

T h u s

M 1

A 1

M 1 ( a n y t w o )

A 1 a l l c o r r e c t

T o t a l 4

2. . . . . . . . . . . . . . . . . . . . . . . . ( 1 )

. . . . . . . . . . . ( 2 )

T h e t h r e e f o r c e s a r e ) .

R e s u l t a n t f o r c e = )

=

M 1 A 1

E i t h e r ( 1 ) o r ( 2 )

c o r r e c t .

B 1 a n y t w o c o r r e c t

M 1

A 1

T o t a l 5

3 .

. . . . . . . . . . . . . . ( 1 )

S i m i l a r l y ,

. . . . . . . . . . . . . . ( 2 )

( 1 ) + ( 2 )

C R i s p e r p e n d i c u l a r t o A B .

M 1A 1

A 1

A 1

M 1

A 1

A 1 T o t a l 7


5/9


---- 5555 ----

---- 5555 ----

A l t i t u d e s o f a t r i a n g l e a r e c o n g r u e n t

4 a .

= 5o

T h e h e i g h t o f t h e t o w e r

=

=

B 1

M 1 U s i n g S i n e R u l e

M 1

A 1 S u b t o t a l 4

4 ( b )

F r o m , c o t . . . . . . . . . . . . . . . . . . ( 1 )

F r o m , c o t . . . . . . . . . . . . . . . ( 2 )

F r o m , c o t . . . . . . . . . . . . ( 3 )

( 1 ) = ( 2 )

E F = C F a n d . . . . . . . . . . . ( 3 )

U s i n g c o s i n e r u l e f o r a n d :

. . . . . . . . . . . ( 4 )

. . . . . . . . . . . . . . . ( 5 )

B 1 ( 1 ) o r ( 2 ) o r ( 3 )

c o r r e c t

B 1

M 1 ( 4 ) o r ( 5 ) c o r r e c t

F

h

P e a k o f t h e h i l l , P2

2 a aE

DC

d

P e a k o f t h e t o w e r , P1

AB


6/9


---- 6666 ----

---- 6666 ----

B u t f r o m ( 3 ) E F = C F a n d ,

S o , f r o m ( 5 ) w e g e t . . . . . . . . ( 6 )

2 x ( 6 ) ( 4 ) ,

. . . . . . . . . . ( 7 )

S u b s t i t u t e c o t f r o m ( 1 ) a n d c o t f r o m ( 3 ) i n t o ( 7 ) :

M 1 s o l v i n g

e q u a t i o n s ( 4 ) a n d ( 5 )

T o e l i m i n a t e

A 1 S u b t o t a l 5

T o t a l 9

5 ( a )

5 ( b )

5 ( c )

( A n g l e i n t h e a l t e r n a t e s e g m e n t )

( G i v e n )

( R e m a i n i n g a n g l e s a r e e q u a l )

S o , t r i a n g l e P A Mi s s i m i l a r t o t r i a n g l e B P Q

( A t l e a s t 2 c o r r e c t r e a s o n s ; o t h e r w i s e N W W - 1 )

( A n g l e i n t h e a l t e r n a t e s e g m e n t )

( A n g l e s i n a s t r a i g h t l i n e )

( R e m a i n i n g a n g l e s a r e e q u a l )

A n o t h e r p a i r o f s i m i l a r t r i a n g l e s i s t r i a n g l e s P B Na n d A P Q.

( A t l e a s t 2 c o r r e c t r e a s o n s ; o t h e r w i s e N W W - 1 )

( T r i a n g l e P A Mi s s i m i l a r t o t r i a n g l e B P Q . )

( T r i a n g l e s P B Na n d A P Q a r e s i m i l a r . )

T h u s , = P Q2 = A M B N

B 1

B 1

O p t i o n a l

B 1

S u b t o t a l 3

B 1

B 1

O p t i o n a l

B 1

S u b t o t a l 3

B 1

B 1

B 1

S u b t o t a l 3

T o t a l 9


7/9


---- 7777 ----

---- 7777 ----

6 ( a )

6 ( b )

6 ( c )

L e t

5 0 0 0

T h u s ,

. . . . . . . . . . . . . . . . . . . . . . . ( 1 )

. . . . . . . . . . . . . . . . . . . . . . . . . . . . ( 2 )

. . . . . . . . . . . . . . . . . . . . . . . . . . . ( 3 )

W h e n t = 1 0 , f r o m ( 3 ) ,

= 2 0 2 3

B 1

M 1

A 1

A 1 W i t h o r

w i t h o u t a b s o l u t e

s i g n

M 1 E l i m i n a t e l n

M 1 s u b s t i t u t i o n

A 1 ( 1 ) o r ( 2 ) o r ( 3 )

S u b t o t a l 7

M 1 ( h i s v a l u e )

A 1

S u b t o t a l 2

M 1A 1


8/9


---- 8888 ----

---- 8888 ----

6 ( d )

, p = 2 5 0 0

t h e r a t e i s h i g h e s t w h e n p = 2 5 0 0

S e c o n d m e t h o d :

C o m p l e t i n g t h e s q u a r e :

T h u s , t h e r a t e i s h i g h e s t w h e n p = 2 5 0 0

S u b s t i t u t e p = 2 5 0 0 i n t o ( 1 )

t = 1 0 l n 4 o r t = 1 3 . 9 y r s

A 1

M 1 A 1

A 1

M 1

A 1S u b t o t a l 5

D 1 C o r r e c t s h a p e

D 1 a s y m p t o t e

p= 5 0 0 0

S u b t o t a l 2

T o t a l 1 6

7 ( a )

7( b )

P ( X = 7 ) = 2 0 0C7 ( 0 . 0 3 ) 7 ( 0 . 9 7 ) 1 9 3

= 0 . 1 4 0

= P(X> 1 1 4 . 5 )

= >= P ( z > - 0 . 7 9 3 9 ) = 1 P ( z > 0 . 7 9 3 9 )

= 0 . 7 8 6 5

M 1

A 1

M 1

M 1

M 1

A 1 T o t a l 6

8 ( a ) k + 0 . 3 + 2 k + 0 . 4 = 1 k = 0 . 1

M 1A 1


9/9


---- 9999 ----

---- 9999 ----

8 ( b ) E ( X ) = 1 ( 0 . 1 ) + 3 ( 0 . 3 ) + 5 ( 0 . 2 ) + 7 ( 0 . 4 )

= 4 . 8

E ( 4 X 3 ) = 4 ( 4 . 8 ) 3

= 1 6 . 2 E ( X2) = 2 7 . 4

V a r ( X ) = 4 . 3 6 V a r ( 4 X 3 ) = 4 2( 4 . 3 6 ) = 6 9 . 7 6

M 1

A 1

M 1M 1A 1 T o t a l 7

9 P (A) =

=

=

P (A | B ) =

=

=

( i ) P ( B | A) = , P ( B) =

P (B | A) P ( B ) , A a n d B a r e n o t i n d e p e n d e n t

( i i ) P (A B) = 0 . A a n d B a r e n o t m u t u a l l y e x c l u s i v e

M 1

A 1

M 1 ( e i t h e r ) M 1 ( b o t h )

A 1

B 1

B 1

T o t a l 7

1 0 . X= n u m b e r o f w o m e n s u f f e r e d f r o m o s t e o p o r o s i s ( a ) X B( 1 0 , 0 . 0 2 )

( i ) P ( X= 2 ) = 1 0C2 ( 0 . 0 2 ) 2 ( 0 . 9 8 ) 8

= 0 . 0 1 5 3

P (X 2 ) = 1 P ( X = 0 ) P ( X = 1 )

= 1 ( 0 . 9 8 ) 1 0 1 0C1 ( 0 . 0 2 ) ( 0 . 9 8 ) 9

= 0 . 0 1 6 2 ( b ) X

n p = 2 0 0 x 0 . 0 2 = 4

X P o( 4 )P ( X 2 ) = P ( X= 0 ) + P ( X= 1 ) + P ( X= 2 )

=

= 0 . 2 3 8 1

M 1

A 1

B 1M 1

A 1

B 1

B 1

M 1

A 1

T o t a l 9

melaka trial stpm 2012-mathst paper 2(q&a)

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