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M A T H E M A T I C S T S M K G A J A H S e c 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 ( 1 + 5 i) p – 2 q = 3 + 7 i, f in d th e v a lu e s o f p a n d q in p a n d q a r e b o th r e a l n u m b e r s . p a n d q a re re s p e c tiv e ly a c o m p le x n u m b e r a n d its c o
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k s ] ] e ] k s ]
G i v e n t h a t t h e s e r i e s i s c o n v e r g e n t , s t a t e t h e s e t o f v a l u e s o f x , a n d [[ 6 m a r k s ] ]
t o i n f i n i t y .
o u t le t r u n s s a le s p r o m o tio n s a t th r e e d if f e re n t v e n u e s , e a c h v e n u s a le s p ro m o te rs . T e a m A s o ld 2 c a rto n s o f p ro d u c t P , 5 c a rto n s f p r o d u c t R . T e a m B s o ld 4 c a rto n s o f p ro d u c t Q a n d 2 c a rto n o a rto n s o f p ro d u c t P a n d 2 c a rto n s o f p ro d u c t Q . T h e b o n u s e s p a a r e R M 2 5 0 . 0 0 , R M 2 2 0 .0 0 a n d R M 2 1 0 .0 0 r e s p e c t i v e l y . x , R M y , a n d R M z to re p re se n t th e b o n u s e s p a id fo r e a c h c a rto n c tiv e ly s o ld b y th e s a le s p r o m o te rs , o b ta in a s y s te m o f lin e a r e q n E l i m a t i o n m e t h o d , f i n d t h e v a l u e s o f x , y a n d z .
R e la tiv e to th e o r ig in O , th e p o s itio n v e c to r s o f th e p o i n ts P , Q j + 9 k O P = = 8 i + 5 j
7 )
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n ju g a t 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 . [[ 6 m 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 . 9 y 2 = 3 6 a n d 4 x 2 – y 2 = 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 [[ 4 m b o l a .
. r − 2 2 r = 1 fin d in te rm s o f x , th e s u m
r e ta il te a m o f c a rto n o s o ld 6 c B a n d C L e t R M R . re sp e G a u s s ia
n . e a c h o f t h e f o l l o w i n g c a s e s :
+ 1 2 c o s x i n t h e f o r m R s i n ( x + ϑ ) w h e r e R > 0 a n d 0 0 < ϑ < 9 0 0 . a p h o f y = 5 s i n x + 1 2 c o s x f o r 0 0 ≤ x ≤ 3 6 0 0 . t h e i n e q u a l i t y 5 s i n x + 1 2 c o s x > 1 2 f o r t h e r a n g e o f v a l u e s o f x b e t w e e n 0 0 a n d [[ 9 m a r k s ] ] e .
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m a n a g e d b y d u c t Q a n d u c t R . T e a m e th re e te a m
o f th e p ro d u c ts u a tio n s a n d b y [[ 7 m
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, O R = - i + 7 j - 3 k a n d O S = - 3 i + 3 j + k .
[[ 3 m a r k s ( a a ) P r o v e t h a t O O P i s p e r p e n d i c u l a r t o O O R a n d O S [[ 3 m a r k s ( b ) C a l c u l a t e t h e a r e a o f t h e t r i a n g l e O S R . S e c c t i o n B [ 1 5 m a r k s ] A n s w e r a n y y o n e q u e s t i o n i n t h i s ss e c t i o n . O n e o f t h e f a c t o r s o f t h e p o l y n o m i a l f ( x ) = x 4 – x 3 + a x 2 + b x – 4 i s x – 1 . W h e n f ( x ) i s d i v i d e d b y x + 3 , t h e r e m a in d e r i s 6 8 , d e te rm i n e t h e v a lu e s o f t h e c o n s ta n ts a [ 4 m a r k s F a c t o r i s e f ( x ) c o m p l e t e l y . H e n c e , s h o w t h a t t h e e q u a t i o n f ( x ) = 0 h a s 2 r e a l r o o t s a n d 2 [[ 5 m a r k s c o m p l e x r o o t s . I f t h e c o m p l e x r o o t s a r e w a n d z , w r i t e w a n d z i n p o l a r f o r m . [[ 6 m a r k s B y u s i n g t h e D e M o i v r e ’ s T h e o r e m , f i n d z 4 + w 4 , s i m p l i f y i n g y o u r a n s w e r .
8 . T h e e q u a t i o n s o f t w o π 1 : x + 2 y + z π 2 2 x – 4 y – z a ) F in d th e v e c to r p l a n e s . b ) F i n d t h e v e c t o r ( 3 , 0 , 0 ) a n d is c ) S h o w th a t th e a :
p l a n e s a r e g i v e n b y : = 4 a n d = 2 e q u a tio n a n d th e C a rte s ia n e q u a tio n o f th e lin e o f in te rs e c tio n o f th e t [[ 7 m e q u a tio n a n d th e C a rte s ia n e q u a tio n o f th e p la n e w h ic h c o n ta in s th e p [[ 4 m p e r p e n d i c u l a r t o t h e t w o g i v e n p l a n e s . [[ 4 m n g l e b e t w e e n t h e p l a n e s π 1 a n d π 2 i s o b t u s e .
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