2 0 1 2 T R IA L S T P M
B A H A R U
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
S e c t i o n A [ 4 5 m a r k s ] A n s w e r a l l q u e s t i o n s i n t h i s s e c t i o n .
1 . T h e f u n c t i o n s f a f a n d g a a r e d e f i n e d a s 1 a n d f : x → , x ∈ ℜ + g : x → l n x , x ∈ ℜ + x ( a ) S t a t e t h e r a n g e s o f f f aa n d g . . ( b ) I f h h i s t h e c o m p o s i t e f u n c t i o n g g o f , f i n d t h e f u n c t i o n h h . ( c ) S h o w t h a t h ( x ) + g ( x ) = 0 . ∞
[ 2 m a r k s ] [ 2 m a r k s ] [ 1 m a r k ]
i
1 0 ⎛ 1 ⎞ 2 . ( a ) G i v e n ∑ p ⎜ ⎟ = ∑ ( 1 + 3 j ) w h e r e p p i s a c o n s t a n t . 3 ⎝ ⎠ j = 3 i = 0
[ 3 m a r k s ]
F i n d t h e v a l u e o f p . (b ) U s e th e b in o m ia l th e o re m
o f x u p t o a n d i n c l u d i n g t h e t e r m
2 ⎞ ⎛ 0 1 ⎜ ⎟ 3 . M a t r i c e s P , Q a n d R a r e ⎜ 1 1 − 2 ⎟ ⎜ ⎟ ⎝ 2 − 2 − 3 ⎠ 1
1 2 H e n c e , s o lv e th e s y s te m
1 − x
a s a s e r i e s o f a s c e n d i n g p o w e r s
i n x 2 , w h e r e x x < 1 .
1 H e n c e , b y s u b s t i t u t i n g x = , s h o w 1 0
r e s p e c t i v e l y . F i n d
1 + x
t o e x p a n d
t h a t 1 1 ≈
6 6 3 2 0 0
.
[ 6 m a r k s ]
6 − 6 ⎞ ⎛ 1 4 3 4 3 3 ⎞ ⎛ 6 ⎜ ⎟ ⎜ ⎟ , ⎜ − 7 − 8 − 3 ⎟ a n d ⎜ − 3 3 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 8 2 2 1 5 ⎠ ⎝ − 2 − 6 − 2 ⎠
P Q R a n d d e d u c e R - 1 .
o f l i n e a r e q u a t i o n s
[ 4 m a r k s ]
x + y – z = 8 - 4 x + 4 y + 4 z = 0 x + 3 y + z = 1 0
[ 5 m a r k s ]
4 . G i v e n t h a t t h e r e a l n u m b e r s r r a n d θ , w h e r e r > 0 , , - π < θ < π , r c o s θ + 2 r r 2 c o s 2 θ + 3 r r 3 c o s 3 θ = 0 a n d rr s i n θ + 2 r r 2 s i n 2 θ + 3 r r 3 s i n 3 θ = 0 . B y w r i t i n g z = = r r ( c o s θ + i i s i n θ ) a n d u s i n g D e M o i v r e ’ s T h e o r e m , s h o w t h a t 1 [ 5 m a r k s ] z = = ( − 1 ± i 2 ) . 3 [ 4 m a r k s ] D e t e r m i n e t h e v a l u e o f r a n d t h e t w o p o s s i b l e v a l u e s o f t t a n θ . 5 . A
c u r v e h a s t h e p a r a m e t r i c e q u a t i o n s x = 8 c c o s θ + 3 a n d y = 4 3 s i n θ w h e r e − π < θ ≤ π . S h o w t h a t t h e c u r v e i s a n e l l i p s e a n d f i n d i t s v e r t i c e s , c e n t r e [ 9 m a r k s ] a n d f o c i . S k e t c h t h e e l l i p s e .
6 . F i n d t h e v a l u e o f α f o r w h i c h t h e v e c t o r s a = 3 i - 4 jj + k a n d b = i + 2 jj + α k [ 4 m a r k s ] a r e p e r p e n d i c u l a r . H e n c e , f i n d a - b .
2 0 1 2 T R IA L S T P M
B A H A R U
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
S e c t i o n B [ 1 5 m a r k s ] A n s w e r a n y o n e q u e s t i o n i n t h i s s e c t i o n .
7 . ( a ) S o l v e f o r x t h e e q u a t i o n s i n 2 x = 3 c o s 2 x , g i v i n g a l l s o l u t i o n s b e t w e e n 0 o a n d 3 6 0 o c o r r e c t t o t h e n e a r e s t 0 . 1 o . [ 5 m a r k s ] ( b ) E x p r e s s 2 c o s x x + 5 s i n x i n t h e f o r m r c o s ( x - α ) , w h e r e r r > 0 a n d 1 0 < α < π . H e n c e , f i n d t h e m a x i m u m v a l u e o f 2 c o s x x + 5 s i n x a n d t h e 2 [ 6 m a r k s ] c o r r e s p o n d i n g v a l u e o f x x i n t h e i n t e r v a l 0 ≤ x ≤ 2 π . [ 2 m a r k s ] S k e t c h t h e c u r v e y y = 2 c o s x + 5 s i n x f o r 0 ≤ x ≤ 2 π . B y d r a w i n g a n a p p r o p r i a t e l i n e o n t h e g r a p h , d e t e r m i n e t h e n u m b e r o f r o o t s [ 2 m a r k s ] o f t h e e q u a t i o n 2 c o s x x + 5 s i n x = 1 , i n t h e i n t e r v a l 0 ≤ x ≤ 2 π .
8 . T h e l i n e l h a s e q u a t i o n (a ) S h o w
⎛ 5 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ r = ⎜ 0 ⎟ + λ ⎜ 1 ⎟ , λ ∈ ℜ . ⎜ ⎟ ⎜ ⎟ ⎝ 5 ⎠ ⎝ 0 ⎠
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 ⎛ − 1 ⎞
⎜
⎟
r . ⎜ 2 ⎟ = − 5 . ⎜ ⎟ ⎝ 0 ⎠
[ 3 m a r k s ]
( b ) F in d th e p o s itio n v e c to r o f A , t h e f o o t o f th e p e r p e n d ic u la r f r o m
t h e o r i g i n O t o l . [ 4 m a r k s ] [ 4 m a r k s ] ( 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 O a a n d l . ( d ) F i n d t h e p o s i t 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 π w h o s e ⎛ 1 ⎞
⎜ ⎟
e q u a t i o n i s r . ⎜ 2 ⎟ = 1 1 .
⎜ ⎟ ⎝ 2 ⎠
M
[ 4 m a r k s ]
A T H E M A T I C A L L F O R M U L A E B i n o m i a l e x p a n s i o n s ⎛ n ⎞ ⎛ n ⎞ ⎛ n ⎞ ( a + b ) n = a n + ⎜ ⎜ ⎟ ⎟ a n − 1 b + ⎜ ⎜ ⎟ ⎟ a n − 2 b 2 + . . . + ⎜ ⎜ ⎟ ⎟ a n − r b r + . . . + b n , w h e r e n ∈ ZZ + ⎝ 1 ⎠ ⎝ 2 ⎠ ⎝ r ⎠ n ( n − 1 ) 2 n ( n − 1 ) . . . ( n − r + 1 ) r ( 1 + x ) n = 1 + n x + x + . . . + x + . . . , w h e r e n ∈ Q , x < 1 2 ! r ! C o n i c s P a r a b o l a w i t h v e r t e x ( h h , k ) ) , f o c u s ( a a + h , k ) a n d d i r e c t r i x x = - a + h 2 ( y – k ) ) = 4 a ( ( x – h ) E l l i p s e w i t h c e n t r e ( h h , k ) ) a n d f o c i ( - c + h , k ) , ( c + h , k ) , w h e r e c c 2 = a a 2 – b 2 ( x − h ) 2 ( y − k ) 2 + = 1 a 2 b 2 H y p e r b o l a w i t h c e n t r e ( h h , k ) ) a n d f o c i ( - c + h , k ) , ( c + h , k ) , w h e r e c c 2 = a a 2 + b 2 ( x − h ) 2 ( y − k ) 2 − = 1 a 2 b 2
2 0 1 2 T R IA L S T P M
B A H A R U
M A T H E M A T IC S T
1 . T h e f u n c t i o n s f a f a n d g a a r e d e f i n e d a s
f : x →
1 x
S M K
, x ∈
ℝ
A C S I P O H ,P E R A K
A N
( a ) S t a t e t h e r a n g e s o f f f aa n d g . . ( b ) I f h i s t h e c o m p o s i t e f u n c t i o n g g o f , f i n d t h e f u n c t i o n h h . ( c ) S h o w t h a t h ( x ) + g ( x ) = 0 .
B 1
R g = { y y : : y ∈ }
B 1
g o f ( x x ) = g = ln
( c )
ℝ +
[2 m a rk s ] [ 2 m a r k s ] [1 m a rk ]
1 ( a ) R f = { y y : : y > > 0 }
( b )
x → l n x , x ∈
D g :
o r – l n x
h : : x � – l n x , , x > > 0 h ( ( x x ) + g ( x x ) = – l n x + + = 0 ∞
M 1 A 1 A 1
l n x
[ 5 ]
i
1 0 ⎛ 1 ⎞ ( a ) G i v e n ∑ p ⎜ ⎟ = ∑ ( 1 + 3 j ) w h e r e p i s a c o n s t a n t . ⎝ 3 ⎠ j = 3 i = 0
2 .
F i n d t h e v a l u e o f p .
[3 m a rk s ]
( b ) U s e th e b in o m ia l th e o re m a n d in c lu d in g th e te rm
1 + x
t o e x p a n d
1 − x
a s a s e r i e s o f a s c e n d i n g p o w e r s o f x x u p t o
i n x 2 , w h e r e x < 1 .
1 H e n c e , b y s u b s t i t u t i n g x = , s h o w 1 0 2 ( a )
t h a t 1 1 ≈
=
O R
6 6 3 2 0 0
.
[6 m a rk s ]
B 1
E it h e r o n e
= 8 + 3 { 3 + 4 + 5 + . . . + 9 + 1 0 } M 1
= 1 6 4
A 1
p = = ( b )
× =
×
=
×
≈ 1 + x + + ½ x 2 =
M 1 M 1
A 1 B 1
C o r r e c t b in o m ia l e x p a n s i o n
2 0 1 2 T R IA L S T P M
B A H A R U
≈ 1 +
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
+ ½
M 1
S u b s t . x = =
A 1
[ 9 ]
≈ 3 × ≈
2 ⎞ 6 − 6 ⎞ ⎛ 0 1 ⎛ 1 4 3 4 3 3 ⎞ ⎛ 6 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ M a t r i c e s P , Q a n d R a r e ⎜ 1 1 − 2 ⎟ , ⎜ − 7 − 8 − 3 ⎟ a n d ⎜ − 3 3 3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 2 − 2 − 3 ⎠ ⎝ 8 2 2 1 5 ⎠ ⎝ − 2 − 6 − 2 ⎠
3
1 F i n d P Q R a n d d e d u c e R - 1 . 1 2 H e n c e , s o l v e t h e s y s t e m o f l i n e a r e q u a t i o n s x + y – z = 8 - 4 x + 4 y + 4 z = 0 x + 3 y + z = 1 0
r e s p e c t iv e ly .
3
[4 m a rk s ]
[ 5 m a r k s ]
P Q R =
=
M 1
C o rre c t Q R
M 1
C o rre c t P Q
=
o r
P Q R =
= =
A 1 ∴
P Q R =
P Q R = - 9 I R - 1 = -
M 1
P Q
= -
=
B y a d ju s t in g t h e s y s t e m
o f lin e a r e q u a t io n s :
A 1 B 1
A b le t o r e a r r a n g e t h e s y s t e m o f e q u a t i o n s
2 0 1 2 T R IA L S T P M
B A H A R U
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
B 1
M 1
=
R e je c t a ll o t h e r m e t h o d s .
A 1 A 1
∴
[9 ]
x = = 1 , y = = 4 , z = - 3
G i v e n t h a t t h e r e a l n u m b e r s r a a n d θ , w h e r e r > > 0 , , - π < θ < π , 2 3 r c o s θ + 2 r r c o s 2 θ + 3 r r c o s 3 θ = 0 a n d r s i n θ + 2 r r 2 s i n 2 θ + 3 r r 3 s i n 3 θ = 0 .
4
B y w r i t i n g z = r ( c o s θ + i s i n θ ) a a n d u s i n g D e M o i v r e ’ s T h e o r e m , s h o w
4
[ 5 m a r k s ] D e t e r m i n e t h e v a l u e o f r r a a n d t h e t w o p o s s i b l e v a l u e s o f t a n θ . r c o s θ + 2 r r 2 c o s 2 θ + 3 r r 3 c o s 3 θ = 0 (1 ) 2 3 r s i n θ + 2 r r s i n 2 θ + 3 r r s i n 3 θ = 0 . (2 ) B y o p e r a t i n g ( 1 ) + i ( 2 ) , r ( ( c c o s θ + i s i n θ ) + 2 r r 2 ( c c o s 2 θ + i s i n 2 θ ) + 3 r r 3 ( c c o s 3 θ + i s i n 3 θ ) = 0 r ( ( c c o s θ + i s i n θ ) + 2 r r 2 ( c c o s θ + i s i n θ ) 2 + 3 r r 3 ( c c o s θ + i s i n θ ) 3 = 0
1 t h a t z = ( − 1 ± i 2 ) . 3
[4 m a rk s ]
M 1 M 1
U s in g D e M o iv r e ’s t h e o r e m
A 1 ∴
O R
z + 2 z z 2 + 3 z z 3 = 0
z = r ( ( c o s θ + i s s i n θ ) z 2 = r 2 ( c o s 2 θ + i s s i n 2 θ ) a n d z 3 = r 3 ( c o s 3 θ + i s s i n 3 θ ) 2 z 2 = 2 r r 2 ( c o s 2 θ + i s s i n 2 θ ) a n d 3 z 3 = 3 r r 3 ( c o s 3 θ + i s s i n 3 θ )
∴
z + 2 z z 2 + 3 z z 3 = 0
M 1 M 1 A 1
U s in g D e M o iv r e ’s t h e o r e m
2 0 1 2 T R IA L S T P M
B A H A R U
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
z ( 3 z z 2 + 2 z z + 1 ) = 0 M 1
z ≠ 0 , z = =
A 1 B 1
| z | = t a n θ =
o r
t a n θ = 5
5
t a n θ =
M 1
t a n θ =
A 1 A 1
[9 ]
A c u r v e h a s t h e p a r a m e t r i c e q u a t i o n s x = 8 c c o s θ + 3 a n d y = 4 3 s i n θ w h e r e − π < θ ≤ π . S h o w t h a t t h e c u r v e i s a n e l l i p s e a n d f i n d i t s v e r t i c e s , c e n t r e a n d f o c i . S k e t c h t h e e l l i p s e . [ 9 m a r k s ] B y u s i n g c o s θ = +
a n d
w i t h s i n 2 θ + c o s 2 θ = 1 ,
s i n θ =
= 1
+
B 1
E lim in a t in g p a r a m e t e r θ U s i n g t r i g . i d e n t i t y
M 1
= 1 A 1
∴
t h e c u r v e i s a n e l l i p s e .
C e n t r e is (3 , 0 ) T h e v e r t ic e s a r e (- 5 , 0 ) a n d (1 1 , 0 ) T h e fo c i a re (7 , 0 ) a n d (-1 , 0 )
y
(3 , 4
●
A 1 A 1 A 1 A 1 )
D 1
●
( - 5 , 0 )
●
( - 1 , 0 )
●
( 3 , 0 )
●
( 7 , 0 )
●
( 1 1 , 0 )
x D 1
● ( 3
, - 4
)
E l l i p s e s h a p e ( d e p e n d e n t o n t h e c o r r e c t e q u a t i o n a b o v e ) A l l ‘ h i s ’ p o i n t s s h o w n a n d l a b e l e d . [ 9 ]
2 0 1 2 T R IA L S T P M 6
6
B A H A R U
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
F i n d t h e v a l u e o f α f o r w h i c h t h e v e c t o r s a = 3 i - 4 jj + k a n d b = i + 2 jj + α k a r e p e r p e n d ic u l a r . H e n c e , f i n d | a - b | . [4 m a rk s ]
B y u s i n g a • b = 0 ,
•
= 0
M 1
α = 5 a - b =
A 1
-
M 1
=
| a – b | = =
7
A 1
A c c e p t 7 .4 8 [ 4 ] 2 ( a ) S o l v e f o r x x t h e e q u a t i o n s i n 2 x = 3 c o s x , g i v i n g a l l s o l u t i o n s b e t w e e n 0 o a n d 3 6 0 o c o r r e c t t o t h e n e a r e s t 0 . 1 o . [5 m a rk s ] 1 ( e ) E x p r e s s 2 c o s x x + 5 s i n x i n t h e f o r m r c o s ( x - α ) , w h e r e r r > 0 a n d 0 < α < π . 2
H e n c e , fin d th e m a x im u m v a lu e o f x i n t h e i n t e r v a l 0 ≤ x ≤ 2 π .
2 c o s x x + 5 s i n x a n d
t h e c o r r e s p o n d i n g v a l u e o f [6 m a rk s ]
S k e t c h t h e c u r v e y y = 2 c o s x + 5 s i n x f o r 0 ≤ x ≤ 2 π . [2 m a rk s ] B y d r a w i n g a n a p p r o p r i a t e l i n e o n t h e g r a p h , d e t e r m i n e t h e n u m b e r o f r o o t s o f t h e 7 (a )
e q u a tio n 2 c o s x + 5 s i n x = 1 , i n t h e i n t e r v a l 0 ≤ x ≤ 2 π . [ 2 m a r k s ] 2 s i n x c c o s x - 3 c o s 2 x = = 0 B 1 c o s x ( ( 2 s i n x x - 3 c o s x ) = 0 M 1 R e je c t d iv is io n b y c o s x c o s x = = 0
t a n x = =
M 1 A 1 A 1
A ll c o r r e c t
2 c o s x + + s i n x ≡ r c c o s x c c o s α + r s s i n x s s i n α r s s i n α = r c c o s α = 2
B 1
B o th
r = = 3 , t a n α =
M 1
r = = 3 , α = 0 . 8 4 1 r a d i a n s .
A 1
x = = 9 0 ° , 2 7 0 °
∴
( b )
o r o r
x = = 5 6 . 3 ° , 2 3 6 . 3 °
x = = 5 6 . 3 ° , 9 0 ° , 2 3 6 . 3 ° , 2 7 0 °
B o th
2 0 1 2 T R IA L S T P M
B A H A R U
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
A 1 s i n x = 3 c c o s ( x – – 0 . 8 4 1 )
2 c o s x + +
∴
m a x v a lu e is 3 w h e n c o s ( x – – 0 . 8 4 1 ) = 1
y 3
�
x = = 0 . 8 4 1 r a d i a n s .
B 1 B 1
( 0 . 8 4 1 , 3 ) ● ● ( 2 π , 2 )
2 ●
y = = x1
D 1
A c o s i n e c u r v e w i t h a m a x . a n d a m in . t u r n i n g p o i n t .
D 1
- 3
M 1 T h e re a re 2 ro o ts .
8
A 1
T h e l i n e l h a s e q u a t i o n
⎛ 5 ⎞ ⎛ 2 ⎞ ⎜ ⎟ ⎜ ⎟ r = ⎜ 0 ⎟ + λ ⎜ 1 ⎟ , λ ∈ ⎜ ⎟ ⎜ ⎟ ⎝ 5 ⎠ ⎝ 0 ⎠
In it ia l a n d a n d ‘h is ’ m l i n e y y = = 1 D e p e n d e n c o r re c t g r [ 1 5 ]
f i n a l p o i n t s a x . p o i n t . t o n a p h .
ℝ .
⎛ − 1 ⎞ ⎜ ⎟ ( 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 r . ⎜ 2 ⎟ = − 5 . ⎜ ⎟ ⎝ 0 ⎠ (b ) F in d th e p o s itio n v e c to r o f A , th e fo o t o f th e p e rp e n d ic u la r fro m ( 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 O a a n d l .
[ 3 m a r k s ] t h e o r i g i n O t t o l .
[ 4 m a r k s ] [4 m a rk s ] ⎛ 1 ⎞
⎜ ⎟
( d ) F i n d t h e p o s i t 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 π w h o s e e q u a t i o n i s r . ⎜ 2 ⎟ = 1 1 .
⎜ ⎟ ⎝ 2 ⎠
[ 4 m a r k s ] 8 ( a ) r
=
+ λ
a n d
r
•
= - 5
B y u s i n g r • n = d ,
•
= - 5
B 1
2 0 1 2 T R IA L S T P M
B A H A R U
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
M 1 - 5 – 2 λ + 2 λ = - 5 f o r a l l v a l u e s o f λ ∈
ℝ .
A 1 l i n e l l l l i e s o n t h e p l a n e .
∴
( b ) =
+ λ 1
•
= 0
•
, λ 1 ∈
ℝ
= 0
B 1
1 0 + 4 λ 1 + λ 1 = 0 � λ 1 = - 2
=
M 1
- 2
M 1
= A 1 ( c ) n
=
×
=
M 1
=
A 1 M 1
∴
e q u a t i o n o f p l a n e i s r •
=
• A 1
r
(d )
S i n c e P i s o n l i n e l ,
•
= 0
o r
– 5 x x + + 1 0 y y + + 5 z z = 0
A c c e p t r •
= 0
2 0 1 2 T R IA L S T P M
=
+ λ 2
B A H A R U
, λ 2 ∈
⎛ 1 ⎞ ⎜ ⎟ G i v e n t h a t r . ⎜ 2 ⎟ = 1 1 . ⎜ ⎟ ⎝ 2 ⎠ ⎛ 1 ⎞ ⎜ ⎟ . ⎜ 2 ⎟ = 1 1 . ⎜ ⎟ ⎝ 2 ⎠ 5 + 2 λ 2 + 2 λ 2 + 1 0 = 1 1 λ 2 = - 1 =
M A T H E M A T IC S T
S M K
A C S I P O H ,P E R A K
ℝ
B 1 M 1 A 1
- 1
= A 1
[1 5 ]
