Kedah Trial Stpm 2012-Mathst Paper 2(Q&A)
Transcript of Kedah Trial Stpm 2012-Mathst Paper 2(Q&A)
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S T P M K E D A H 2 0 1 2 - M T P A P E R 2 C H U / S M K K
1 P r o v e t h a t ( ) ( ) A A A A 3 s i n
4 1
6 0 s i n 6 0 s i n s i n = + . [ 4 m ]
2 G i v e n A B
= ( 2 k 1 ) i i i i j j j j a n d A C
= 6 i i i i + k j j j j a r e t w o v e c t o r s l i e o n a s a m e
p l a n e . F i n d t h e v a l u e o f k
( a ) i f t h e p o i n t s A , B a n d C a r e c o l l i n e a r ,
( b ) i f B A C = 9 0 o . [ 4 m ]
3 T h e d i a g r a m s h o w s a c i r c l e . P Q i s a t a n g e n t t o t h e c i
s t r a i g h t l i n e P R c u t s t h e c i r c l e a t S . S h o w t h a t
( a ) i f R Q S = Q P S , t h e n t h e l e n g t h o f c h o r d Q R i s t w i c e to f t h e c i r c l e ,
( b ) i f R Q S = 3 0 o , t h e n t h e l e n g t h o f c h o r d R S i s e q u a l t o t h c i r c l e .
[ 3 m ]
[ 3 m ]
4 F i n d t h e g e n e r a l 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 n2 5 0 .
d y x y y
d x + =
D e d u c e t h a t t h e p a r t i c u l a r s o l u t i o n f o r w h i c h y = 1 w he x p r e s s e d i n t h e f o r m a n d s k e t c h t h e c u r v e .
S k e t c h , i n s e p a r a t e d i a g r a m , t h e s o l u t i o n c u r v e w h i c
p o i n t ( 1 , 0 ) .
[ 3 m ]
[ 2 m ]
[ 3 m ]
5 I n a p a r a l l e l o g r a m A B C D , X i s t h e m i d p o i n t o f A B a nd i a g o n a l A C a t P .
G i v e n t h a t A B = a , a , a , a , A D = = = = b , b , b , b , A P = = = = A C a n d D P = D X , e x p r e s s A P i n t e r m s o f ( a ) , a a a a a n d b b b b . ( b ) , a a a a a n d b b b b . H e n c e , d e d u c e t h a t P i s a p o i n t o f t r i s e c t i o n o f b o t h A
[ 2 m ] [ 3 m ] [ 5 m ]
6 E x p r e s s ( c o s x + 1 ) 2 + ( s i n x + 3 ) 2 i n t h e f o r m a + b c o s ( x ) , w h e r e a >
0 , b > 0 a n d 0 < < 1 2
. H e n c e ,
( a ) 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 f o l l o w i n g e
( c o s x + 1 ) 2 + ( s i n x + 3 ) 2 = k 2 ,
[ 4 m ]
[ 3 m ]
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( b ) s o l v e , f o r 0 < x < 2 , t h e e q u a t i o n
( c o s x + 1 ) 2 + ( s i n x + 3 ) 2 = 5 + 2 2 ,
( c ) f i n d t h e v a l u e s o f p a n d q s u c h t h a t 2 ( c o s x + 1 ) 2 + ( s i n x + 3 ) 2
l i e s i n
t h e i n t e r v a l ( p , q ) a n d t h e c o r r e s p o n d i n g v a l u e s o f x .
[ 3 m ]
[ 5 m ]
7 T h e r e s u l t o f a v e t e r i n a r y s u r v e y o n 1 2 5 r a b b i t s , e a cp a r t i c u l a r d i s e a s e i s s u m m a r i z e i n t h e f o l l o w i n g t a b l
D i s e a s e d N o t D i s e a s e d T o t a l F e m a l e 2 5 3 5 6 0 M a l e 2 0 4 5 6 5 T o t a l 4 5 8 0 1 2 5
T h e c a s e h i s t o r y o f e a c h r a b b i t w a s k e p t o n a s e p t h r e e d i f f e r e n t r e c o r d c a r d s a r e s e l e c t e d a t r a n d o m
c a l c u l a t e t h e p r o b a b i l i t y t h a t
( a ) a l l t h r e e r e c o r d c a r d s r e l a t e t o r a b b i t s w i t h t h e ( b ) o n e r e c o r d c a r d r e l a t e s t o a f e m a l e r a b b i t w i t h
m a l e r a b b i t w i t h t h e d i s e a s e a n d o n e t o a f e m a l e r a
t h e d i s e a s e .
[ 3 m ]
[ 3 m ]
8 T h e p r o b a b i l i t y t h a t a s h o o t e r s t r i k e s a t a r g e t i n o n
p r o b a b i l i t y t h a t n o t l e s s t h a n t h r e e o f f i v e s h o t s f i r e d ,
A f t e r t h e s h o o t e r h a s c o m p l e t e d a o n e m o n t h s h
p r o b a b i l i t y t h a t h e s t r i k e s t h e t a r g e t i n o n e s h o t i s 0 .
a p p r o x i m a t i o n , f i n d t h e p r o b a b i l i t y t h a t i n a s a m p l e o
s h o t s s t r i k e t h e t a r g e t .
[ 3 m ]
[ 4 m ]
9 T h r e e r e d b a l l s a n d t h r e e w h i t e b a l l s a r e p l a c e d i n a
b y o n e , a t r a n d o m a n d w i t h o u t r e p l a c e m e n t . T h e r a
n u m b e r o f w h i t e b a l l s d r a w n b e f o r e t h e f i r s t r e d b a l l i
( a ) S h o w t h a t P ( X = 1 ) = 1 0 3 , a n d f i n d t h e r e s t o f t h e p r o b a b i l i t y d
f u n c t i o n o f X .
( b ) F i n d E ( X ) a n d s h o w t h a t V a r ( X ) = 8 0 6 3
.
[ 3 m ]
[ 4 m ]
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1 0 T h e t a b l e s h o w s t h e d u r a t i o n s o f 6 0 j o u r n e y s o n t ht h e v a r i a t i o n s i n j o u r n e y t i m e s b e i n g c a u s e d b y v a r y
C a l c u l a t e , t o t h e n e a r e s t m i n u t e , e s t i m a t e s o f t h e d e v i a t i o n f o r t h e d u r a t i o n o f t h e j o u r n e y s .
W h e n t h e t i m e s f o r 4 0 o t h e r j o u r n e y s w e r e t a k e n , i t wa n d s t a n d a r d d e v i a t i o n f o r t h e t i m e s o f t h e s e 4 0 j o u
a n d 1 8 m i n , r e s p e c t i v e l y . F i n d a l s o , t o t h e n e a r e s t mm e a n a n d s t a n d a r d d e v i a t i o n f o r t h e d u r a t i o n o f a l l 1
T i m e T i m e T i m e T i m e o f o f o f o f j o u r n e y j o u r n e y j o u r n e y j o u r n e y i n i n i n i n h o u r s h o u r s h o u r s h o u r s
N u m b e r N u m b e r N u m b e r N u m b e r o f o f o f o f j o u r n e y s j o u r n e y s j o u r n e y s j o u r n e y s
4 . 6 4 . 8 2 4 . 8 5 . 0 7 5 . 0 5 . 2 1 6 5 . 2 5 . 4 2 1 5 . 4 5 . 6 1 2 5 . 6 5 . 8 2
[ 4 m ]
[ 5 m ]
1 1 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 i s t h e d i s t a n c e ,
k i l o m e t e r s , t h a t a p a r t i c u l a r c a r w i l l t r a v e l o n a f u l l t
t h a t
= , 1
, 2 4 1 0
, 0
) ( 2 x x x X P a a
w h e r e a i s a c o n s t a n t . S h o w t h a t a = 1 . ( a ) 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 .
( b ) S h o w t h a t E ( X ) = 3
1 3 .
( c ) T h r e e i n d e p e n d e n t o b s e r v a t i o n s o f X a r e t a k e n .
t w o o f t h e o b s e r v a t i o n s a r e l e s s t h a n 4 . 5 a n d o n e i s g
[ 2 m ]
[ 3 m ]
[ 3 m ]
1 2 A f a c t o r y p r o d u c e s b o t h m e t a l r o d s a n d t u b e s . T h e
o f a m e t a l t u b e i s d i s t r i b u t e d N ( 2 . 2 , 0 . 0 0 0 9 ) . T h e d i a
r o d i s d i s t r i b u t e d N ( 2 . 1 5 , 0 . 0 0 0 4 ) .
( a ) F i n d t h e p e r c e n t a g e o f t u b e s w i t h i n t e r n a l d i a m e( b ) F i n d t h e p e r c e n t a g e o f r o d s w i t h d i a m e t e r g r e a tt h a n 2 . 2 c m . ( c ) A r o d a n d a t u b e a r e c h o s e n a t r a n d o m . F i n d t h e
w i l l n o t p a s s t h r o u g h t h e t u b e .
( d ) T h r e e p a c k e t s , e a c h c o n t a i n i n g a r o d a n d a t u b e
F i n d t h e p r o b a b i l i t y t h a t t w o p a c k e t s c o n t a i n r o d s t h
r e s p e c t i v e t u b e s a n d t h e o t h e r w i l l n o t .
[ 3 m ]
[ 3 m ]
[ 5 m ]
[ 3 m ]
x
x
x
5
, 5 4
, 4
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S T P M K E D A H 2 0 1 2 - M T P A P E R 1 - M A R K I N G S C H E M E
1 . 2 x = 3 y
2 x = 3 1 x
x l n 2 = ( 1 x ) l n 3
x ( l n 2 + l n 3 ) = l n 3
x = l n 3 l n 6
.
2 .
3 .
2 s e c = d d x
,
t a n 0 = 0 , t a n 4
= 1
( 1 + x 2 ) 2 = ( 1 + t a n 2 ) 2 = s e c 4
( ) =
+ 4
0
2
4
1
0 2 2 s e c
s e c
1
1
1
d x x
d
= 4 0 2 c o s
d
( ) ( ) + = +
1
0
4
0 2 2
2 c o s 1 2 1
1
1
d d x x
= 1 2
4
0
1 s i n 2
2
+
B 1
M 1 A 1
A 1
B 1
M 1
M 1 A 1
M 1
A 1
A 1
M 1
B 1
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= 1 8
[ + 2 ] .
4 .
5 . c o s 3 = x y a x
c o s 3 = y x a x
d x d y
x x y 3 2 ) ) ( 3 ( + = a x a s i n
d x
d y x y x 3 2 3 + = a x a s i n
( ) ( ) ( ) ( ) a x a x d x d y
d x y d
x x y d x d y
x c o s 3 2 3 2 2 2 2
3 2 = + +
+
a x a d x d y
x d x
y d x x y
d x d y
x c o s 3 6 3 2 2 2 2
3 2 = + + +
a x a d x y d
x x y d x d y
x c o s 6 6 2
2
2 3 2
= + +
S u b i n t o
3 2 2
2 3 2 6 6 y x a
d x y d
x x y d x d y
x = + +
0 6 6 3 2 2
2 3 2 = + + + y x a
d x
y d x x y
d x
d y x
D 1 : x y =
D 1 : s h a p e o f 2 = x y
D 1 : p o i n t s ( 2 , 0 ) , ( 0 , 2
T h e s o l u t i o n s e t i s } { < < x x x , 4 1 : B 1
A 1
1
2
1 2
M 1 A 1
M 1 A 1
M 1
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( ) 0 6 6 2 2 2 2
2 =
+ + + y x a y
d x d y
x d x
y d x x
( ) 0 6 6 2 2 2 2
2 = + + + y x a d x d y
x d x
y d x
6 .
L e t ( ) ( ) 4 :
4 :
2
1
+ = =
x m y l
x m y l
i n t e r c e p t 2 5 5 4 = + y x a t p o i n t F ( x 1 , y 1 ) a n d G ( x 2 , y 2 ) .
( ) [ ]
( )
m m
x
m m x
m m x x
x m x
5 4 2 0 2 5
2 0 2 5 5 4
2 5 2 0 5 4
2 5 4 5 4
1
1
1 1
1 1
+ + =
+ = + = + = +
+ + = 4
5 4 2 0 2 5
1 m m
m y
m m y 5 4
9 1 +
=
+ + +
m m
m m
F 5 4
9 ,
5 4 2 0 2 5
( ) [ ]
( )
m m x
m m x
m m x x
x m x
5 4 2 0 2 5
2 0 2 5 5 4
2 5 2 0 5 4
2 5 4 5 4
2
2
2 2
2 2
+ =
+ = + = + = + +
+
+ = 4
5 4 2 0 2 5
2 m m
m y
m m
y 5 4
4 1 2 +
=
+ +
m
m
m
m G
5 4
4 1 ,
5 4
2 0 2 5
A 1
B 1
( a ) M 1
A 1
A 1
( l 1 o r l 2 )
A 1
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8 5 4
4 1 5 4
9 5 4 2 0 2 5
5 4 2 0 2 5
2 2
=
+
+ +
+
+ +
m m
m m
m m
m m
2 2 2
8 5 4
3 2 5 4
4 0 =
+ +
+ m m
m m
6 4 2 5 4 0 1 6
2 6 2 4 2
2
= + + m m m
0 1 6 4 0 1 6 2 = + + m m
0 2 5 2 2 = + + m m
( ) ( ) 0 2 1 2 = + + m m
2 , 2 1 = m
7 . ( a ) | w | = 4 1 ,
a r g w = t a n 1 4 5
= 2 . 4 7 r a d . ( 3 s . f . )
( b ) z w
= 3 + 2 i 5 + 4 i
x 5 4 i 5 4 i
= 7 4 1
2 2 4 1
i
( c ) Z ( 3 , 2 ) , W ( 5 , 4 ) , Z * ( 3 , 2 ) , P ( x , y )
m i d - p t o f Z Z * = m i d - p t o f W P
( 3 , 0 ) = ( x 5 2
, y + 4 2
)
x = 1 1 , y = 4
p = 1 1 4 i .
8 .
( b )
M 1
A 1
M 1 A 1
M 1 A 1
A 1
A 1
M 1
B 1
5 l n 2 l n x y =
5 l n 2 1 =
d x d y
y
5 l n 2 y d x d y =
2 5 l n 5 2 x
d x
d y =
M 1
A 1
M 1
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( ) 1
0
2 2 2 5 l n 5 d x x x
= ( )
1
0
2 2 5 l n 5 2 5 l n d x x x
= 1
0
2 2 2 5 l n 5 2 5 l n 5
d x x x x
= [ ] 1 0 2 2 5 2 5 l n 5 x x x
= 5 0 l n 5 2 4
H e n c e , m = 5 0 , n = 2 4
9 . ( a ) D o m a i n : } { 5 , : = x x x g
R a n g e : } { 0 , : = y y y g
( b ) ( ) 3 ) ( + = x x f g
( ) ( ) 3 2
5 3 2 + = + + x b x a
( ) ( ) [ ] 2 2
3 2
5 3 + = + + x b x a
1 2
= a
a = 2
0 2
5 = + b
b = 5
B 1 B 1
M 1
M 1 A 1
A 1 A 1
B 1
B 1
M 1
A 1
A 1
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( c )
S i n c e t h e l i n e p a r a l l e l t o t h e x - a x i s i n t e r s e c t s t h e c u r v e y = f i s o n e - t o - o n e f u n c t i o n .
f ( x ) = 2 ( x + 3 ) 2 5
2 ( y + 3 ) 2 5 = x
2 ( y + 3 ) 2 = x + 5
( ) 2
5 3 2
+ = + x y
2 5
3 + = + x y
2
5 3
+ = + x y
2 5
3 + = x y
5 , 2
5 3 ) ( 1 + = x x x f
1 0 .
=
+
+
2 2 2
2 5 1 2 2 4
1 0 0
0 1 0 0 0 1
3 2 2
1 2 1 1 0 1
3 2 2
1 2 1 1 0 1
3 2 2
1 2 1 1 0 1
b a
=
+
+
2 2 2
2 5 1
2 2 4
0 0
0 0
0 0
3 2 2
2
0
9 1 0 1 0
4 6 5
4 2 1
b
b
b
a a a
a a a
a a
=
+ + + + + + + + + +
2 2 2
2 5 1 2 2 4
3 9 2 1 0 2 1 0
4 2 6 5 4 2 1
b a a a
a b a a a b a
B 1
S h a p e D 1
A l l c o r r e c t D 1
s i n c e 3 y
M 1
A 1
M 1
A 1
f ( x )
x - 3
- 5
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6
1 5
= = +
a
a
1 1 4 1 = = + + b b a
( )
= + +
2 2 2
2 5 1
2 2 4
3 2 2
1 2 1
1 0 1 2 b I a P P P
=
6 0 0
0 6 0
0 0 6
( ) I b I a P P P 6 2 = + +
=
2 2 2
2 5 1
2 2 4
6 1 1 P
=
5
2
3
3 2 2
1 2 1
1 0 1
z
y
x
=
5
2
3
2 2 2
2 5 1
2 2 4
6 1
z
y
x
=
0 2 1
3
z y x
0 , 2 1
, 3 = = = z y x
A 1
A 1
A 1
B 1
B 1
A 1
M 1
B 1
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1 1 .
( a )
( b )
C o n s i d e r ,
S i n c e
H e n c e
H e n c e ,
M 1
A 1
M 1 A 1
A 1
M 1 A 1
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1 2 . ( a ) t h e a s y m p t o t e i s x = 5 4 .
( b ) f ( x ) = 4 x + 4 4 x + 5
.
= 4 ( 4 x + 1 ) ( x + 1 ) 4 x + 5
f ( x ) = 4 + 4 ( 1 ) ( 4 x + 5 ) 2 ( 4 )
= 4 ( 4 x + 7 ) ( 4 x + 3 ) ( 4 x + 5 ) 2
W h e n f ( x ) = 0 , ( x , y ) = ( 1 , 2 ) o r ( 1 4
, 1 8
+ l n 4 )
M 1
A 1
A 1
c )
0 1 1
6 1
1 3 1
1 2 1
4 2 3 4
= +
+
x x x x
B 1
M 1 r e f e r t o ( a )
A 1
s i g n o f f
( x ) :
s i g n o f f
( x ) :
1 1 4
x
3 4
+ +
5 4
+ +
B 1
M 1 f o r f a n d f
M 1 A 1 f o r s t a t i o n a r p o i n t s
M 1 d e t e r m i n e n a t u ro f s t a t i o n a r y p o i n t s
A 1 f o r f o r f
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l o c a l m a x i m u m a t ( 1 , 2 )
l o c a l m i n i m u m a t ( 1 4 , 1 8 + l n 4 )
( c ) W h e n f ( x ) = 0 , x = 3 4
{ 7 4
i s n o t i n t h e g i v e n d o m a i n }
i n f l e x i o n a t ( 3 4
, 9 8
+ l n 2 )
( e ) m a x i m u m f o c c u r s a t x = 1
m a x . f i s 2 + l n 9
A 1
A 1
M 1 A 1
A 1
B 1
D 1 f o r s h a p e
D 1 f o r a s y m p t o t e
D 1 f o r p o i n t s i n ( b ) & ( c )
x O
y
( 1 , 2 + l n 9 )
( 1
4 ,
1
8 + l n 4 )
( 1 , 2 )
( 3
4 ,
9
8 + l n 2 )
4 5 = x
( d )