sabah trial stpm 2011-mathst paper 2(q&a)
TRANSCRIPT
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J A B A T A N J A B A T A N J A B A T A N J A B A T A N P E L A J A R A N P E L A J A R A N P E L A J A R A N P E L A J A R A N N E G E R I N E G E R I N E G E R I N E G E R I S A B A H S A B A H S A B A H S A B A H
E X C E L E X C E L E X C E L E X C E L T I N G K A T A N T I N G K A T A N T I N G K A T A N T I N G K A T A N 6 666 A T A S A T A S A T A S A T A S 2 0 1 1 2 0 1 1 2 0 1 1 2 0 1 1 ( S T P M ( S T P M ( S T P M ( S T P M 2 0 1 1 ) 2 0 1 1 ) 2 0 1 1 ) 2 0 1 1 )
S I J I L S I J I L S I J I L S I J I L T I N G G I T I N G G I T I N G G I T I N G G I P E R S E K O L A H A N P E R S E K O L A H A N P E R S E K O L A H A N P E R S E K O L A H A N M A L A Y S I A M A L A Y S I A M A L A Y S I A M A L A Y S I A
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
T h i s T h i s T h i s T h i s q u e s t i o n q u e s t i o n q u e s t i o n q u e s t i o n p a p e r p a p e r p a p e r p a p e r c o n s i s t s c o n s i s t s c o n s i s t s c o n s i s t s o f o fo fo f 5555 p r i n t e d p r i n t e d p r i n t e d p r i n t e d p a g e s . p a g e s . p a g e s . p a g e s .
( K e r t a s ( K e r t a s ( K e r t a s ( K e r t a s s o a l a n s o a l a n s o a l a n s o a l a n i n i i n ii n ii n i t e r d i r i t e r d i r i t e r d i r i t e r d i r i d a r i p a d a d a r i p a d a d a r i p a d a d a r i p a d a 5 555 h a l a m a n h a l a m a n h a l a m a n h a l a m a n b e r c e t a k . ) b e r c e t a k . ) b e r c e t a k . ) b e r c e t a k . )
I n s t r u c t i o n s I n s t r u c t i o n s I n s t r u c t i o n s I n s t r u c t i o n s t o t ot ot o c a n d i d a t e s : c a n d i d a t e s : c a n d i d a t e s : c a n d i d a t e s :
A n s w e r a l la l la l la l l q u e s t i o n s . A n s w e r s m a y b e w r i t t e n i n e i t h e r E n g l i s h o r M a l a y .
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H J A B A T A N P E L A J A R A N S A B A H
9 5 4 / 2 9 5 4 / 2 9 5 4 / 2 9 5 4 / 2 S T P M S T P M S T P M S T P M
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
T h r e e T h r e e T h r e e T h r e e H o u r s H o u r s H o u r s H o u r s (((( T i g a T i g a T i g a T i g a j a m ) j a m )j a m )j a m )
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A l l n e c e s s a r y w o r k i n g s h o u l d b e s h o w n c l e a r l y .
N o n - e x a c t n u m e r i c a l a n s w e r s m a y b e g i v e n c o r r e c t t o t h r e e s i g n i f i c a n t f i g u r e s , o r o n e
d e c i m a l p l a c e i n t h e c a s e o f a n g l e s i n d e g r e e s , u n l e s s a d i f f e r e n t l e v e l o f a c c u r a c y i s s p e c i f i e d
i n t h e q u e s t i o n .
M a t h e m a t i c a l t a b l e s , a l i s t o f m a t h e m a t i c a l f o r m u l a e a n d g r a p h p a p e r a r e p r o v i d e d .
A r a h a n A r a h a n A r a h a n A r a h a n k e p a d a k e p a d a k e p a d a k e p a d a c a l o n : c a l o n : c a l o n : c a l o n :
J a w a b s e m u a s e m u a s e m u a s e m u a s o a l a n . J a w a p a n b o l e h d i t u l i s d a l a m b a h a s a I n g g e r i s a t a u b a h a s a M e l a y u .
S e m u a k e r j a y a n g p e r l u h e n d a k l a h d i t u n j u k k a n d e n g a n j e l a s .
J a w a p a n b e r a n g k a t a k t e p a t b o l e h d i b e r i k a n b e t u l h i n g g a t i g a a n g k a b e r e r t i , a t a u s a t u t e m p a t
p e r p u l u h a n d a l a m k e s s u d u t d a l a m d a r j a h , k e c u a l i a r a s k e j i t u a n y a n g l a i n d i t e n t u k a n d a l a m
s o a l a n .
S i f i r m a t e m a t i k , s e n a r a i r u m u s m a t e m a t i k , d a n k e r t a s g r a f d i b e k a l k a n .
J a b a t a n P e l a j a r a n N e g e r i S a b a h 2 0 1 1
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1111 E a c h s i d e o f a u n i f o r m t e t r a h e d r o n A B C D h a s a l e n g t h o f 4 a. T h e p o i n t P l i e s
o n B C s u c h t h a t B P = 3 a. F i n d t h e c o s i n e o f a n g l e A P D. S h o w t h a t t h e
p e r p e n d i c u l a r d i s t a n c e o f t h e v e r t e x A f r o m t h e p l a n e B C D i s , a n d h e n c e
s h o w t h a t t h e a n g l e t h a t t h e l i n e A P m a k e s w i t h t h e p l a n e B C D i s .
[ 1 0 m a r k s ]
2222
F i n d t h e g e n e r a l s o l u t i o n o f . [ 6 m a r k s ]
3333
D i a g r a m 1
D i a g r a m 1 s h o w s a t a n g e n t o f l i n e A B t o u c h e s t w o c i r c l e a t p o i n t M . G i v e n
a l s o l i n e P R t o u c h e s t h e s m a l l e r c i r c l e a t p o i n t Q . M N i s t h e c h o r d o f t h e
s m a l l e r c i r c l e e x t e n d e d t o p o i n t P , w h e r e a s p o i n t P t o u c h e s t h e c i r c u m f e r e n c e
o f t h e b i g g e r c i r c l e . R M i s t h e c h o r d o f t h e b i g g e r c i r c l e a n d Q M i s c h o r d f o r
t h e s m a l l e r c i r c l e . P r o v e t h a t
( a ) B M Q = P Q N + Q P N ,
( b ) R M Q = P Q N ,
( c ) R M Q = Q M P .
[ 4 m a r k s ]
[ 3 m a r k s ]
[ 3 m a r k s ]
4444 T h e p o s i t i o n v e c t o r s o f t h e p o i n t s P , Q a n d R a r e xxxx + yyyy, 3xxxx 3yyyy a n d 4 xxxx + kyyyy.
F i n d t h e v a l u e o f ki f
( a ) P , Q a n d R a r e c o l l i n e a r ,
( b ) t h e l i n e P Q i s p e r p e n d i c u l a r t o t h e l i n e Q R a n d .
[ 4 m a r k s ]
[ 5 m a r k s ]
BA M
P
Q
R
N
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5555 E x p r e s s 3 s i n x + 4 c o s x i n t h e f o r m r s i n ( x + ) w h e r e r i s p o s i t i v e a n d
i s a n a c u t e a n g l e , g i v i n g t h e v a l u e o f t o t h e n e a r e s t o1.0 . H e n c e , f i n d t h e
m i n i m u m a n d m a x i m u m v a l u e o f f o r t h e r a n g e o f
.
[ 6 m a r k s ]
6666 A c o m p a n y h a s t h r e e p r i n t i n g m a c h i n e s , A , B a n d C . M a c h i n e A a c c o u n t s f o r
3 7 % o f t h e t o t a l o u t p u t a n d 0 . 6 % o f o u t p u t f r o m m a c h i n e A i s s p o i l t . M a c h i n e
B a c c o u n t s f o r 4 2 % o f t h e t o t a l o u t p u t a n d 0 . 4 % o f o u t p u t f r o m B i s s p o i l t .
M a c h i n e C i s k n o w n t o h a v e s p o i l t r a t e 1 . 2 % . A c o p y i s f o u n d t o b e s p o i l t .
F i n d t h e p r o b a b i l i t y t h a t t h e c o p y i s f r o m
( a ) m a c h i n e A ,
( b ) m a c h i n e B o r m a c h i n e C .
[ 3 m a r k s ]
[ 3 m a r k s ]
7777 T h e r a t e o f d e c a y o f a r a d i o a c t i v e m a t e r i a l i s p r o p o r t i o n a l t o t h e m a s s o f t h e
m a t e r i a l r e m a i n i n g g i v e n b y . I f o n e t h i r d o f t h e m a s s o f t h e
m a t e r i a l i s l e f t a f t e r 1 2 y e a r s , f i n d t h e p e r c e n t a g e o f t h e m a t e r i a l t h a t w i l l
r e m a i n a f t e r 2 0 y e a r s . [ 8 m a r k s ]
8888 I t i s f o u n d t h a t t h e n u m b e r o f w o r k e r s t a k e a d a y o f f i n a c e r t a i n d e p a r t m e n t
f o l l o w s a 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 . 5 . A s s u m i n g t h a t t h e w o r k e r s
t a k e a d a y o f f i s i n d e p e n d e n t e a c h o t h e r , c a l c u l a t e :
( a ) t h e p r o b a b i l i t y t h a t a t l e a s t 4 w o r k e r s t a k e a d a y o f f o n a p a r t i c u l a r d a y ,
( b ) t h e l e a s t n u m b e r o f c o n s e c u t i v e w o r k i n g d a y s i f t h e p r o b a b i l i t y o f a t l e a s t a
w o r k e r t a k i n g t h e d a y s o f f e x c e e d s 0 . 9 9 9 9 .
[ 3 m a r k s ]
[ 5 m a r k s ]
9999 T h e r a n d o m v a r i a b l e Xi s d i s t r i b u t e d b i n o m i a l l y w i t h t h e p a r a m e t e r n = 1 0 a n d
p = 0 . 5 w h i l e Yi s a l s o d i s t r i b u t e d b i n o m i a l l y w i t h t h e p a r a m e t e r n = 2 0 a n d
p = a. F i n d t h e v a l u e o f a i f P ( X= 0 ) = P ( Y= 0 ) .
[ 5 m a r k s ]
1 01 01 01 0 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 Xt a k e s t h e v a l u e kw i t h t h e p r o b a b i l i t y
, w h e r e c i s a c o n s t a n t .
( a ) D e t e r m i n e t h e v a l u e o f c,
[ 2 m a r k s ]
[ 1 m a r k s ]
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( b ) T a b u l a t e t h e v a l u e s o f ,
( c ) C a l c u l a t e t h e m e a n a n d v a r i a n c e o f X,
( d ) S k e t c h t h e g r a p h o f t h i s f u n c t i o n .
[ 3 m a r k s ]
[ 2 m a r k s ]
1 11 11 11 1 E v e r y m o r n i n g C h o n g s j o u r n e y t o s c h o o l i n v o l v e s a w a l k , t h e n a b u s r i d e .
T h e t i m e f o r w a l k i n g , w a i t i n g a t t h e b u s s t o p a n d t h e b u s t r i p a r e a l l n o r m a l l y
d i s t r i b u t e d w i t h m e a n t i m e s o f 8 m i n u t e s , 7 m i n u t e s a n d 1 0 m i n u t e s
r e s p e c t i v e l y . T h e i r c o r r e s p o n d i n g v a r i a n c e s a r e 1 , 2 a n d 5 r e s p e c t i v e l y .
( a ) W h a t i s t h e p r o b a b i l i t y t h a t t h e t o t a l j o u r n e y t a k e s m o r e t h a n 3 0 m i n u t e s ?
( b ) F i n d t h e p r o b a b i l i t y t h a t t h e t i m e s p e n t o n w a i t i n g a t t h e b u s s t o p i s g r e a t e r
t h a n t h e t i m e s p e n t i n t h e b u s .
[ 4 m a r k s ]
[ 4 m a r k s ]
1 21 21 21 2 T h e f o l l o w i n g d a t a s h o w t h e m a s s e s , i n g m o f m a n g o e s h a r v e s t e d i n a
p a r t i c u l a r d a y .
D i a g r a m 2
( a ) F i n d t h e m e a n , m e d i a n a n d m o d e ,
( b ) F i n d t h e s t a n d a r d d e v i a t i o n ,
( c ) F i n d t h e f i r s t a n d t h i r d q u a r t i l e s a n d t h e s e m i - i n t e r q u a r t i l e r a n g e ,
( d ) 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 o n D i a g r a m 2 ,
( e ) C o m m e n t o n 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 .
[ 5 m a r k s ]
[ 3 m a r k s ]
[ 5 m a r k s ]
[ 2 m a r k s ]
[ 1 m a r k s ]
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1
K n o w n t h a t B C D i s a n e q u i l a t e r a l t r i a n g l e , B C D = 6 0o.
=
=
T h e r e f o r e ,
=
B e c a u s e
a n d
=
=
D 1
M 1
M 1
A 1
B 1
M 1
4a
4a
4a
4a
A
D
B
C
N
PO
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=
=
=
=
= ( s h o w e d )
T h e a n g l e t h a t t h e l i n e A P m a k e s w i t h t h e p l a n e B C D i s A P O =
=
=
( s h o w e d )
M 1
A 1
M 1
A 1
2
B 1
A 1 + M 1
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, k=
M 1
M 1
A 1
3
D i a g r a m 1
( a ) B M Q = P Q N + Q P N
B M Q = M N Q ( a l t e r n a t e a n g l e o f c i r c u m f e r e n c e )
M N Q = Q P N + P Q N ( e x t e r i o r a n g l e e q u a l t o s u m o f t w o o p p o s i t e i n t e r i o r
a n g l e s o f a t r i a n g l e )
B M Q = Q P N + P Q N ( p r o v e n )
( b ) R M Q = P Q N
R M Q = B M Q - B M R ( c o m p l e m e n t a r y a n g l e )
K n o w n t h a t B M A i s t h e t a n g e n t t o c i r c l e P M R
B M R = M P R ( a l t e r n a t e a n g l e o f c i r c u m f e r e n c e )
K n o w n t h a t M P R = N P Q
R M Q = P Q N + N P Q - N P Q
D 1
B 1
M 1
A 1
B 1
M 1
R
Q
P
N
BA M
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R M Q = P Q N ( p r o v e n )
( c ) R M Q = Q M P .
P R i s t h e t a n g e n t t o t h e c i r c l e M Q N
P Q N = Q M N ( a l t e r n a t e a n g l e o f c i r c u m f e r e n c e )
K n o w n t h a t Q M N = Q M P
F r o m ( b ) R M Q = Q M P ( p r o v e n )
A 1
B 1
M 1
A 1
4 ( a ) O PO PO PO P = x + y O QO QO QO Q = 3 x 3 y O RO RO RO R = 4 x + ky
P QP QP QP Q = mQ RQ RQ RQ R
O QO QO QO Q O PO PO PO P ==== m( O R( O R( O R( O R O Q )O Q )O Q )O Q )3 x 3 y - ( x + y ) = m [ 4 x + ky ( 3 x 3 y ) ] 2 x 4 y = m x + m ( k+ 3 ) y C o m p a r e : 2 = m
- 4 = m ( k+ 3 )- 4 = 2 ( k+ 3 )
- 2 = k+ 3k= - 5
( b ) I f P Q i s p e r p e n d i c u l a r t o Q R , t h e n
P QP QP QP Q Q RQ RQ RQ R = 0
[ 2 x 4 y ] [ x + ( k + 3 ) y ] = 0 K n o w n t h a t x y = 0 , x x = 1 , y y = 1
2 x x 4 ( k + 3 ) y y = 02 4 ( k + 3 ) = 0 2 4 k 3 = 0 - 4 k = 1
k =
B 1
M 1
M 1
A 1
B 1
A 1 + M 1
M 1
A 1
5 3 s i n x + 4 c o s x = r s i n x c o s + r c o s x s i n r c o s = 3 , r s i n = 4
r = 5 o r 42
43 +
t a n =3
4
= 5 3 . 1 o
H e n c e ,
3 s i n x + 4 c o s x = 5 s i n ( x + 5 3 . 1 o
)
6 s i n x + 8 c o s x + 5 = 1 0 s i n ( x + 5 3 . 1 o
) + 5
- 1 1 0 s i n ( x + 5 3 . 1 o
) + 5 1 1
T h e r e f o r e , M a x v a l u e = 1 1
B 1
M 1
M 1
A 1
M 1
A 1
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M i n v a l u e = - 1
6
( a )
= 0 . 0 0 6 4 2
=
=
( b )
=
=
B 1
M 1
A 1
B 1
M 1
A 1
7
G i v e n
, A = c o n s t a n t
I f
W h e n
B 1
M 1
M 1
M 1
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W h e n t= 2 0
T h e r e f o r e , t h e p e r c e n t a g e o f t h e m a t e r i a l l e f t a f t e r 2 0 y e a r s i s 1 6 . 0 % .
A 1
M 1
A 1
B 1
8 L e t X b e t h e n u m b e r o f w o r k e r s t a k i n g a d a y o f f , .
( a )
= 1 -
= 0 . 2 4 2 4
( b )L e t t h e n u m b e r o f c o n s e c u t i v e d a y s o f f b e n a n d t h e n u m b e r o f w o r k e r s t a k i n g
d a y s o f f b e a s Y , .
B 1
M 1
A 1
B 1
B 1
M 1
M 1
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n = 4
A 1
9
= - 0 . 1 5 0 5
1 a = 0 . 7 0 7 1
a = 0 . 2 9 2 9 = 0 . 2 9 3
B 1 + M 1
M 1
M 1
A 1
1 0 ( a )
( b )
X= k 0 1 2
P (X= k)
( c )
=
=
=
M 1
A 1
A 1
M 1
M 1
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= 1
=
=
( d )
A 1
D 1A 1
1 1 L e t , a n d ( a )L e t T = A + B + C
= 8 + 7 + 1 0
= 2 5
= 1 + 2 + 5 = 8
== 0 . 0 3 8 5
( b )
N e w
= 7 1 0 = 3
= 2 + 5
B 1
B 1
M 1
A 1
B 1
0
1
P ( X = k )
X
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= 7
== 0 . 1 2 8 4 = 0 . 1 2 8
M 1
M 1
A 1
1 2( a )
M e a n =
=
= 5 6 . 2 9 = 5 6 . 3
M e d i a n =
=
= 5 3
M o d e = 5 2
( b )
=
= 1 7 . 4 7 5
= 1 7 . 5
( c )
= 4 4
M 1
A 1
M 1
A 1
A 1
M 1 + M 1
A 1
M 1
M 1
A 1
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7/31/2019 Sabah Trial Stpm 2011-Mathst Paper 2(Q&A)
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