Maths T STPM 2014 Sem 1 P1 Trial Pahang

3 .

( i ) 4 2 6

5 0 4 − 3 − 4 1

( i i )

( c )

S t a t e t h e v a l u e o f

( b )

x 2 + x 3 + z 5 = y 2 − 4 z =

3

1 2 3

0 5 4 − 3 − 4 1

[ 2 m a r k s ]

[ 4 m a r k s ]

a S n d o l v A e − t h . 1 e f o l l o w i n g 3 x s y + s t 4 e y m = o f z l + i n 1 e a r e q u a t i o n s b y m e t h o d o f m a t r i c e s t h a t i n v o l v e s A

( a )

F i n d A −

1

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

T h e m a t r i x A

=     

2 1 3

5 0 4

− 3 − 4 1     

h a s a n i n v e r s e A

M a t r i x P i s i n v e r t i b l e i f P ≠

0 w h e r e

P

i s t h e b e d c e a t u e r s m e i n 2 1 3 a n 5 0 4 t o f − 3 − P 1 . 4 −

1

D e t e r m i n e t h e r a n g e o f v a l u e s o f x s o t h a t t h i s e x p a n s i o n i s v a l i d .

( b ) E x p r e s s ( 2 x ) 4 1 +

3 x

[ 6 m a r k s ]

( b ) m G i a v x e i m n u t h m a t v t h a l e u f e u o n f c b t i . o n g

∑ 0 1 3

0

(

−

r

3

r − 1

, g i v e y o u r a n s w e r i n t h e s t a n d a r d f o r m A

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

1 0 .

1 .

)

i n t h e f o r m

= −

[ 3 m a r k s ]

r = 1

−

2 . ( a ) E v a l u a t e

o

f ( x ) i s d e f i n e d , f i n d t h e m i n i m u m v a l u e o f a a n d t [ h 3 e m a r k s ]

×

1 0 n .

[ 3 m a r k s ]

( a )

d B e y f i n f n e i d d i n g t h e v a l u e s o f g ( 1 ) a n d g ( 3 ) , e x p l a i n w h y t h e i n v e r s e f u n c t i o n g ( 1 x [ ) 2 i m s a n r o k t s ]

6 .

( b ) ( b −

a ) × ( c

F i n d :

( a ) a • b

−

a )

T

h = e − p i o + i c

4 j − 4 k r e s p e c t i v e l y .

n t s A , B a n d C h a v e p o s i t i o n v e c t o r s a = +

i j + 2 k , b =

3 i + 2 j + 4 k a n d [ 3 m a r k s ]

[ 1 m a r k ]

g f : : x x

a a

x 2 – 4 x ,

x + 4 ,

∈ ∈ ℜ ℜ

x x

, , x a ≤

≤

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 b y :

F o r t h e h y p e r b o l a , s t a t e t h e e q u a t i o n s o f t h e a s y m p t o t e s .

[ 6 m a r k s ]

A n s w 2 e S 0 r e 1 a c 4 l t l i 1 q o u n P e A A s t i [ H o n 4 A s 5 N m G i n a t r h k 9 i s 5 s 4 ] / s 1 e c t i o n

4 x ≤ b ,

5 . a S s k y e m t c p h t o t h t e e s c o u f r t h v e e s h 4 y x 2 p e + r 9 b o y l 2 a . = 3 6 a n d 4 x 2 – y2 = 4 o n t h e s a m e a x e s , s h o w i n g c l e a r l y t [ h 4 e m a r k s ]

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

S h o w t h a t t h e t w o c u r v e s 4 x 2 + 9 y 2 = 3 6 a n d 4 x 2 – y 2 = 4 h a v e t h e s a m e f o c .i

H e n c e , s i m p l i f y

( 3 +

i ) . 5

( b ) θ E x i p s r t h e e s s a t h r g e u c m o e m n t p l o e f x t h n e u m c b o m e p r l e x 3 n + u i m i n b t h e e .r f o r m r ( c o s θ +

[ 5 m a r k s ]

i s i n θ ) , w h e r e r i s t h e m o d u l u s a n d

4 . ( a ) G i v e n p ( 1 + 5 i ) – 2 q = 3 + 7 ,i f i n d t h e v a l u e s o f p a n d q i f p a n d q a r e b o t h r e a l n u m [ b 3 e m s r . a r k s ]


2

8 . ( d ) i F s i n d r t h      e 2 2 1      p o = s 1 i t 1 i o . n v e c t o r o f t h e p o i n t P w h e r e l m e e t s t h e p l a n e .

( c ) F i n d a n e q u a t i o n o f t h e p l a n e c o n t a i n i n g O a n d l .

π

[ 4 m a r k s ]

w h o s e e q u a t i o n

[ 4 m a r k s ]

( b ) t o F i n d l . t h e p o s i t i o n v e c t o r o f A , t h e f o o t o f t h e p e r p e n d i c u l a r f r o m t h e o r i g i n O [ 4 m a r k s ]

( a ) S h o w t h a t l l i e s i n t h e p l a n e w h o s e e q u a t i o n i s

T h e l i n e l h a s e q u a t i o n =

r      5 0 5      + λ      0 1 2     

,

λ ∈ ℜ

r

.

     0 2 −

1

     = −

5 .

.

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 c o t θ

( c ) P r o v e t h a t c o t θ −

t a n θ =

2 c o t 2 θ .

−

t a n θ =

1 2 f o r 0 o

≤ θ ≤

3 6 0 . o [ 3 m a r k s ]

[ 4 m a r k s ]

[ 3 m a r k s ]

t h e r a n g e 0 <

x

< 2 π

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

c o r r e c t t o t w o d e c i m a + l 2 p s l i a n c e x s . a s w e l l a s t h e c o r r e s p o n d i n g v a l u [ 5 e o m f a x r i k n s ]

( b ) E x p r e s s c o s x +

2 s i n x i n t h e f o r m

R s i n ( x + α

) w h e r e R i s p o s i t i v e a n d 0

< α < 2 π

.

7 . ( a ) I T f h Q e ( p x ) o l h y n a s o m a i q a u l a Q d ( r x a ) t i i s c d f e a c f i t n o e r d ( x b ² y + 5 Q ( ) x a ) n = d a x ³ z + e m r o x o ² f + – 5 2 x , – f i n n d . t h e v a l u e s o f m a n d n . [ 3 m a r k s ]


A n s w e r a S n y e c

o t i n o e n q B u

e 1 s [ t i 5 o m n a i n r k t s h ] i s s e c t i o n

3

Maths T STPM 2014 Sem 1 P1 Trial Pahang

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