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ICSE 2015: Important Notes and Formulae for Mathematics

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P ( l + ST - x - b - W hi Ca lc u late V A T o n M a n u fa c t u r in g C o st : T a k e Ca lc u late V A T o n step 3 : T ak e Ca lc u la te V A T o n P r o n t 2 o n ly ste p 4 : T ak e Calc u la te V A T o n P r o n t 3 o n ly In S In step 2 In In ( Se l l i Ba = M + A d d ) an u f ac tu r i n g c o st + p r o f i t 1 + p r o f i t 2 + a ll to P r o fit 1 o n ly g et T o t a l V A T P ro f it 3 e tc . , + T o ta l V A T o klng n t : Sa y i t dYTi W h il , : i e ta k i n g th e e n tr ie s y o u h a v e to b e a r in m in d th a t (f r e s h ) T ak e th e l a t e s: T ak e t h e l e a st e n t r y E o m e n try on o r b efo r e t o th 1 1 th t o t h e l a st d a y o f ev ery m o n th o f th e m o n th \ T ak e th e le a st o f th e s e tw ," - - W h ile g iv B - 1 18 c o m p u t in g th e in te r e st a l w a y s t a k e t i m e a s 1 / 1 2 i . e n . V A T le s T a x & a x A , (w ) - = 1 + 15 / 10 0 ) it ia l v a lu e g gg " + R / 10 0 : ii) Th e P ( 1 = e g e t d im i n i sh e d a s tim e p a s se s r e , n A . e a n , lu a p o p u latio n g ro w th p ro b le m s i) T he Sa s - a l V a lu e o f m a ch in e . 1 2 / 10 0 ) ( r e c ia t i o n C e r ta in i te m F in 46 + k n o wl e d g g l l b a s i c tq iR m fo r th e c o n se c u ti v e y e a r s j u Dep a . v o u r m so m e to c o m p u t e th e in te r e st se m i , A h e m a t i c s Fo nn Tn l a e & + R a tel 10 0 ) No t e : I f y o u ar e a sk e d be m o d i f i e d by t a k i n g a ia : iw . P R T / 10 0 = Fo r . I n t e rN . # n . Ba s e . e in m e n t P R T , K a / 10 0 lk a j l . Ta k e t i m e a s 1 / 1 2 N D 19 . . , in , ir r e s p e c t i v e o f t h e to ta l n u m b e r t) 1 s te a d o f to ta l n o o f m o n t h s Sa m m u . nj ay K u m a r (9 8 9 9 19 9 8 7 6 ) w w w . s t s tuto r la ls . c o m Scanned by hx c i li St e p 4 t l i qD i e m i e n s T g Sg c c e s s T u t o r i a i s If n o . . . e n 9 w w w k n o w l e d lsg l t _x r : z ( m . . 2 - _ . n a 9: v t o t a k e t h e l a s t e n t r y o f th e p m to u s y ou ha e ti o n ) m i s ta k e , c ho o se t h c v a l u e f ro m t h e q u e s - e n t ry o f a p a r t ic u la r m o n th i s n o t g iv en t , he n m o n th (H e r e I f yo u a r e a sk ed to f i n d th e a m o u n t th a t w i l l b e o b ta i n e d o n c l o s in g th e ac c o u n t T he n a t tim e s th ere is c h an c e o f m ak in g ( Bu t D 0 (g - a tu r i t y V alu e o J e# p : P O it R: d y N o o f m o n th , IM er g E T o tal = o f N o n l pm E o n l y in t e r e st i s a sk e d t h e n In t er e st k D t v i d e GE B & l D iv id en d a No o f x E = y T o ta l e s ha r In co m e o j I n a o In c o m = l X e _ U sin g b = (a ) t h e. _ Pm l R e c e i v e d X 10 0 I n v e stm en t 1 ) (a (b ) + b) (v iii b) - 2 b _ (v i) b ) - 3 a ) a 2 + 2 ) 2 (i i i ) a (a + b ) ( a b ) ( a + b ) 2 2 (a + b )( a + a b + b ) 2 2 (a b ) ( a + a b + b ) _ 3 + b _ 2 b _ z i (a _ - bx a b) - 2 2 = b _ 2a b _ A 3 b 3 2 . - OR - u se r ia l In i , th e se (v ) (v i i ) ( T o co m p an y D iv = T o ta l i) (a 4 e m si 10 0 OP e stm en t E i th e r b y sp l i t t i b) a by th e D iv p o ly n o a) (l v ) E 1a l s tfie g i c m j Fac to riz e h ere X a r k e t V a lu e o f o n e sh a r e In v H C F & L C M g F p g lv jn t (w sh a r e s o f N o a p er so n m ay b e tak en a s th e f th e r e f o r e I n c o m e y st pl a l u e V F a c e o o f M In T q a Sh a m 0 step Co m m o n term s th o f ) f th e o f in d th e v alu e o f u n l e s s sp e c i f i e d t h e m e th o d X s t it, he i r he t H C F . H C F v a lu e w h ic h g iv es y o u th e v a lu e o f L C X # t w o p o ly n o m ia l s . fa c to r iz ed b y u sin g m ay ad o p t eith er yo u X a Om be s p l it tin T ri a l " & E rm r g th e m id d l e te r m o r m e t ho d . fo r m u l a m e rh o d . Z 2a B - _ ta k e th e T o ta l p r in cip a l ) N O T r: r n n T n m E D : E s i t E M + I n t er e s t o b t a i n e d th e m o n th in w h ic h ac c o u n t h as b e e n c lo sed ) (o f tak e l a st e n t r y . 1 18 , B a s e m e n t , K a lka j i, N D 19 - . Sa nj ay K u m a r (9 8 9 9 19 9 8 7 6 ) w w w . s t s t u t o r ia ls . c o m Scanned by . . T ft - St m T g Su c c e s s If m E q u a tio n : x I f D isc r i m in an t D isc r i m in a n t 2 ( _ th e n O, in a n t o , O, t hc n i , hen . . . z o ri T n DJ QW k u M k dRg lm i J l : su m o f th e ro o lx ) u n eq u a l o r u n iq u e eq ual l in e n , (n o t th c r o o ts a re I m a g i n a ry real l in e s , o = 1 ( ro d u c t o f th e rn o ts) p , X th c r o o ts ilrc R ea l & t . . t h e r o o t s a r c R c i\ l & O he n = 3 n L W r o o ts o f an e u atio n a re q g iven D is c r i m i e n s T u to rl als Qu a d r a t i c Re i 4 t f i qo e a rc in tc rs c c t in g nrc c o in c id e n t . . lin es ) pa n t l l c l , f le c t lg n A fte r p lo ttin g th e p o in ts, / ln X - V ln - 1 In o r ig in Y a is x I n v a r i a n t " the re is a - - ( y c o o - o c , o i n t din a t e r o d in a t e r t p o i n n y : A re m a i n s sa m e b u t X & - y r d thc n a y , is - g iv en c o d in a te r o c h a n g es i n s ig n ) a to r e s p e c t i t h l i n e i v e n g , O , O r it & 44 i w a lies o n th e l in e . i a i f x di s u s c r u l e r 1a m e a s u r e t h e v e r t i c a l he n t , ) co o rd in a te c h a n g e s in s ig n - w h en X to b c p o in t is c o o r d in a te b o th c h an g e s i n s ig n ) i n v a r ia n t w i s - " ir r o m st lie o n X e g i f a p o i n t i s i n v :1 r i a n t o n X a x i s t h e n i t m u Fo r N o te : ha t t - (0 O) ( X p c X c o o rd in a te r e m a in s sa m e b u t y ( ax is a s s t l m po th e lin e in t fro m , th e r s id e to o b ta in it s a n d t h e n ta k e th e s a m e d i sta n c e o n th e o Ra t rt io n P r oM io & Du p l icate Tr ip a . Pr o d u c t g " No If i b If = a/b There f o In c ase /d c = is e g iv e n 3 : - 4 p r o 2 " lf hen t , K are g iv en d k - C . . J i v i d C k2 c ( Ei th e r t , e t h gp b oif & = b l( t ag t i n g c e a s a/ b g &y R , u b st i tu te e , C p = b/C - = c /d = /f e " th e v a lu es o f k hen c e , a bk b - a . = c , d - a n d C k th e r e f o r e th ese v a l u e s e q u a t i o n c a n b e so l v e d ) d . k = b ) r o p o r t i o n a l s en d v a lu es W . ni 3 b . ) ro o t idd le v a l u e i s r e p e a te d ) a d o p t it in th e fo l lo w i n g s i t u a t i o n s " m e do is (M C, : 2 a + b Cu b e r o o t ta k e y o u h av e to nd x tr e m e s = nd ) Sq u a r e ta k e y o u h av e to b o r t i p v alu es o f c o n t in u ed p r o p o r ti o n :W - t io io du p l - j ia : P rop o i l j dl i i Co n t in u ed , ( M i dd l e to ta k e b k, - re a c an g et a Re ean s ( In c as e o f S u b 3 b ( In c a se o f S u b W n ere te : M 2 : b 0: 3 2 2 2 . . d : d nd 1 of C b : st a a : b is lica te ratio o f Pr o p o r t i o n a a : b is ratio o f in th e g iv en p r o b lem . th e v a lu e o f b w e p u ttin g ) m ain d e r t h I f (X - 2 ) is v a lu e in p p (2 ) : iv a (X 5x = 5( 2 z ) + 6 - 6 - 3 x e n ex p re ssio n he n t tak e X - 2 0,t her e f o = re X - 2, t hen su b st i tu t e th i s as 0 ( H er e = , tak in g - ta k i n g - O is v e ry im p o r ta n t . 4 is v ery i m p o r ta n t . If n o t ta k e n a n sw e r c a n If n o t t ak en a n sw e r c a n t be f ou t be found n rem ain d er o f 4 ) W e m u lt ip t t h o s e Fo r m a w r e X W n p 1e th e t r in g m 1 18 , B a s e f t im e a Ch e c k ( : :) ej o x 5 m e n t , +@x g) 7 x 7 5 + K a @ j lk a j i , t h e m a m b e r o f c o h m ao s i n t h e s e c o n d a l a l r i g th en t a k e th e & . ) st m a t r i x , l @x - 4 ( H er e = m a l zi c e s , o a n b e r f B 6 - her e; th e r t h e r e a r e t gq o c o l o t m Jn s i n t h e s e c o n d m a ir & e f or e w r i t e b e Gr st m atr a s t w i c e . : tg ks) N D 19 - . Sa n j ay Ku m ar (9 8 9 9 19 9 8 7 6 ) w w w t st u t o r i a ls Scanned by . s . c o m c M e St e p So 4 ft t i qD m m e t im e s. Ya u tl l a y m m am x . . h e a sk e d to f in d : A + as A (H j ) i A B + A B is 7 + : 7 1 v o u r k n o w le d g e m g iv e n H e r e I is , h a v e to a ssu m e i t yo u , m a tr i x th c I d e n t rt ) , Se c t i o n W q zg u I \n w h ic h a l l th e p r i n c ip a l d ia g o n a l v a l u e s a r c a n ce & . ra f M i d e n = 4 o n T q Su c c e s s T u t o t l a l s Iden b ty Di s t i e n s A . O . d n th c re st a r c Zero . r m u la e D istan V - ce (X : : ) x l - (y : . - y i ) ( T he . se g m e n t , To v e p r o CQ - lin e a r ity A r e a o f tr i a n g le Se c fo rm ed ri X 1 + X 2 n po i , If + n a , d C Y , y 3) a . R l X : (y 3 y i ) I f tw l Dep a ) d + x 2 + x 3 m he n t , , 1 y + + z y = S lo p e ( m ) t Si m j l a r , lin es are llel Pe r p en d ic + C w , he r e , Pa r a t o C is e a y g le lm f in d y : ) n e t c . - g ad T g O = (y i ru B . - b j i ly - : . , 3 m 3Q - he n V h c wi g e n d in g u p o n th e q u e s x n 3 slo p e ar e g iv e n - y a n , o lin es a re o 3 y 2 o p o in t s a r e g i v e n a p o in t If tw x t a c so lv e th e q u to : " e h a v e to o u , " Y X # . - r , u \ z y l B qu a re li t le n g t h ln e a t i o n in m I f tw . m , , s u se th is c o n d it io n m r x a + m i y x l Eq u C OR A l m a ng o f a oid B C y) . 2 C e n tr = th e se p o in ts : x l(y : h y tio n fo rm u la : p o in t (X M id + A B A th r e e p o in ts o t th e g iv e n th e n u se th e c o n d i t io s ide s o f a t r i a n g l e th e f in d to sa m e fo r m u l a i s to b e u s e d X j i sl o p e s a r e e q u a l i Eo . e m m i z o th er th en p ro d u c t o f th e ir slo p es is . e 1 i . . e m l X m 1 - : ay h a v e to u se e q u a tio n o d st r a ig h t l in e a s , th e rcep t . OR it y : I f tw i . e th e n o tri an if A A B C - /( PQ R then A B . R a = pQ A PQ R t h e n If A io t BC = A B C rea o f A P . AC QR A rea o f A A o f th eir sid e s a r e e q u a l PR . Z Si d e l S id e = QR Z AB _ p Q 2 = BC QR 2 _ AC Z P R Z Z a: h ic h a 3 5 7 B - . . . d iv id e s t h e g iv e n f ig u r e in to tw o id e n t ic a l p a r ts i s k n o w n n a n g le h a s o n e l i n e o f sw w w e tr y a r 2 . e c ta n g le h a s 2 l in e s o f sw w w e tr y A Rho m maequ b u s h a s 2 l i n e s o f syr n m e t r y il . t 1 1 8 . Ba s e m e n t r ian n , Ka g ha s lkaj l N D 19 . 6 . 3 l in e s o f sw w we tr y - , 4 . . . a squ ar e a s l in e o f h as 4 An . l 8 A C ir c . m e try l in e s o f sw w w e try A Pa r au . Sy m E 1o g r a m h a s m e l e s N o . lin es o f sw ha s m e try O n e l i n e o f S :l r n m e t r :r le h a s I n f in i te l in e s o f s m m e tr y y Sa nj ay . (9 8 9 9 1 9 9 8 7 6 ) K u m a r w w w . . s tst u t o r i a ls . c o m Scanned by . c i li 4 t fi a) i e . St e n T g Su 9 ha s la r P o l im o n U M H m 5 n o . . . . m T IFL sM VQY ( sid es h as l in e a o f n sy m m c try R d R e H o r l go m nm Fo r , ex : A : R eg u la r p e n tag o n ( 5 s id e s ) 5 l in g s o f sy m m etr y Ho l e : ( 2n - A n g le o f a reg ular poly gon . 4) 1 n Lo i e n s c c ess T u t o rl a l s A lm , m 9 0 (t i e r e , fe r s r e n n u m b e r o f s id e s o f a p o ly g o n , ) cl The L o c u s o f a l in e seg m en t is it j iB s i f fc m la c b i sc c m u a Fo r so lv in g m o st o f th e L o c u s pro b lem s, he t o ab o v e tw o p o i n ts a rc g o l co ns n o w le d g e o f g eo m c tr ic a th e se p o in t s , Y o u sh o u ld h a v e th e b a s i c k giv Cl r c le # e n f i g u r e in te rm s o f e ith er i T aD G n t s & lin e seg m en ts g le s n a o r . er u id i s ta n t f r o m th e c e n t c h o rd s o f a c ir c le are eq is e c t b th e c e n tr e o f a c i rc le , he p e rp en d ic u la r d ra w n fr o m Do u b le th e a th e c e n tr e b y a n a r c l e s u b t e n d e d at h e a n , Eq u a l l # T a T n A # g le s nce o f a r t o f th e c ir c u m fe r e . If g t he n an g le m ad e . w a y s 18 0 ra . su m g r e a o f a C ir c l e # A a Pe r i m A m a te s eg m en t g s o f a cy c lic q u ad r il o f o p p o site a n g le a mi g re n c e & A ce g of T he , th e p o i n t o f c o n ta c t , and a c h o rd is d a ta n g en t is d ra w n A n g le m ad e u1 d t h e ta n e n t be t w e e n th e c h o r d a n m eq u al e am e a rc in th e sam e s su b ten d e d b y th e s c i r c l e , a T o c lm , g th e c ir c l e # c irc le e ter o f a l Ci r c w H P - - m rea o f sec to r Leng th o f an arc A rea o f ring D el ista n c e m o v ed b y a w h e Num To b er o f N el fe ren ce o f th e w h e . rea o f an eq u i o e; . an ce m ov ed ta l c m Cir c u m A fe r en c e o f th e w h e e l Cir c u m in g M en su ra tio n pr o b l e m s t a , . . n a c irc le is g iv e , t he n f i n d t h e ra d iu s f i r st r l ier b y ta k in g e m i sta k e e a d a ta o f th e en tire e c k th e u n its q c a re o f th e fo l lo w in g ke ar The re fo re D i a m e te r - 14 cm , a n . If the d a s r a d iu s u n its are d if fe ren t, d I Je i g h t - 3 00 cm d a n (H a v e t so l v e d th e p r o b l e m hen c o n v er t th em 7 ) to th e sa m e u n its i stak e 7 v e r c o m m i t t e d su c h m y ou e Sa n j 9 1998 76 ) ay K u m a r (9 8 9 Scanned by ) . i li c 4 t ft e St ep T g S u So q D i m i e n s c c e ss T u t o r l a l s 6 n o . . (B w id e n . . h o r iz o m g c k n o w le d m m B i lj d s 1 . EJ l n d e c V o l u m Cu r v e d To tal - e o f a c y l in d er 2 su r f a c e a r e a / 2 n r h + 2n Z n n R h - c y l in d e r e o f ho llow h n r h 2 su r f ac e a r e a V o lu m l n l - (h n r + r ) h T SA c y l in d er o f h o llow Ou ter - 2 ( V olu m e o f a C o n e C SA , . o f a C one - T SA 2 Cqge: of a C on e 4 3 Vo h rr n yi n r 2 R T SA o f h em isp h er e 2 n r 3 - S j A lcz c u f a Bn ) m ay yo u w an t, t h eig h t ) ) &w h c a se o f S 3 2 Sl a n the y a r e sa m e ) e . sp h e r e rr ak e rr ak e + r p 2 3 _ l n 3 t X _ In to + r v r / o f h e m isp h e r e u lt a u c f er s r e n . C SA V P / ( 4n - lu m e o f h e m i sp h ere o f r in 2 n r h If yo u co u rse, = n r l + n - A rea 2 . 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Su r fa c e a r e a w i l l n o t b e a sk e d ) o f u sin g th e id en t itie s a Tan e = Se c e (i i i ) - Co z se e c e Co t - z e = in term s o f S in e an d C o s e e w r it t e n a s 1/ C o s e m ay Ra t i o n b e w r itte n a s S in e / C o s e m o f u sin g (a + b ) 2 , (a a l i z e th e d en o m in a t o r a s e y p a r a te t h e d e n o m 2 - b ) EI f a + b ) (a , + b i n a to r , F o r 3 (o r ) a E x (a , : - b ) - b fo S i n e B a s e m e n t , Ka lkaj i , N D 19 - . 3 L CM fo rm u lae etc . , rm at i s g iv en i n th e d e n o m in a to r + Si n 1 18 , S A m a y b e w r i t t e n a s 1/T a n e T h in k - t (u s u a l l y th en th in k t a k in g th e ir e r e v e r f r a c t i o n a l p a r ts a p p e a r s B . . I / S in e e o u ) en th en y o u h av e to tak e v a l u e s o f th e g i v e n p r o b l em w r it te n as itg vt : j sp h er es o f h e m i sp h er e C SA th ink pp e a r s t . 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Additional Info : Notes for Math Exam on 9 March!

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