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ICSE Prelims 2018 : Mathematics (Chatrabhuj Narsee Memorial School (CNM), Mumbai)

54 pages, 415 questions, 29 questions with responses, 36 total responses,

Aameya Kulkarni
St. Xavier’s College, Mumbai
XI-XII Arts (Humanities)

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( 1) M G r i 8 % u p ta p u r c h a s e d fin d th e , s a le ic e (3 ) M c re a m Is R 3 7 0 S har m a rs ia 6 * , an d in h e re l u r n o a r tic fo re a m c o n fe c r$ n s : s a le 7 9 10 in s s h v in , S ha r m a a *m o f th e n$ o !1c ra isct (A (5) T h e p r ic e o f a t a rtl le in c lu s iw is re d u c e d to S a le a T m (A m a r ke d m p r ic e - ru p g R , pe e s r a le o 7% R s , ho w l 80 0 . o m uc If th e p r in g R s 230 ill te tu m h d o e s h e s s d o e s th e c u s to m e t w 5 ra te o h w e to ill p a y fo r th e w ha t m a x r a te o f o ney sho pa y an p k e e p e r to y th in g m o re n e c d e in th e c o s t 6% ) 2 0 6 F in d its , u s !o m e r f S a le s T h i h th e a s ks rc u tio n a s p e r e n t f 12 % is 13 % ) o n w f e o R s 7 2 10 ) TI - a x d p c e p ke e p e r - no \ p r o n t o c o s l p r lc # o h ic h th e n o te to th e s h o w te f sa e s ta x on w c o s tin f S a le s T R s 6 50) f S a le s T a A il 6 % Ca a tc th e cu f S a le s ta x r a te o a r tic le r a te o 16 5 e t T h e a ho p k e p c r n s :. Re d u c tio n a s f th e T he ns c h a n e x te n t th a t o a t r a d io s c t th a n R s 1, 6 96 in c lu d is i S a le s T x p r ic e c o s tin ? (A o If th e d ir 8 S a le s T a x f v e ' h u n d r e d . d u e e p r ic e p r ic e a , s o (4 ) S m ith b u y s a r a d i m fo r R # 1 6 9 6 r c lu o n e r y g o o d s I a h e g iv ts M R r k (A f i n d r a e o f s a e s T a x IA t X h S a le a T a x is 10 % w ; f th e o purch so m e fb r R s 7 0 2 in c lu d in g S a le s ta x a n a r tic le (2 ) ( ie e ta p u re h a e d fa c e fa c e V A L U E D E A X N A T ) . m a rke p a y fo r th e a r ti le - R s d p r ic e Il th e a rtic le ? 9 0 le s s i Scanned by (6 ) A tr a d e r fr o m ds R spe n pe r c en c u s to m e r sho (7 ) A , ho w is 8 % R s (A (A , ill a w 2 , 4 5 2 5 0 in c lu d in g 9 % (8 ) T h e P r o fit% c a ta lo 2 5% = g u e p r ic e S a le s T (i) th e ax a m o u n fa H 8% is at c e T he n s: (9 ) D in e s h b o u g h t a n an lA (1 0 ) a s a le s ta x o d m a rke n s A s ho ( 1 1) A sho th e R f 10 % s - a r tic le e to a a r tic b e in g R s R se t is R ta x , f a l ta x e s ) fro If h e d e s ire s e tc he K m a f S a le s T ? r a te o an pu r H e p ro fit o f 15 p a id b y th e 5,9 6 ] 6 0) s d s pe n d s 2 0 % 4 5 ,0 0 0 T h e s o n th e r e m a in in g am o u n h a s to p a y , fo p u te r to p a y ax fo r R s - r th e c o m R 3 74 s , f th e o its p a c k in g c o st o n p r ic e If h e se ls th e a r tic le fin d h is p r o fit a s p e r , sho p k e e p e r g iv , fo r cen t w th e s 4 00) a n a r tic le a t a r e b a te o a n a r tic le a as t a d is c o 2 , 0 0 0 ) a n d p a id pe r un s a le s an d a F in d th e d is c o m a rke o n th e f 20% A fte r c h a r g in g o n th b a la n c e cen t (A ns: t o f 3 0 % fr o m - un R to s f 15 % d p r ic e o H f 7% o w e v e r. f 7% o n P ro fit % th e w a t th e r a te o f 8% n s: o n th e m a r k e d f th e H = 19 p r ic e a r tic le e s pe n ds R th e p r in te d p r ic e l s 40 he , o n se lls % ) h o le s a le r (th e p r in te d p r ic e T he s ho c h a r g e s ta x a t th e s a m e r a te ) p a id b y th e s h o p k e e p e r (A unto 4 3,39 0 0 8 ) p r in te d p r ic e a s a le s ta x o ta x d is c o se t h ic h in c lu d e d p r ic e es a F in d t 3 , 2 14 0 8 ; th e fin a l p r ic e re d u c e d o n b u y e r a t th e p r in te d p r ic e (V a lu e a d d e d an R - th e p r ic e m a r k e d ax o n 1 , 0 7 0 F in d h is g a in pke e pe r bu ys c u s to m e r 1, 5 0 0 s th e a f tic e 7 T r a x a c u s to m e r tr a n s p o r ta t iq n o f th e a r tic le th e a r tic le fo r p a y fo o , fu r th e r o s e a s o n d is c o u n t o f 4 % es a ha rge d a r tic le p ke e p e r bu y s R r p u te r t o f S a le s T d p r ic e 3 6 0 0 (in c lu s iv e s m a r k s th e a r tic le a t a c e r ta in com t o f S a le s T am o u n fo S a le s T g iv (ii) th e fin a l p r ic e h e h a s (A R r ) o th e lis te d p r ic e er y p a id b y a n a r tic le T he n he e tc fo tr a n s p o r ta tio n d o th e a r tic le , c u s to m n s: m on e p k e e pe r bu y s n s: o n h a n a r tic le in g o n tr a v e m u c tr a n s p o r ta tio n bu ys e e ru t 1 2 00 s t M V A T p a id b y th e p ke e p e r se o f lls th e F in d th e V A T s ho p k e e ie r - R s 48) Scanned by (1 2 ) sho A pke e pe r f 12 % fr o o (in c lu s iv (13 ) A se m p ro fit o (i) th e fR s lls re r se tr a d e a a n a r tic le a t its m a r k e m 1f th e er f t? ) p a id b y th e fa c tu lls it to 1s t h e c u s to m e o an u se r a a m o u n t o a w a s h in t a p ro 1, 8 0 0 re r a n (R s 7 ,5 0 0 ) a n d p ke e pe r pa ys a V A T p ke e pe r h in g m ac fit o f R s 1,2 0 0 If th e fV A T th e m a n u fa c tu sho s ho r a te o (i) R 1 200 ; fo r R bo fo R s g ht by a d is tr ib u to r fo so (iii) R 96 ; ld th e (15 ) A m an u V A T If th e (i) V A T p a id b y d is tr ib u to n s: fa c tu r in g co m T h e d is tr ib u to r a te o s a le f ta x p r ic e o a r so so ld th e f th e co m m o r ld R - r (A th e s a le p r ic e - R s s t th e r a te se T he lls it to w h o le s a le r a co n su m e r a t a co m m o d ity fo r f th is h in e fr o ld it to a tr a d e r m ac m R (e x c lu d in g 15 , 0 0 0 s to a r e ta ile r fo r R s ta x ) H 2 2 0 0 0 (e x , r a te w a s e so c lu d in g 10 p e r ta x ) F in d c en t 50 0 ) a co m m o f V A T p a id b y th e arn o u n t o a 6 ,7 2 0) s 15 , 0 0 0 , x ) 19 , 4 4 0 s a r tic le t e a c h s t a g e is 10 % (ii) th e n s: pa n y R s ta 1 8 0 ; c a lc u la te th e p r ic e e o o n th e s a th e V A T p a id b y th e d is tr ib u to r a n d b y th e tr a d e r if th e ta x (A R s a le s th e m a c h in e r O i) T h e tra d e r 2 0 ,0 0 0 s u r in tu m h o le s a le r w (A n a r tic le w a s , d th e t th a t th e c o n s u m e r p a y s , d th e tr a d e r an d b y th e S ta te G o v e r n m e n t o u n (14 ) A to a w h o le s a le r fo e s th e p r ic e p a id n s r e c e iv e am s IA fR o ha rge s c f V A T is 8 % fin d (ii) th e n s: d p r ic e d ity to its d is tr ib u to r fo d ity to , w a r e ta hat w ile r fo r R s r R s 2 2 , 0 0 0 in 2 5 , 0 0 0 p lu s ta x c lu d in g (u n d e r V A T ) a s th e th e m a n u fa c tu r e r 7 r e ta 2 0 ,000 ; A ile r ? m ou n to fV A T = R s 300) Scanned by ( 16 ) T h e p r in te d p r ic e fan o to th e s h o p k e e p e r a r tic le T he s ho S a le s ta x (u n d e r V A T ) is (i) th e c o s t to th e s h o is R p ke e pe r (A p k e e p e r in c lu s iv c o s t to t h e c u s to m e r (i) R ns: s lls th e se h a r g e d a t th e c (i ) V A T p a id b y th e s h o p k e e p e r (iii) th e 6 0 ,0 0 0 T h e s 5 0 , 8 80 ; to th e in c u s iv (ii) R s 7 2 0 f 6% tev a e r y s ta f 2 0% un t o p r in te d p r ic e g e F in d f ta x Government e o f ta x (iii) R ; d is c o s a a r tic le to th e c u s to m e r a t th e r a te o e o ho e sa e r a llo w w 6 3 ,6 0 0 ) s . 2 B A N K IN G in te r e s t is 1 0 % p e r y e a r ; c a lc u la te th e ie (2 ) M R (3 ) r At th e tim R K A hm e a ir N e c u r r in e o g D e fm a tu r ity g e ts R p o s it A If h e g e ts R 6 455 . cco u n a r e c u r r in d ha s 2 y e a rs s (A a t n s: t th e 66 2 50 , am o u n t en d o th a t R fon a t th e tim e o j ill r e c e iv i tge t = a t th e r a te o in s ta m ban k fm j a e sh w R a e sh w year i t in a acco m e o n t l ly F in d th e m g d e p o s it s a m q u n t th a t H , s n s: s th e e n d o f 2 ye a rs 7 950) , f 14 % p e r d e p o s its R e a tu r ity (A en t R e at in an n u m In s ta m e n 2 500 pe r , t = R m o n s a 50 0 ) th tb r fin d (i) th e in te r e s t p a id b y th e b a n k (ii) th e (4 ) M o n ic a r a te o f in te r e s t ha s C D th e m a tu r a ity fin d th e tim e A v a lu e o (in cc o u n t in th e U f th is a c c o u n t is y e a r s ) fo r w n io n R h ic h th e s B an k o 24 9 30 , acc o u n f In d ia an t w as d th e h e ld an d d e p o s ite d R r a te o (A 600 p e r s m o n f in te r e s t is 1 0 % p e r n s : th e tim e - th {f an n u m 3 y e a rs ) Scanned by ' (6 ) M o h a n d e p o s ite d R 80 pe r s th e a m o u n t p a y a b le to (A R ns: s h im m o n R s 2 5 s h a r e s fo r R (2 ) M ano (A in c o m jb u y s (i) th e e R s R - , 2 5 0 in s 3 125 s 3 00) m a r k e t v a lu e o n s : m a r k e t v a lu e d lv ld e n d $ 4 % m an o o R - f th e first y e a r f d iv ?d e n d s h a re s o ac h , , fa com f in te re s t is 6 % p e r r a te o F in d an num s s ( i) h is , f a ye a r f th e s h a re on (A (A c ha s e n s: , n s: r e c e iv es R a te o a c o rn e in v e s tm e n an nu m t l in c o m e ; o l R s 50 d iv id e n d a t th e R s 10 , F in d (iii) h is p e r c e n ta g e in c o m R s 3 , 000 ; p e r c e n ta g e In c o m - f he b o u gh t p a n y p a y in g 15 % d iv id e n d 2 50 s h a re s e a c h o f fa c e v a lu e p. th e m an n u a 2 4 , 0 0 0 ; a n n u a l in c o m v a lu e o o n th e s h a r e s fro e 2 0 p r e m iu m in a t th e e n d o e e r p a n y p a y in g 6 % d iv id e n d p e r fin d h is in c o m F 2 0 0 s h a re s b o u B ht 5 0 0 s h a r e s d u r in g the if th e d e p o s it a c o u n t fo r s ix y e a rs S H A R E S A N D DIVIDEI ) 10 0 s h a re s a t R (3 ) F in d th e d iv id e n d d u e (4 ) A a c u m u la tiv e 6 ,8 11 20 ) (1) R a v i in v e s te d R s 6 ns: in o n m a tu r ity , 3 (A th o fa e ac h en c e rta in d o , e - e 12 5 % ) it th e h a ll y e w ly f th e y e a r - R s 1, 0 0 0 ) b u sin e ss c o n c e rn a n d R s 400 a s d iv id e n d o n his s h a res n d th e m f d iv id e n d , = 8% ) Scanned by (S) M k u l in v u fa c e v a lu e R in t R s 150 a o w m uc h he w a n ts b u y ? lA (7 ) A pa n y co m f 15 % o T he an n u a o u n t o l in m an t su lA c h r e c e iv e bu ys a V f (9 ) ja y o w n s d e c la r e s L:L d o n a R s f S a lm ly 10 % o n f d iv id e n d 40 = in s h a re s a - R s fno ? If h e e v a lu e w s 50% se n s I, 5 0 0 ) b e in g R s 10 0 ) - o F f h is ? R n a s ha re o he n ? l in c o an n ua by R m e l in c o m s 15 0 , ? e ho w 8 , 1 8 4 ; a n n u a l in c o m R m in a l v a lu e s o fR s l lo m an e each - R y s e x tra s h a r e s s h o u s h a re 4 6 5 ; e x tr a d e c la r e s an an n u a - d he 2 0) l d iv id e n d h o h o ld s 8 8 an w h is in v R s , pa n y , w in th e fin d th e p r ic e ua in co m e co m o pa n y f S a lm p a id fo an 1, 4 52 ; p a id s h a r e - h ic h h e p a y s 10 % d iv id e n d h is in v o n s h a re s e s tm e n t H e r e ac - h s h a re R s 16 5 ) b u y s th e s h i u A t w h a t p r ic e d id h e b u y th e s h a re ? , 56 0 sha re s o d iv id e n d o lA d iv id e n d th a t A f in te r e s t T h e fa c e f a m pan y on ja y v a lu e o f e a c h s h a r e is R s 2 5 T he co m pa n y C a lc u la te f 9% r a te o . e s tm e n t 6 6 ,0 0 0 ; a n n a co m pa n y co m 2 5) (ii) th e ! 13 2 (p a r s f d iv id e r d p a id b y th e (i) th e d iv id e n d th a t ja y ns o l in c o m ann u m h is g a in in th is tr a n s a c t io = p r ic e th a t h is p r o fit is 16 % M n s in v e s t e to co m e o n s : th e a m o u n t o (8 ) A a he ha v ith 4 0 0 0 s h a r e s la m to ta (iii) If h e (A e st R at h a t w i l b e h is w , f 6% pe r k lu k g a in in th is tr a n s a c tio ila b le av a ill h a v e to in v w u n a t is W , an n u a o C a lc u la te (i) T h e O i) 200 to in c r e a s e h is a n n u a n s w s d iv id e n d a h is n a t is W R s 36 0 ; M - (ii) f th e d iv id e n d is 7 5 % (iii) If h e R to b u y 6 2 s h a re s m a n w a n ts to (i) H com e p a n y p a y in g a co m , 10 0 s ta n d s s n s: an n u a (6 ) A 9 0 0 0 in s h e n th e p r ic e r is e s s h a re s w (A R e s ts w ill g e t h is in w v e s tm e n t ill g e t = R s , if A ja y h a d p a id R l , 2 6 0 ; ra te o s 3 0 fo r eac h s h a re f in te r e s t - 7 5 % ) Scanned by ( 0 ) A d iv id e n d o f9% S 7 5% ret urn m (ii) th e a m o u nt (A M n s: u e o a rk e t v a V o b e in v to fa s h a re (12 ) A T h e fir s t in v n s: m a n se in v e s ts in co m e (A c n s: (13 ) M r R lls 6 0 , R if th e 15 s se c o n p a l in R s 75 tR s 41 s h a re s a t sh a re s a en t : 12 % s h a re s e lin s f th e p r ic e a t a c e r ta in eac h R an s d iv id e n d t to b e in am o un ra te o f 8% r at v e s te fR o d s R = 630 s 8 40 0) , 90 ? b e tte r ) fa p a n y p a y in g 12 p e r co m f a n o th e r co m d iv id e n d a p a n y pa y s - an an n u a t 12 0 o a s h re s o 6 s co m v e s te o 12 0 ; th e in R d b ta in s s h a re s o co m e G to o R - h a n g e in in am d e s te e s tm e n t is th e p ro c e e d s , 10 0 s s h a re f th e (1 1) W h ic h is b e tte r in v e s tm (A R o n c a lc u la te , (i) th e d e c la r e d w as o pa n y a f 8 pe r cen t t R s 9 eac h h R s 21 F in d t h e c h a n g e in d iv id e n d , at eac an d cen t 4 0 8 0 le s s ) 8 0 0 0 in 7 % R d R s d in v e s te d th e p r o 10 0 s , cee s ha re s a t R s A fte r 80 a year he d s (in c lu d in g h is d iv id e n d ) in 8 % so R , s ld th e s e 25 F in d (i) h is d iv id e n d fo r th e fir s t y e a r (ii) h is an n u a l in c o m e in th e se c o n (iii) th e p e r c e n ta g e in c r e a s e in h is (A (14 ) (i) n s: A sh o k an R s d S a n d e e p in v sh o k b u y s 7 5 ho R a t a p r e m iu o f d iv id e n d A (ii) 700 ; m o s f20% r e c e iv e R s e s ts 10 0 d ye a r r e tu r n o n h is o r ig in a l in v e s tm e n t 9 0 0 ; (iii) 2 5 % ) R s 18,0 00 s h a re s a t a If b o th eac d is c o u n t o f 2 0 % r e c e iv e e d by S an de e p h in b u y in g s h a r e s (A n s: , d q u a l d iv id e n R a te o w o f tw o d iffe r e n t h e re a s S a n d e e p b u y s R a t th e e n f d iv id e n d d o r e c e iv e f th e y e a r p a n ie s co m , s 5 0 s h a re s fin d th e d by S a n d e e p - r a te 1 1 2 5% ) Scanned by (15 ) Jo h n h a d 1 0 0 0 s h a r e s o f a H e so ld p r e m iu at a f th e s e so m e o o m f 50 % pa n y co m , s h a re s a t a an d is c o D iv id e R d is c o in (1 7 ) 4 0 , 6 0 8 in to tw s d th e u n t an co m es , o th e r p a r t is in (A n s: th e tw o p a r ts a r e R A m an has c p a r ts o b o th th e in o m h o ic e to in (A su c v e s tm e n s t v e s t in hu n ila b le at a p r e m iu co m pan y B a re a v a ila b le at a d is c o m u c h in , a ll , d o co m es p a n ie s h e in v W (1) If th e (i) 3 x (2 ) If th e p la c e m re + 4 < 24 (i) 5 x + lA (i) s o lu tio n s 4 5 t se n se an ? (A se t o t = o o d in an If th e in 9 % R a re e , un R s ru o m so n s: T he d p a y in g 8 % d iv id e n d g e in h is in c o m ld b y Jo h n v e s te R - d in 8 % R is R e s s 2 0 sh a re s 19 2 , fin d 600) s s ha re s a t 100 s 8 % p r e m iu m 8% th e a n n u a l , 2 0 ,7 36) o f tw s h a re s o pee t o f 10 % su m an d th e p r o c e e d s in R s ha re s a t 10 0 5 40 v e s te c han f s h a re s s qual f20% d th e 2 4x {0 , f n a tu r a l n an an p a n ie s A d it p a y s 8% d iv id e n d , is R s in fr o m th e re t u r n in v e s ts a ll R = m s an d B S ha re s h e re a s s h a re s w d it p a y s 7 % d iv id e n d f th e m an o co m in If t h e m an 936 fin d h o , o o f f v e s ts w 12 , 9 6 0 ) f w h o le 2x u m b e r s (N + n um fin d th e , b e r (W ) f n d th e s o lu t io n s e to f , s o lu tio n se to f 10 1 , 2 , 3 4 } ; (ii) s o lu tio , ) 2 4x x t is th e s e t o O i) f 10 % R I N E O U A T I O N S (ii) 8 16 r e p la c e m e n e st is th e e n t se t v a lu e o n h th a t if o n e p a r t is in dre d a re a v a N n s: 19 , 8 7 2 & pa n y A q u a lly in b o t h th e u nt o v e s te d c o m e fa c e a d p a y in g 12 % d iv id e n d th e n u m b e r o f s h a r e s s o ld b y Jo h n (1 6 ) ith w n se t - 10 , 1 , 2 , 3 , 4 5 }) , Scanned by j) i.? ! If t h e . re p la c e m e n t s e t is th e s c t o Of lA 0) 6x 9 (i) s o lu tio n n s F th e (4 ) re p la c e m 0 ) 5x (A 1> 3x (ii) l (ii) S o lu t io n (5 ) S o lv e : 2 (A set - 4 , ( ) W h e n , > + E R an d 8 , fin d th e so iu titJ n se t 3 c , , 3 2 2 8 {x : x {x : x w he re n 2 > 4 a 2y (re a l n E R an is x (R ) fin d th e s o lu tio n s e t o , l 2x an d 3 u m d x E R < } R x E p o s itiv }) e o d d in te g e r (A S o lu t io n s: n se t - ( I 3 5 }) . , y + 1 5 4 y + 7 ; if b e rs) {ln te g e r s } s o lu tio n y E 6 e en , 3 4, 5 6, 71 in g in e q u a t io (In te g e r s } (ii) y E n s: se t ( - (r e a n u m b e r s ) s o lu tio n 1, 0 2, se t - b , 1, 2 , 3 2 s : } v 4 an 2 S x < < d y R j) p lify (7 ) S im (8 ) 8 t- 3 (ii) W h e n y G se Jc 5 5 v e th e fo llo w (6 ) S o (i) y {2 - 3x (t o r z ) b e tw , t s e t is th e s e t o f r e a l n u m b e r s e n (i) S o lu t io n ns: (ii) 1 5 + x se t f in te g e rs 1 3 @ ril x R 3 2 ra p h th e v a lu e o f x o n th e r e a l n u m L is t th e se t o s o lu tio n s e t o b ta in e d (9 ) S o lv e an on a nu m d g r a p h th e 3 (2 x in e be r (A t 50 u tio n 5 so in c (A be r 5 )< 25 , g iv en n s : r e q u ir e d s o t o f - 2 1 2x 6 n s : s o lu tio n th a t lu tio n or 2x W x se t + - - A ls o 17 , re 3) p r e s e n t th e s o lu tio n } 8, 9, 5 Z 13 ; w h e r e x E R Scanned by (1 0 1 G iv e n p Q e lA ( l) S o lv e 2 2 \ (13 ) F in d ti e lA ns (14 ) F in d o d < x Q < n s 23 < R ;Q 4 3 2 s o lu tio n s e t o n th e n u m . w 2 0, 2 3 , fx , an h ic h s a t is 5 2, ' u ir e d I {in te g e r s } = b e r lin e s n um ls = x < W r ite 5 ; w he r e do w n th e e le m e n ts i;P x n Q = p n Q {4 E W x be r lin e fy th e in q " G ra a u tio n p h th e s o lu tio n s e t o n th e n u m b e r lin e d 4) th r s m a lle s t c o n w R I} x d if e r e n t x 5 v a lu e o R } x , n 4 2 f th e la r g e s t a n d lA 4x + o n tw o s 6; th e in e q u a t io + 1 5 11 15 3 an 3 = h th e 2 P t P ns 2x < { re a n u m b e rs } a n d = p re s e n 5 : {x = W h e re R R {x = o n e s n a fi h u lle s t c tiv e w h o le ru m b e t s s u c o f th e sm a lle s t is o n s e c u tiv e w h o le a h th a t th e d i f c e b e t w e f o u r t h tle a s t 3 n um b e rs a re : 5 0 , 5 a n d 5 2 ) Scanned by v e so + 2 ( A (2 ) F in d th e q u a d r a tic e q u a tio n (3 ) U s e th e s u (A n s b s titu t io I (4 ) W ith o u t s o lv in n o r , x W ith o u t IA o r n o t (6 ) F in d th e (7 ) lf m x 2 = an d (8 ) If o n e n x I is - f k fo r w e so v e q u a tio IA o r n ot e q u a t io n x v a lu e o k ( n s n s l to v + s th e q u a d ra t ic lv in g so 3 2 o r x fo r y , l) s e s o lu t io n s e t is ho w = { 2, 3 } if 5 ( 3 y (A )2 + n s: x 6 (3 y + z x 8 l) + 6 = = 0 0) ! y (ro o t) o f th is e q u a tio (5 ) - n s x z n s x 1 + x x = - 2x 1 is 2 is 1 0 = , w h e th e r x e n e q u a tio n ro o t s o lu ( tio n fin d w h e th e r f th e g iv e n 3 x a s o lu tio n o 0 ; fin d = f th e g iv n o t a ro o t o h ic h 3 2 n 1 is - x x + x z 2x a ro o t o l a so lu t io n l = 0) f th is e q t a t io n 0) = ) o f e q u a tio n k c z 1 is - + 2x 3 = 0 = an d n (A ro o to - 3 n s: m x a r e ro o ts o - 7 5 an f th e q u a d r a tic th e o th e r ro o t IA ns d f th e n - 3 2 e q u a tio n 2m + 2n - 0 lin d th e v a lu e s o f 9) e q u a t io n th e o th e r r o o t 2x - z + ? 6 = 0 is 2 , fin d th e v a lu e o f a A ls { , lin d ) 2 Scanned by (9 ) S o lv e h ea c (i) 5 2 f th e fo llo w in s e q u a tio n s b y o 3 2x (1 0 ) W ith o u t s o lv in g (i) 5 2 (iii) 2 x lA + + 7 6x + 3 (12 ) F in d th e (4 + eac 2 d e c im (i) x 2 n s :. f p 6x + 2 + a h o (m ' 2 + (iv ) 3 2 im ' , 9 , f th e fo llo g in a ry w f th e if th e if th e = p 5x + e = 0 q u a t io n s 0 = 2 = 0 ro o ts a re ra tio n a u n equ a l; (iv ) T h e l (r e a l) a n d ro o ts a r e r a tio n a e qu a l la n d a tic e q u a tio n a r e e q u a u n equ a l) l 4) - r o o ts o 0 IA 9 F th e f th e jEo llo w in g q u a ? ro o ts o n s: r o o ts o O i) T h e ; i u la (iii) X 2 + 1 1 X + 6 + 6x irr a tio n a l a n d 0 (A = ' m a x g th e tb m 77 e th e m a tu re o 1 )x + 1 + 18x . 0 ' f 2 (ii) r o o ts a re v a lu e o )x m ( 3 ) S o lv e (A 2 + 3 )x (p = x 0 = o lu e v a in r o o ts a r e (iii) T h e (1 1 ) F in d th e (ii) e x am , 6x (i) th e n s 0 = u s in f th e fo llo w in g q u a d ra tic ns in g e q u a tio n s m = fo r x 5 an or m q u a t io n a re e q u a 3) = d B iv e e , in eac h case . Yo u r a n s w e r c o r r e c t to l p la c e s 10 x + 6 9 36 (i) = an 0 i) 0 d 0 64 ; O i) 1 09 an 3x d z + 5x 9 = 0 2 76 (14 ) S o lv e th e fo llo w in s e q u a tio n - x (A n s: 8 2 6 G iv e y o or 2 2) u r a n s w e r c o r r e c t to tw o s ig n ific a n t fig u re s Scanned by v e (1 5 ) S o (i) 2 4 (1 7 ) F in d th e (i) (A (1 8 ) x E so S o lv (2 0 ) S o lv e w ne : n n s x x f th e , (A a bx 1 b o r n s: x b , 1 x 3x a + b w - 4 h e re = )2 u m {3 } ; W o r x a + - 2 (x 8x Q (r a t io n a l n E a + + 3 2 th e s o lu t io n s e t 25 2 (ii) (x e q u a t io n (ii) x E Z _ 0 = e n se t o 3 )2 (2 x a x (A lu t io 3 + (in te g e r s ) Z n s: 5 2 3 3x + ) 6 = 0 , x E R 0 ; w he n = be rs) he n x Q , th e s o lu tio n s e t - {3 , }) 1) b + O , b 0 ) Scanned by m s o V ?N G E R Q B m s @ & (I) n d (A tw o n a tu ra n u m N ns: um (2 ) F iv e tim be r b e r (A num (3 ) D iv id e 8 in to (4 ) F o r th e (A 9) h o le B ( ) t h e h yp o u rs 20 w fw 3 an o rk f a r ig h t tr ia n g le is 13 re p F e se n ts th e th e tr ia n g ie ' a (A (6 ) T he le n g th o f a x ' a s th e e n g th b o v e s ta te m n s: O n e s id e o m e q u a l to th e n u m e r ic a l v a lu e o (i) T a k in g ' x ' a s th e f w ho re c b re a d th s e sq u a re s is 1 17 o f th e be r n um 15 ip r o c a ls is o co m h tim cm an e w ill B w o rk If to g e th e r th e y a lo n e m o re f th e 5 cm w o rk o s id e s , een th e o t e r tw o w r ite a n e q u a tio n fin d th e tw ; o th e r s id e o in id ' x ' s th a t o u n lm o w n s id e s o f th e tr ia n g le - 12 cm is f ) th a n its b re a d th T h e n u m e r ic a v a lu e o f its a r e a is f its p e r im o - p le te th e ta k e to c o m p le te th e d th e d iffe re n c e b e tw f th e s h o r te r o f th e tw f th e tr ia n g le com ) e n t a n d a ls o s o lv e th e e q u a tio n to d a h is 3 v e ra n f th e ir p le te th e te n u s e o T a k in g o m uc l ta k e 3 0 h r s To i cm o A ta k e s 6 h o u r s e s s th a n B , one w a 7 su m d 5) in u te s ; fin d h o w m d th e ho e n u m b e r is 3 ) s u c h th a t th e s u m a re an b e r is e q u a to th re e le s s th a n tw ic e th e s q u a r e num R e q u ir e d n s: sa m e a m o u n t o in 13 h o n s: and R e q u ir e d p a r ts n s: w o rk w tw o p a r ts Q A I D R A T U A T I O N h ic h d iffe r b y 3 b e rs e s a c e r ta in w F in d th e (A 6 a re Q e te r v e ra n da h , w r ite a n e q u a tio n in ' x ' th a t re p r e s e n ts th e a b o v e s ta te m e n t So v e th e e q u a tio nr (A ns: (i) x 2 x o b ta in e d in (i) a b o v e 6 = an d h e n c e fin d th e d im 0 ; (ii) th e le n g th o f v e r a n d a h = 6 m e n s io n s o f th e ; its b r e a d th = v e ra n 3 m da h ) Scanned by (7 ) B y in c r e a s in g th e s p e e d re (A (8 ) C d u c e d by 36 T he n s: ar o in u te s m o r ig in a fa F in d th e l s pe e d A tr a v e s x km fo b y 1 0 k m /h r ca r f th e o o r ig in a ca r litr e r e v e ry o , th e t im e o f jo u r n e y s p e e d f th e o fo r a d is ta n ce o f 7 2 km is c a r 3 0 k m /h r - f p e tr o l , h ile w car B tr a v e s (x 5)km + fo lit r e r e v e ry o f p e tr o l (i) W r ite d o th e be r n u m fcar A (ii) q u a tio th e u se s n in 4 lit r e C x an jo u r n e y (A se llin g d n s: a n a r t ic le a lc u la te th e c o s t ( 10 ) T h e su m if th e (1 1 ) A S o be r n s: fo f p e tr o l u s e d b y o A car f fir s t n u m n it to d e te r m in e th e re c o v e r in ber d B in car c o v e r in N o o itr e s f o f p e tro l u se d by car A N o o f litr e s o f p e tr o l u se d by car B r R e o s 24 b e r is n q u ire d tw su c o C P ln u - o m o o se s f th e se d by as m uc e is a r tic b e r s is g iv o o 400 g a d is ta n ce en car 60 s b y th e w r ite do w n d by car B fo se an r litre x 400 itr e = B + 5 l litr e s ) = c e n t a s th e c o s t h pe r R , f p e tr o l u = x f p e tr o l u a tra d e r , n s: / n a tu r a n s: f litr e s o n u m g th e 4 0 0 k m f litr e s lv o r r e la R s p r ic e o f th e a ic le 40) tio n s ; S = n (n + 1 ) F in d n , 2 3) h th a t th e p r o d u c t o f its d ig its is 12 th e d ig its in te r c h a n T he B in o r e th a n c a r so p r ic e (A is 2 7 6 (A su m twoigit n u m (A f litr e s f p e tr o l m s o N (9 ) B y o f400 km o e w n g e th e ir p la c e s d ig it nu m be r - F in d th e nu m W h e n 3 6 is a d d e d to th is be r 2 6) Scanned by ( 12 ) F iv e y e a n b e i w , ic e \ha t o f h e r tw (i) th c a g e (ii) th e p r (A a w o m a n ! a g e w a i th e r q u a r e o ' g o o f th e (i) T h e a g e n o bo a t , m o to r b a c k ir a (A ns: f th e f th e f c lo th o 2 e s s ; th e R s w h a t is th e ( 5 ) S o s pe e d o m e s tu jo in ho y (16 ) A w m an sho c o st o p ke e p b u y s s s , th e tm n be r o a w om an 5 y e a rs - 3 0 y e a rs ) l w a te r s ti g o e s ]2 k m d o w , a n d co m es n s tr e a m o f th e s tre a m 3 k m /h r) - if t h e p ie c e w o u ld h a v e ? (A m e te r ns: w as 5 o n ge r m , th e c o st o w en t fo r th e le n g th f th e fo o d fo p ic n ic (A a c e rta in n u m be r o o f th e n s: N o o lA x , w r ite a ri e q u a 20 = o w lo n fc o th A s c - f th c w o u ho c o st w en m o re s d 10 ) F in d t fo r p ic n ic pc r b o o k ld b e 4 R an ig h t o f th e m b e r in c r e a s e d b y R s 10 w c o s ts B is th e p ie c e : ra te p e r m m R s 4 80 f s tu d e n ts f b o o k fo r R S 9 6 0 H ged w as r e ac h m e m e a c h m e te r o d c lo th f b o o k s th a t c o u ld b e b o u g h t ib r R s 9 6 0 s x an re m a in e d u n c h a n T h e b u d g e t fo r th e fo o d p ic n ic o be R l c o s t o f e a c h bo o k - - F in d th e s p e e d o r ig in a na a go is 9 k m lh in R s 20 0 l r a te p e F d e n ts f th e f th e p ie c e th e p a r ty s tu o f th e s tre a m d e n ts p la n n e d fa i e d lo 5 y e a rs so n f 3 h o u rs c o s ts o r ig in a w om an s pe e d h o se to a l tim e o T he 4 ) A p ie c e ( w he r a g e ce F ir d ' o n 1 a g e (ii) T h e p re i e n t a g e ( 13 ) A y e a r h m Tm ' so n 1 a l e liv e y e a r a 8 o so n e n l a ge o f he r w as - 16 ) Rs 8 T a k in g th e v e it tio n in x a n d s o 4 8) Scanned by f 2x ( \ ) ( i) (2 ) r d tA b = 49 n s (3 ) T w fin d th e n at 3 : : 9) qu in th e a re b e rs IA (6 ) W tlc ( h do 4 )? ys n th e th e tic lA k e t fo a re g im fE c e r s re 8 im 24 3 x : y 7b ) 6 8 is 5 : a dde d to ea c h be r nu m th e r a tio , be c o m es 2 3 d 40) an adde to e a c h te rm d 3)m (x fa r th e ra e b a ttle 6 d th e o O ; in ( 2 x en jo t m e y be c a m fn um ffic e r s b c r o n u b e fo r e ? O) in th e fa l e c r e as e 3 a c te d fr o c o f th e m eac n' l)? ys + y by e = be r an R R o ns s r a t io 8 an h te r m 1 5 s o th a t it b e co m es e q u a to f th e o ra tio b a s o th a t it ) d th e f a n a ir lin e r w a s w o rk IA x do n s b y (2 x ne x - l) m + en in 6) in c r e a s e d in th e 1 , 4 2 1 F in d th e in r a tio 7 th e 5 c o st o f in th e fi r e c re a s e 4 06) f o ffic rs so ld ie rs n o w is 1 d 22 d ie r s f so b a n te IA s d v a lu e o t io 3 : 10 F in d th e c e r ta in jo u m bm su n s: e l th e ra tio o an n e o be ? (A in th e a re T h e in n s by ne fa r e b a n te In e o r a tio m u st C :d m es w o rk + 8 b ) : (6 a + be 2 9 Fin d 5 ? bec o (x \8 = ns a n t ity m u s t (ii) W n a t q u a n t ity (5 ) T h e 5y + Fin d ( 5 o 3 5 n u m 3x : b e rs o nu m (4 ) (i) W 3y + T he n o o to th e n u m b e r w e re : f s o ie rs k ille d T h e 13 F in d f o ffic e rs - n um r a tio be r 2 1 ;n o 3 w as b e tw 3 1 b e fo r e a be r o e e n th e n u m o f o ffic e r s o f s o ld ie rs an - d so f ld ie rs in th e 2 17 ) Scanned by \ b a (8 ) If b (A c + a + c n s: (i) 3 a O i) lA po b a 2m , : a + ra tio b tr ip lic a te F in d : , (a po u n n s: a r e m a in a , 2 , IO an an d de d o d e rs c a re d b d b = b e r s h o u ld b e a re c o n tin u e 2 + d d a r a tio - 4 a m x 2 b4 4 (A a re an re 1 6 25 an d 18 c o n t in u e d 2 (a b 2 )2 5 6 , th e : q u ire d d r e q u ir e co m re c po com po u n ip r o c a l r a tio un de d r a t io ra t io d f25 o 1 - : l) I : 42 an d th e ) d 4 5 een l to 1 2 in r a tio T he n s: : b n y ; (ii) : f th e d u p lic a te p r o p o r tio n a l b e tw an d O 16 p r o p o r tio n ; fin d th e v a u es o f a an d b 50) su b tra c te d fro (A in p ro p o r tio n (14 ) W h a t s h o u ld b e a d d e d an un 2 b tio o f 2 1 6 : 3 4 3 m ean 0 4 - n at n u m (15 ) If (a 2 + : a th e th ir d p r o p o r tio n a (12 ) Q u a n titie s in d is e q u a to : 3y 4x po an th a t e a c h g iv e n ra tio w 1) , (iii) a re d - f b )2 + co m co m ra an O ; s ho c (i) th e fo u r th p r o p o r tio n a l to 3 6 (ii) th e (13 ) W b (i) r e q u ir e d (1 0 ) F in d th e (A : n b+ a + is r a tio s r a t io o d un d an b a + f th e g iv e n o co m : 2b n s: su h eac (9 ) F in d th e (1 1) c = to e a c d p r o p o r tio n h o (A f th e + in p r o p o r tio n (A a , n s: n um n s : re ) (a b + c d ) a n d (b 2 14 x ns: d , b, f th e an n u m d b e r s 13 , 17 n um ) a r e in c o 84 - q u ir e d 2 h eac m an d d be r x an - c o n tin u e a re b e rs 2 3 , 3 0 , 5 7 - an d 7 8 ; s o th a t th e 6) d 22 s o th a t th e r e s u tin g nu m be rs 3) d p r o p o r tio n p ro v in p r o p o r tio n e th a t a , b , c ) Scanned by (16 ) ir p q (1 7 ) If a * b p ro po (A (18 ) If (A (19 ) If q an d r tio n a n s: ' c a + c n s: ' = a , q is t h e p 2 : r p r o v e th a t p , l b e tw is th e b b, c een m ean an an m ean 2 + q - b is th e d u p ic a te a : m : r d ' ' a p r o p o r tio + a re p r o p o r t io 2 d d d r an _ ' q b , p ro : f ra tio o 2 q IA n s: p : r an d b an d a + c + c ) , p ro v e th a ' t c ' is th e m ean ' n a b e tw = , ' een ' a e th a t a p ro v , b , ' c b ) ' an d d a re in p ro p o r tio n in p ro p o r tio n ) na 4 l b e tw v e th a t e a c een h B iv an p 2 p (2 0 ) If z p 2 + r , p ro v e th a t 2 q e n ra d r tio d an d is e q u a l to Scanned by (2 2 ) 6 is th e x an (2 3 ) If (2 5 ) If m F in d th e d y 8x + p r o p o r tio ea n l3 y nu m 9 " 8x 13 y 8a 5b 7 8a ' b e tw n b e rs fin d + 8c 5d + (A : 5b ' p ro v (A ns: e th a t : x (2 6 ) If P = x + y ' v a lu e s o an d y f + p : d 4 8 is th ir d p ro p o r tio n 3 an n s - o f d 12 ) 1) d b 2x 13 - y a re an _ 2x fin d th e x n s : th e re q u ir e d n o s y 5b , 8c x b e rs e e n tw o n u m p p + 2y 2y (A 2) Scanned by (2 9 ) r ( ) n x e ac (A , h ns; R (2 ) F in d th e e m a in de r v a l o h e th e r (x w (3 ) F in d th e f5 o (A + n s: Q = yn o po ' a a ? x f c to fB c to r (A if (h e d iv is io n ' Bx 3a b x + = O , 4 is d iv id e d b y 2 x + 1 + r o k n s: o f f ax x ' + 2 2 1 3 : (x - " 9x ' + kr 5 ) is + 10 + a en ce , d e te r m in e th to r ) 10 b y 4x H x + 8 is d iv \d e d b y c 3 le a v e s a re m a in d e r 2) m ia 2 x i6 14 F in d th v a lu e (5 ) T h e p o ly n o m ia ls 3 x w he n w 2 ) is if ( x a a ls o a 5 ) is f ' der bx 5 8 - f k v a lu e o n e n th e (4 ) W r e m a in F in d th e c ase v e th a t : p ro - he n d iv i{k d b y x o ' lc r + ' f' k ' (A z a Jr + ns: 5x 3 F in d th e 3) (5 k k - 13 ue o v a x 2 th e , r e m a in der 2) d (a an f ' a ' + (A 1)x ns: z Q 7x - + 5 le a v e th e s a m e r e m a in de r 5) Scanned by \ W f (x ) n en d iv id e d b y (A (7 ) n s: W ha t a be in lA (9 ) If n s (10 ) G iv (A r sh o u ld b e 1 is n s: fa c to a m r o r o 2 + 7 - ia l 14 a fa c to f x 2 d an d b a x (13 ) S h o w ex th a t p r e s s io 2x + p r e s s io n c o m (14 ) U s in g th e R 7 is IA n a p le te ly e m a in . u s in de r T he o 5 x fin d th e rs o ' ' a 3 2 f 2 3 m f d ' (x x + b d z e ro a s he n w ' x 4 + 3 x re s u be r 2 x ltin g p o ly n be to + x a 1 dd e d 3) = o r n o t ia l is om ? ia l) 3 + b f (A a n s: a - b ; c a lc u la te th e a c + + 5x r der (A ' 2 ) is 5x g th e fa c to re d an th a t w h e n t h e so e s re m a in an th a t sho w o ' a d e r is 5) v a lu e s o f a a n d b ) 6 le a v n s fa c to r , fa c to a re v a lu e o f th e re q u ir e d n u m n s : 6 x x v a lu e o 2a + bx , (A f o z + 3x 7x 6 th e r e m a in , v a lu e s o m = 2 + f it F in d th e en ex F in d th e f th e g iv e n p o ly n o 3 x 2 x ) fa c to r a r o (12 ) U s in g th e F a c to r T h e o re m th e g iv 30 d d e d to 2 3 1 is an 3 x d e r is = x n o t a fa c to a 8 is d iv id e d b y th e r e m a in d e r is 3 ? , h e th e r e n th a t x (1 1) P o yn o a d b e w 2 is x an 2 bx r e m a in , x x 2 x 1 th e 7 - n u m e te r m 3 + x x + d iv id e d b y (8 ) D = 2 z (x ex IA h e n d iv id e d b y w a fa c to 1 lx th e o r e m fa c to r is e th e ns: a = 8 2 - r o an z f 3x 2 ) (3 x H ns (2 x + 2 3 + n an d x 2 is 5) = 5x 2 H ence , fa c to r is e l) + 4 p r e s s io d b 1 x en ce fa c to r is e th e g iv 7 x ) (x 2 2 ) (x 2x 1 + en ) co m p le te ly Scanned by (15 ) F in d th e (x + g iv en v a f o ' 5 ) a s its fa c to p o ly n (1 6 ) If (x 2 ) is (i) fin d th e O i) lu e w ith t h e fa c to d an Fo rs ia l c o o m a ' a m r o b ' ' th e v a p le te ly (A r f 2 3 th a t th e so lu z x f es o n s: a f p v a lu e o [ 2 x a f - b 2 B . 1 A = x a ' 13 - an m d an ' 3 + ia l x b as o ' , d b a x bx (x 4 5 ha s a bo v e b ta in e d 31 ; = 2 + 1 ) (x + , (x 1) a n d fa c to r is e th e 9 ) (x + 5) 2 f p v a lu e o v a lu e s o ' px , fa c to r is e th e a D M ( 1) F in d th e yno po , y , a an [ ] d b IA , n s bo v e ex p r e s s io n co m e te ly p A T R IC E S if . - 2 , . - 3 , - 2 an d b = 4) [ Scanned by o ) f A 4 = = B (4 ) f po (A A + m a tr ix A ib le n s: A t _ A + X = : ] (i) A + A [ 1 3 ' _ . B .. d B [ = ] ; fin B [: = O i) t d t B + ] fi d B [', a _ n ; t' " SP . . . A t, i. E s t A an d B t [f t " ; A " n o t p o s s ib le an d B " " po s s ib le l a) 4 i 2 d c ] C ]) [ fin d , an = B + B (i) B A ss ,l; B [: ] 0l 1 (ii) X B : = , th e n s o lv e fo 2 A + B = r 2 m a tr ix X t5, ' suc h th a t ; On x = Scanned by [5, ] (7 ) G iv e n ' " i ' " - . (9 ) IF A (1 o ) = L etA G iv (A = ] , , ns. A B - .n If y e s d B , . . Tr i' A e v a lu a te [ ] e a re a s o n d = 2 3A [a i , l, ] " ; + 2 l, s th e w n d . . Tr i. h e r e l is X .. c h a u n t m a tr ix a o M A . 2X = B f o rd e r 2 ib le ? p ro d u c t A B p o s s fin d A B 12 1, Scanned by ( 3 ) If B an d C a r e tw o m a tr ic e s s u c h th at th e m a tr ix A s o th a t B A (14 ) n d th e W (A r ite do n s: m a tr ix w n : M su c . (i) th e th e o r d e r o f th e = 1 V W Q e r ify th e e a c 14 j n r ite ne ra h fo llo w T o = [: l ) " n s [ f th e m a tr ix ' , m a t r ix X = a x ] X b [ - , 16 O i) th e 2 = ] IA m a tr ix n s: M = [ 4 50 X 1 [] [:] - m (Q lA h th a t M rd e r o th e m a tr ix x C = B A e r in g in m an r it h m ? g P rQ m s io rm u ) A P an d th e n w h e re w r ite its g e n e r a l te r m 13 th e fir s t th r e e te r m s o f th e A P = 5, d = 3 Scanned by ( 1 \ 3 W n 4 W n ic 5 F in d th e ic h te r m h te rm o f th e A P 5 3 , 4 8, 4 3 , o f th e A P 3 , 1 0 , 17 . id d le te m 6 F in d h e th e 7 F in d th e \2 th te 8 F in d w ' k te r m s o 9 ' , r 5 5 is ill b e 8 4 o en m o f th e A P 3k + 6 6 an te e ? rm th a n its 1 3 h te r m ? 37 0 4, 2, 4k + 8, 2kz + + , f th e A P 7 , 10 , 13 , d g a t iv m o re f th e A P 2 13 , 2 0 5 , 19 7 , a te r m 2 w n e r n o t d 3k 2 w h ic h te r m it is 100 , . If y e s , f n d + 4k + 4 a re c o n s e c u tiv e th e th r e e fan A P if th e 6 1h te r m F in d th e A P If th e 9 th te r m 11 If th e o su m 14 th te rm is T he o su m o f a n A p is f th e 3 rd (i) F in d an is 19 z e ro d th e 8 th an , d l th te r m th e n p ro v te r m s o is 15 e t h a t its m o re th a n 2 9 h te rm f a n A P is 7 an d th e 1 1 th te r m is tw ic e its 1 9 h te rm su m o f th e 7 th an d th 3 , th e n fin d th e 10 te r m f th r e e n u m V n e II (O n M m a 13 o fr o rm th a t k so 10 12 rm Is th e fir s t th e s u m o 3 b e r s in A P is an d th e ir p r o d u c t is 8 F in d th e n u m b e rs n s (s )) ttw nr . f th e fir s t 2 4 te r m s o fA P : 0 6, 1 7 , 2 8, Scanned by J c i ) F in d th e 14 H If a n 3 - 16 fan A P 17 F in d th e 15 a re 18 F in d th e 4n 19 m 21 9 + + su m an o d th e n s h o w n = 12 30 su m o T h e fo + 67 + u r an Y o u sa v e R s f th e th e n ex t th e 2 0 0 m g le s d R te r m s o t iv e s o th a t th e ir la s t s Fo rm a 3. an A P A ls o fin d s zo ln fan A P ho w se 4 1h te r m a n d 9 1h te r m s y u m o ro b e r s b e tw 10 1 een an d 9 99 w h ic h a re d iv is ib e b e a gj f a q u a d r ila te r a l fo a n g le s 10 in th e fir s t w s n tn a r e s ta c , a 2: , ld b e ta k e n 1 ), th e n f n d th e A P + n a tu ra P n a ll t h e 2 7 5 If in 2 00 o gs re spe c 1 sho u d 5 an fin d (4 n th a t a f th e fir s t 1 7 E E ? c a tio th e n 20 , if S n , b y b o th 2 T ym f:7 o m a n y te r m s o f th e A P 5 4 5 1 4 8 , , . is 5 \3 ? A ls o fin d th e s u m o f h t e te r m o w te r m 15 Su m abo v e ee k o rm an A P If th e co m m o n f th e y e a r th e n in c r e a s e d d if e r e n w ee k ly ce sa v is 15 , in B s b y th e w e e k y o u r s a v in g s b e c o m e R s 5 9 5 F in d n , k e d in th e fo o w in g m it 1 8 in th e n e x t, a b o v , lo s s p la c e d an d ho w m a 20 o g s in th e b o tto an n er e o gs y it an a re d so o n [n h o w th e re in th e to p m an ro w m y ro w ro w , 19 in a s a re ? Scanned by 22 A m an fo rm e r p a y s R s 9 7 5 in by R s 5 T he a e n t ir e a m o u n t m o n th T o u n e 1 V e r ify th a t e a c 2 W I (O r ite lm s ta e n ts , e a c f t rs t in s ta lm e n t h in s ta llm is R s 1o o E n e r a T h o th e fir s t fiv erm G E M E ? 4 5 F in d th e 7 th te r m W h ic h te o rm s o G p an f th e G p ho w an se n o f th e G P 0 4 0 8 , S S lO N (G U , d th e n th w r ite th e n e x t th re e te r m s is te r m f th e G P F in d th e 5 th te r m fr o 27 ' Is . th e e n d o f th e G P m F in d x so th a t x + 9 7 P ro 8 T h e fir s t te r m v e , x + 6 th a t 5 th , 8 th , l l th te r m s o F in d th e 6 h o te rm f a G P is 8 1 o d 4 an a re fa G P an d th e 1 n 5 n + l 1 6. , ? 2 : 27 6 n b e in g e s s th a n th e w h a t tim e w ill th e ) f th e fo l o w in g is e te r m C PR O G ? 2 3 e n t b e p a id ? 11 T yo ly in to 2 I 9 2 , = 3 ' , 16 2 th r e e s u c c e s s iv e te r m s o f a G p a re th r e e sq u a r e o s u c c e s s iv e te r m s o f th e th ir d te r m is fan equ a a n o th e r G p l to its ls t te rm f th e G P Scanned by 9 T h e p r o d u c t o f th r e e n u m b e r s i n a G P is 2 1 6 a n d th e n u m b e rs k t i a e n n p a ir s is 15 6 , F in d th e n u m b e r s 10 T h e th ir d te rm e e I O (i) 11 0 15 12 F in d th e 13 F in d 14 T he n + o 0 0 15 + d S n if 6 + a l f n te r m o 0 0 0 15 f:3 o 5 - s o + , F in d th e p r o d u c t 3 4 + + f th e p r o d u c ts f its fir s t fiv e o f th e te r m s + 15 3 6 + 320 - n To n te r m s To 8 te r m s + 12 a o o (S e rm s f J + su m sum an su m f a G P is 4 Su m O fn T n F in d th e (ii) o su m f a p r o g r e s s io , r (2 " is n 2 - 1 ) S h o w th a t it is G P F in d its a c o o n r a tio 15 G iv en a 16 T he su m v 17 c A bo y ith w f th r e e o F in d th e T G P a 0 b e r s in P b e m a m sp e n ds R fa s ce ss T7 - D e te r m in e d 5 7 64 G P is 2 1 ro an d th e su m f th e ir s q u o a re s is 18 9 s) 15 in th e fir s t w w s q u a re m u c h is 10 c m , in a ee k, R s 4 5 in th e l w ill h e T he m spe n id p o in ts o d a f its se c o n t th e s d en id e s w ee d are o k, R f8 , w ho se m s w ee jo in e d id p o in ts a r e a g a in jo in e d to fo rm o n e is c o n tin u e d th r e e m o r e tim e s F in d th e s u m o f th e a n o th e r s q u a r e T h e p ro a d b e rs nu m n e s id e o an n u m th ir d w e e k F in d h o 18 729 - 13 5 in th e ks to fo rm m o re s q u a r e a re a s o f a ll s q u a r e s Scanned by C O T he A B C do w n (i) (2 ) tr ia n g le A A an , B , P tr ia n , D g le A B C is n e c te d in th e co o r d in a te s o fP co o r d in a te s o f th e im , re B , re o O M E T RY re R E F L E C T IO N fle c te d in th e lin fle c te d in th e o r ig in o n to f (ii) A (8 6 ) ; C 4 ) a n d C (3 , 7 ) is fir s t d C an = R , th e c o o r d in a te s o (i) F in d th e (ii) F in d th e n s: O (\ 2 ) B (4 d th e n A p o in t P is (A - a x is x o r d in a te s o a ge o r d in a te s o o o fp f th e im y - 0 o n to tr ia n g le A tr ia n g le B C ax is W te d C an C e u n ge a o f its im (8 a g e a re f e c tio n in th e y de r re fp u n de r re lle c t io n , 6) a x is in th e y - ( 8 6) , (3 ) P o in ts ( 5 0 ) a n d (4 0 ) a r e in v a r ia n t p o in ts u n d e r r e fle c t io n in th e lin e L ; p o in ts (O 6) a n d (0 5 ) a r e in v a r ia n t o n r e fle c t io n in th e in e L 2 (a ) N a m e o r w r ite e q u a t io n s fo r th e lin e s L I a n d L 2 (b ) W r ite d o w n th e im a g e s o f P (2 6 ) a n d Q ( 8 3 ) o n r e fle c tio n in L N a m e th e im a g e s as " an d Q r e s p e c t iv e y (c ) W r ite d o w n th e im a g e s o f P a n d Q o n r e fle c tio n in L z N a m e e im a g e s a s P a n d Q , , . , , , ly r e s p e c tiv e (d ) S ta t e o r d e s c r ib e a s in g le tr a n s fo rm a tio n th a t m a p s Q O in e L r is x a x is w h o s e e q u a tio n is y (A n s : (a ) T h e o n to Q - , in e L 2 is y T he (b ) P (4 ) (i) F in d th e (ii) F in d th e (A n s: (2 = , 6 ); Q a x is w e q u a tio n ho se is x u e c tio n o re fle c tio n (i) P (5 3 ) ; (ii) Q (2 O - ( 8 3) . , f th e p o in t P ( 1 3 ) in th e lin e o f th e p o in t Q (2 , 1) in th e lin e y re , , , , x + 2 a 3 = 0 7) Scanned by (5 ) T h e p o in ts P (5 , 1) a n d Q ( 2 im a (A n s: (6 ) U e s P and th e Q o f p o in ts P c o o rd in a te s o 2) a re r e d Q re s p e c , an fP . - tiv e ly in in e r - 2 U g ra p h p a p e r to fin d th e 2 T a k e 2 c m e q u a l to 2 u n its se , g r a p h p a p e fo r th is q u e s tio n P lo t th e p o in ts P (3 2 ) a n d Q ( 3 , , a x is x x ( 1 1) se a o n th e fle c te d ln lin e (T a k e tw o d iv is io n 1 u n it o n b o th th e a x e s ) 2 ) F ro m P a n d Q d r a w e r e n d ic u a rs P M a n d Q N p p - (a ) W r ite th e c o o r d in a te s o f p o in ts M a n d N (b ) N a m e th e im a g e o f P o n r e tle c tio n in th e o r i in g (c ) A s s ig n th e s p e c ia l n a m e to g e o m e tr ic a l fi u re p M Q N a n d fin d its a r e a g (d ) W r ite th e c o o r d in a te s o f th e p o in t to w h ic h M is m a e d o n re fle c ti i on n pp (i) x ? is O i) y a x is i i i i i ( )o r , (A g , C o o r d in a te s o fM C o o r d in a te s o fN n s: (3 , 0 ) a n d = ( = 3 , 0) (7 ) U s e g r a p h p a p e r fo r th is q u e s tio n T h e p o in ts A (2 , 3 ) B (4 5 ) a n d C (7 , 2 ) a re th e v e r tic e s o f A A B C (i) W r ite d o w n th e c o o r d in a te s o f A , B C if A B C is th e im , , , re f e c te d in th e n th e c o o r d in a te s o (ii) W r ite d o w w (iii) M he n en A BC C m (1) F in d th e re o e c te d in th e t io n th e s p e c ia (2 . B is an _ (4 fA B :, C if A A B C is th e im f th e q u a d r i a te ra B C C B and fin d its , e o , 5)a n d C . (7 is o s c e le s tr a p e z iu m ; a re a o , th e r a tio 1 : 2 o n th e x (A w , he n ns: a ge o fA A B C , a x is w a re a 2) f q u a d r ila te ra l B C C c T I o N F O R M U L 1 CI M p o in ts ge A B C a x is x ln a m 3 ), B , a o r ig in B . 2 1 sq on x ) IPAMWFORMULAE h o s e d is ta n c e s fr o m th e p o in ts A (7 , 6 ) a n d B R e q u ir e d p o in ts u n it a x is a r e : (9 0 ) a n d ( , , ( 3 4 ) a r e in , O) Scanned by (2 ) P o in t P (x O (3 ) x F in d th e a re \ (4 , 8y - co o r d in a t e s o 2 (5 ) F in d th e lA (6 ) In n s: ha t w co : C n s: a lc u (A H n s: e nce f th e o r d in a te s o 5 (A ve d th a t - c ir c u m c e n tr e o w f p o in t P ( = r a t io r a t io is th e lin e o r d in a te s o T he 3 r a t io , 10 (A n s: A ls o in w an d th e fin d th e - lA th e r e q u ir e d p o in t , th e o co th a t 21 ) - c ir c u m c e n v e r tic e s A , B d C an tr e o f th e g iv e n tr ia n g le jo in t o f A (4 e re p o in ts (4 , 2 ) a n d (3 o jo in in g f in te r s e c tio , - (3 3 ) , 5 ) a n d B (6 3 ) in th e , , p o in ts (2 , )a n d 5 ) d iv id e d b y th e (7 , 6) jo in in g th e n ax is ? A ls o , n - p o in ts (4 , 6 ) a n d ( ( , 5 , O) 4 ) is d iv id e d b y th e f th e p o in t o f in te r s e c tio n o r d in a te s o f in te r s e c tio x n q u ir e d p o in t o f in te rs e c tio in e h ic h th e lin e y v e 2) f th e p o in t 2 ;5 - r a t io jo in in g , ) 9 , is 3 4 ) P ro , f th e tr ia n g le A B C ; w h o s e h ic h d iv id e s th e w 8y x , q u ir e d co ( 2 0 ) a n d B (3 h ic h th e p o ir t (5 4 ) d iv id e s th e lin th e r e la te th e n s p ro 3 p n s: in ra t io fin d th e (A 21 th e p o in ts A m 6 ) , (0 . 4 ) a n d (6 , 2 ) r e s p e c t iv e ly F in d th e r a t io ) is e q u id is ta n c e fr o y - ( , 3) Scanned by (8 ) T h e o r ig in O , B ( 6, 9) a n d C th e r a tio 1 : 2 (A (9 ) p o in t P n s: S ho th a t w A (4 F in d th e P (3 (10 ) F in d th e (1 4 o r d in a te s o co (1 2 ) T he m n s: m , m id p o in t o u r th v e r te x (A n P ( C fA C 2 , 5 ) is ( 4 ) A B C is C (A = (a n s: tr ia n g le a n d G a , a 1), fin d = ' 7 ; b a = ' an Q f th e lin v a lu e o f ' e se gm sho w , th a t p q :B C = BC ) = gm en (A n s: en t jo in ' m e se A ls o t jo in m - in g th e p o in ts 5) in g th e p o in ts P (4 , 6)a n d v e r tic e s o jo 2 th e n s in in g (3 m An d n - 6)a n d , (0 C v e r te x F in d th e A B C D f a p a r a lle lo g ra m 2) , ( 4 3 n ) is (1 2 m , , ) F in d th e O) id p o in t o f A B o r d in a te s o , Q (2 , 4 ) is f v e rtic e s A m , id p o in t o f B C B an d R ( 1 2 ) is an , d C P (4 d m _ lA C en t d p o in t p d iv id e s 0 8 in 1) , th e c o a lc u la te ; n o , gm In tr ia n g le A B C m f tr is e c tio (8 2 ) a r e th r e e 2)a n d D an f tr ia n g le O B C 1) ; p o in t p Q , fin d th e (1 e se d f f p o in ts P (4 = v e r t ic e s o id p o in t o f th e in m f Th e lin an v a lu e s o (13 ) o f th e Q o , M id p o in t f th e fo id p o in t en ce , 2 ), B (6 , , p o in t a (1 4 ) H n s: , A ) is o r d in a te s o co Q ( 2 4 ) (A (1 1 ) 5 m , 2)an d 8 , ; p o in t , 3) a re , o r d in a te s o co ( 2 3) = (12 ' , b 5 ; B C 3 ) is th e c e n tr o id o e n g th F in d th e ' = 5 u n it s f th e o tr ia n g le If A = (1 , 3) , B = (4 b ) a n d , f s id e B C ) Scanned by O a u i ti w (t h s i A B \H n fwe q m - \B D E 3 A a d A D (i m \R a re A C A B c is , p c M a r ia n hat a A B D is 3 6 = c m a n s im i a r d C D a g le k o d b ) Bc u k r ig h t a n M e A D g le d a t v e n e x & c A D oo 6 4 m d ic u a r m er p e n 4 = is a l t i t B D u l D E m m e f p n ) a i u / ; fin d th e e n g th o f A D c m / A ls o lA (4 ) In lin A PO B (5 ) In o in ts o n D E (A 3 re , , an A B n s : a re a - A D E) (a A g iv m g le B B C c m a n ) A d A C If P O 90 = 6 a re a o , A B = r e s p e c t iv e 54 cm A rc - 6 d an cm a cu a te th e a r e a o 2 r g s id e , p B fin d th e = A r (A A D E ) (t r a p e z iu m B C d ED L \ ) ) a o n A B c m 2, an C c m r l 5 = en t gm 12 0 = d A B C = A D E e se (& fin d n s t h e fig u th e P , \ , fA 12 ly QA QO = QO A em an su c 2 ) 9 p e r p e n d ic u la r s to cm (A an a re a o 270 n s: d A C d = 15 cm 2 an d th e n th e f ) D cm h th a t a n g le A E D f A B c cm a re = an 90 a re a o d E an a re d f& A D E ! Y . Scanned by r (6 The scale o fm a p is 5e 000 In the m ap i) dtm en sio n s i A B = 2 cn , BC - 3 5 u l n an ca lcu a e iiihe , lo w n g o t A B C o flan d ha s the fo ar p m \ a lri d an gle A BC 90P = le n @ o fside bc in km . . o f the lan d (ii) tbe area o f he plo t in Km IA ns (i) l75km ; (ii)O 875 sq Km ) C ' m A x an g e tan k has len gth = re j 4m tan k is m a de w itl ca\ ity 240 c m w , idth = 3 m 3 Fin d and e\ 3 o fthe ity 30 m A sm allm o de - fi) d e dim ension s o fthe m o de . i m ode (ii) dbe ra io betw ee the totalm rf e airea o fthe tan k and ts . lA ns (i) 8cm x 6 cm 5 cm x (ii) 2500 1) . (81 In the f gu re giv en a lo n gside / , A n gle Qps = an gle R pT and an gle PRQ = e PT S an g R and PST are sim ilar (i)pro v e d at u iangles PQ (ii)If PT ST 3 = . \ QR : Pk 4 ;fin d the ratio betw ee Scanned by (iv ) M h p o in t a rk w h ic h i d f e c o rd M *n *k A # tr (4 ) C o n m c o n {u c lio n O e (ii) p i e q a idia (5 ) A p s tra ig h t lin e \id \# w BC w iM A B *w h th a t AB m M a r k th e tw o D e s c r ib e th e 7c m - an d A B is 8 c m \o n * (i) a lw a y s 4 c m fro m \c e and I e q u fo n m i t d f PB th e \e n * th o f p o in t P w e f n m A B , l c D BL Bc M ra w 8 . cm a nd a n g le M L l r o td th c e a s u re a ru th e lo c u s a n d d e s c rib e o - \c n g th a M )" \ n e M t by \ fP t in t w h ic h is lim A B th e p o in ts x an d Y , w h ic h a re 4 cm fro m A B and equ id i# tn n c e fro m A and B gu n A X B Y n b o th th e a x e s 2 c m - \ u n it o k T e a i n o t e s u i f th s q (6 ) u * g ra p h pa pe r o r 2 7 1 \) b (5 , 3 ) a n d C ( , ) o t th e p o in ts A ( (i) p A and B i t e u id is ta n c e fro m h e lo u s o f p o n s q t t c u t r s n c o ii ( ) f m A B and A C e q u idis ta n c e ro t i a n f o o s u l c p o h (iii) c o n s tr u c t t e AB d P is e q u id is ta n c e fro m h at pA - pB a n h t c p u s i t n (iv ) L o c a te th e p o h P A in c m o r d th e le n g t (v ) M e a s u re a n d re c , , an d AC Scanned by (7) C t r ia n o n s tr u c t a ( i) C p le te g le B C P g i v B C e n th e r e c ta n (a ) P s e q u id is ta n c e fr (b ) P is e q u id is t a n c e fr o m C e a s u re a n d t h e le n th o o m iJ) M d re c o r g le A B C D 5 A B o m g B P c m h th s u c a t a c h (2 ) fn o p th e A E (A (3 ) A o c f le in t E fig th n s : ho n r o p c e n tr e 42 c m o d A B a c m is d r a c m g n n s : C h p is in O g le p B C 4 5 " e h c o rd E ra s u c d iu C D ) h ic h e te r w m a t te r s e c ts a t th a t ro v e d ia a fin d t h c m a n C IR C L E S rd A B h th a t A B s u c d f A B c e n tr e o C D e is 3 it h w a n d D a n g le A E D a n id s t E If C E c m g th 4 8 h th f th s o t a e c ir c le ) C c m w n is a t in th e d is t a n c e s a n e c o f 1o ir c le fr o c m th e c e n tr e o m f in d it s d is t a n c e fr o f th e c ir c le th e c e m n A tr e o n o f th th e r c e ir c le c h o rd ) a n e c ts (A lo a n e n t c jo in in g th e id p m o in p a ra ts o f t w o p a l e lc ho rd s o o th e r a n d lie ra f a c ir c le p a n o p po a s s e s th r o u g h th e it ro v e f th e n ir c le c rd C D f le gm (5 ) C h o r d s A B a b is e 1 c m se e o f o 4 g th 2 0 (4 ) T h e in o r 4 r d h g iv h c B E l A n s 24 c e n tr e c u re e d C D If O E rd A B e e ts m o a n c m d B C a n 16 ( J) C h o r d A B 4 a t d C D o ir c le ; fin d t h e ra f a c If A B d iu s o ir c le 36 = l th a re a e c c m ir c le , C D (A le ] t o = n s : e a c 48 R h c m = 30 a n d t h e d is t a c m n c e b e tw s it e e e n s id e s o th e c f th e h o r d s is ) Scanned by (6 ) A B d C D a n tT A B a r e tw o e d C D a n th e c ir c c pro (? ) T w lin (8 ) O ith t o tw P rov u ne (9 ) In th e A D C A a n q u a l c h o rd s F in d a , ( 2 ) In th e A B hn fa ith c e n tr e t \ 1 ln t P M eet nt u \ h id * an A R lA re h rse c t e a c n J p Q pe rw c ir c e Th e o th e r \t w d ic n u a r ly h ig g e r h o H In t p (A n s : is = C , g le A o c an j a d o in in BC (i) L D E C an d If L g fig u L (iii) 3 5 0 PQ PR = an d L PR Q = s a c ir c le th ro u B A C (ii) L 6 7 ; A C is re A ED = = 1 18 D A B a , p rtw e th a t th e u n \r \ h lq e c \s p 9 t to th e ce n tre 1 6 th e c \r c \e 7 0 g h th e p o in ts f n d : L D B C d ia m {) c a lc u la te 40) n s d D \ 1o = n n c \o s e r (iii) L O A C , n (b ) p l = p l) d B in te (ii) 12 5 ; d jo in in g fi u g Q L figure (1 1 ) T h e g iv e n fig u r e s h o w B o c lr c c w pC - co m m o n c (ii) L A B C (i) 5 5 n s: , rn pm d u c e d (a ) p A c e n tr e d jo in in g a ( 1 0 ) In th e A n js n it e L o c \m u a b e in g A B b is e c ts th e e (i) o n v e th a t o c ir c e s w u lA , q e te r o + L f th e DC B c ir \e C a lc u la te lA n s (i) 2 8 (ii) 73) Scanned by \ \ t\ \ \ L \ 1 14 1 n u e * r h n th a t A t is w gu re G i\ t n c in k a m i a n n i t \e 5V f th e P tw \ t s ln th e g i\ A BC L : t an d (i) A an n s ( 16 ) If n o an o th e r m 2L g le B C D (i) 2 5 s id e s o s fa id e s a r ig h t a n 6 0 tr e o e f tr ia n g IF a n g \e B A C f th e 50 = (iii) a n g le B C tii) a n g le IC D c D (iii) t 5) l p ro y ic lic q u a d r ila lie r a l a r e p a r a lle a re e C A Fin d (ii) 5 5 ' = c e n tr e m e e t s th e c ir c u m c ir te o 7 0 = O is th e 14 0 = p o in t D at g le A B C r e te r I is th e in c e n re 17 ( I) A B C is gk A O C p n c le A B C n m n mgsi y fig u m a F in d 5 , v e th a t th e qua l T A N G E F T S A M ) IN T E R S E C T IN G C H O R D S g le d tr ia n g le w ith A B = 6 cm an d Scanned by (2 ) A , B an d C a re B A p ro d u n \ e e ts th re e p o in ts c e d 48 by A B f th e (3 ) T a t c e n tre o w o c ir c le s w d ir e c t (4 ) T h e co m m ith o n c e n te r s o T a t th a t a n g e A C T ra d G iv e c a lc u ' ii 2 5 (A c m a n (A n s : o c ir c es w n s: d 9 30 ith ra th e tr a n s v e r s e c o m m o n t a n e n t g (5 ) In th e fig rr e g iv e n an d Qc a re v a lu e s o (A (6 ) In n s: ta n g e n t s to th e f (i) a n g e Q o p (i) I 12 th e g iv C h o rd o g s id e a lo n ; (ii) 6 8 f g u r e e n f a c ir c e su c O at R D ia m If a n g le S P R x o + 2y fig u en = re e te r = x , Q = g le h th a t w d ge e g su n t a t c 36 a n = d b te n d e d 9 6 ) c Ln to c m ) c m n s 6 QR , ith h u c e a c a n d 2 c m ) a n O a re cm g le R c e n tr e o th e r h , Q P IO c m a p a rt F in d th e le C a lc u n g th o la te th e le f th e n g th o f 68 , P C = c a lc u externally a te th e Qcp ) an ce P T to sQ an IA A B is th e d ia m , a n ta n g le A T C d ii 6 c ir c le w (ii) a n A t C in te r s e c ts A B p r o d u ( ) In th e g iv p , T he n th a t a n la te th e c ir c e ta n B e n t f tw a c ir c e o n d u c e te r a n g le B A C a tD he s P ro g le QR 3 0 o T h e ta n g e n t P = ith m e e ts y B C th a t v e a c ir c le w h e n p ro d u c e d an = d A C is th e , cen PT sho w = B D A [ tr e at P th a t 90 R Scanned by (8 ) In th c g iv PA = fi8 u en A M P ro (9 ) in PB X a r ig h t tr ia n A C in P ( 10 ) A B C is P ro (] I ) 7 e g iv an P ro a ta n v e th a t to u c d in te r s e c ts e te r o h e s th e g iv e n fa v e ( 13 ) F r o m th a t :a n g e po a n e x te r io r p o in ts A (i) na s id e Q = p o in t P d B lie e te r is d r a w s id e Bc c ir c le A A B in p o in t p c ir c le a t S ho th r o u n g h B to th a t w to in te r s e c ts th e h y p o te n A B u c = hes 4 x s id e A c at u se it A p E at A C = p Q in t A (A n s: he n ith cen tr e 0 p o in t R d B an d ia m as A C = h th a t A B c ir c le w c ir c le a t to th e c ir c le th r o u h A g P ro ir c e an d is o s c e le s tr ia n le g a c ir c le s u c g e n t to th e ith a b ith A B w D E is p a r a lle l to B C (j 2 ) A B is th e d ia m c g e n t a t p , b is e c ts th e g le sho w s a n A B C in s c r ib e d In If D A E is a c ir c le w sc e e s tr ia n fig u r e e n g e n t to th e P v e th a t th e ta n id d le p o in t D m a ta n z g le a b c is o an is sc e es M B = PM , v e th a t (i) P M B is is o (ii) P A re an d , ce an v e A lin e d) pQ th e ta n g e n t s c u ts p o in ts P pro d Q r e s p e c tiv e ly 90 , a ta n ge n t P T o n th e c ir c u m fe r e n c e an o d a se c a n f th e t P A B a r e U a w n to a iv e n c ir c If e g c ir c le , p ro v e th a t tr ia n g le s P A T a n d P B T a re s im ila r (ii) P A PB = PT 2 Scanned by & C O N W U IQ E I tc lr s ) ( ) D c ir c le e a c (2 ) D a c ir c c o ra w h ta n ge M c m th c g iv e n a rk pn in ir c le fm m c t a d is m n c e t h is e x t e r io r f n c m p o in t M o n i m th e c e n t r e n f th e th c e n g th e a s u re f n n t ir c e o n ts is 6 0 a 12 e te r t a n 8 c n ts t o ra w c ra w ta n g e (3 ) C D f d ia m f o n s tr u c t a n e u d iu ra i a te 4 5 s ra D c m ta n g e n ts to th is c ir c le s o th a t th e a n g e M tw o ra w tr ia n g c A B C ith w s id e 6 D cm a c ir c le c ir c u m ra w sc r ee n w th e ib in g th e r ia n g k A B C (4 ) C o n s tr u c t a c (5 ) D (6 ) C (7 ) D ra w in a n sc ir c e in A b in g th e c ir c le o a C c e n tr e e n g th o fo ir c le c a r he o n s tr u c t a r e g u ra w s c r ib in f ra o x a go s 3 5 d iu g a n fa e re f s id c m M is l l o c m lA H n s n e f th e ta n 40 cm i a tc r a t r ia n g e 4 ar x a C c m k a go n o ith s id e 5 f s id e 5 8 6 c m c m o n s tr u c t a c irc le c irc u m p o in t P o u ts t o th e fom P w id e th e g iv en s c r ib in c ir c le a t a c ir c le M B th e h e d is t a n e a s u re a n c e o d w r x a f6 go fr o c m ite d o n w n m th e ge n t c u rv e F in d = e o n s tr u c t tw o ta r g e n ts C O N E ? (S a re a o u a r he gu n o 19 ( I) T h e q : d s u r fa c e o u r ta e fa c u m e - 3 8 50 0 , A w El y lin d e r is 4 , 4 00 (i) th e h e i h t o f th e ; v o SP H E R E c cm y lin d e r ] cm l O i) th e an v o d th e lu c ir c u m m e o f th e fe r e n c ce o f its b a s e y lin d e r ) Scanned by r (2 ) T h e b a rr e o t fo a u n ta in pe fu l b a r r e l o f in k in t h e e p m a n y w o rd s w o u (A (3 ) A y lin d r ic c cm th e m e t a (4 ) T h e ra F in d th e (5 ) F in d w a n d 63 lA n s G iv lu m e an m e - p 0 0 e a re st at b o t h th e fc o h e ig h t f th e a re a o 3 an v as m ;a T he w m p 2 in m c o st c a n v a s re , - w R s ta e c m r it in g 3 10 4 5 0 0 de f o m e ta he v e ry w a lo nRh o n n s : f d e c im f o l IA f th e 2 cm id t h is in d ia m a n a v e ra m e te r ge H A o w ) Th e in te r n a C c m 30 6 2 n s m litr e ? a o rd s w d 5 o rd s o n w is 0 4 re an g n a r c o n e a re r ig h t c ir c u a re a o lin d th e n w is 7 ta in in g a m e o 704 - he ) (A fa o ha pe c o n e a c p o ne re a A so ; u d d s is e n d th e h e ig h t c m s w o rd s d th e to ta l s u r fa c e 5 94 = a n u se in b o tte o f in k a c o r r e c t to , 12 3 2 s a n t le n g th o ca f th e v as u se a so h e ig h t 2 4 lid (A c a n v a s an ly lid cm n s: c is d in fo ld s 9 ,4 2 8 p a c ity r e s p e c tiv e so in m (i) n s (8 ) F r o m lu n il b e w a l d ia m a lc u cm 7 c m 3 e te r o a te f th e tu b e th e v o lu m n e o f ) a n d 24 r e s p e c t iv e c m ly c o n e ) re q u ir e d to f th e c o st o a m ke a co n ic a l te n t 1 2 c a n v a s a t t h e r a te o fR s in d ia m m 5 p e e te r r m e te r 8 ,9 1 0 ) q u ir e d to m a ke a c o n ic a l te n t 1 4 m h ig h a n d 96 in d ia m m e te r e n th a t (ii) c a n (7 ) T h e pe f th e b a s e h a t le n g th (i) 2 0 % lA v o v o (6 ) F in d th e o l in t h e tu b e s o u n in d r ic y c , d its e n g th is 2 c m a n d iu n s: l tu b e a u se th e n s n w e r c o r r e c t to is I 2 (A ld n 4 7 2 m u se a n ; d d in fo ld s s t itc (ii) W d th e b a s e a re a o 7 fa c u rv e y lin d e r f h e ig h t 3 6 o is d r ille d v o lu m e - o ut s t itc 2 o cm f th e 9 ,0 5 1 _ s u r fa c e a r e a o 2 h in g c u rv e 3 1 m cm an d ra f th e v o lu m e a n a re a v e s se d iu s 14 4796 d s u r fa c e a r e a o cm lv e sse l IA , n s l a re 9 8 5 6 220 0 a c o n ic a d th e to ta l s u r fa c e cm 2 f th e te n t 2 r ig h t c ir c u la r c o n ic a F in d th e 2 0 94 4 d h in g is 2 0 % F in d th e d an cm cm 2 an d 6 16 cm 2 ) l c a v ity a re a o 3 o f r a d iu f th e s 7 cm r e m a in in an d g ) Scanned by (9 ) f th e s u r fa c e fa a re a o phe s is 6 \ 6 c re 2 m N d its n lu v o m IA e \4 3 7 n s ( O ) T h e internn r e s p e c t iv e lA n s ly bu ke t c h e a p is 2 4 (i) th e (ii) th e lA s la n t n s r G iv c m s a n g ro u n ra co u n (1 5 ) A ra (1 6 ) IA d iu L n s s o ppe r ifo rm m e ta d iu s ro f a d o l llic w d = m a he w a c o n 2 c m in h e ig h t ke s is p h e r e m te r c u r v o u 32 5 3 2 5 c m s p h e re e w f d ia an e yo an e d v e sse s u r fa c e l a re 4 2 c m a n d 45 s d She c m a re a ) d 18 ic a l h e a p f th e o in ra sa n d c m d iu ith s w sa n F th e h e ig h t e m f th e o p tie s \c a \ c o n so It is c m an t to o n e a r e a ls o th e a m s 10 5 an e te r 8 an e ss o cm d d ra w 972 n s p la c e f d e c im o a \ c a lc u e la te th e s la n t h e ig h t o f is m e is d r a w cm ir e w lt e d n u m an be r d external d ia m cm in to n ir e a w f d ia m o e te r 2 m F in d m ) m f th e F in d th e c m f internal d d le n g th 8 F in d th e th ic k n f b a s e d ia m lid s te e m IA e te r s f r a d iu e r c o rr e c ) cm 1 u r a n sw ) c m s e tw o cm d h e ig h t 3 sphe re o a co n e o m m e te r sph e re o cm f th e is 9 ir e in G iv 43 3 32 9 8 = th ic k n e s s 3 5 A h o llo in to a n ls o its a a re a f th e h e a p o cm th e le n g th o f th (14 ) A d ho lo a ) d h e ig h t th e c o n e ] d lb u c ke t 32 th a t th e v o lu m e s o e n (\ 3 ) T h e a c it y a n cm f e te r s o : n d , 36 - pa 19 , 4 0 4 - o n th e d iu ra c a y lin d r ic a c l d ia m te r n a e x F in d its p a c it y c a ( 1 ) A B ir l n ils th e d a n c m 3 TA fco e te r s n s d th e n o in to n 4 15 a w ir e cm ) o f le n g th \ 8 r e c a s t in to s m n es M u s cm an d 8 F in d th e h e ig h t o f e fo r m cm m o f aH c o n e s e a c h o e d IA n s r e s p e c t iv e f 12 6 1 ly \s m e \te d co n e Scanned by ( 7 ) A is in th e fb to d ia in tir e o d tA 4 14 4 8 c m c inder w 8 fba a r J o fin d v es se th c \ o 3 ? o pe e o ho 6 s - 5 " o a re a f th e e v e r te d d iL a n e ra h e ig h t is 8 c m it h t h e i sa m d iu The s D e t e r \ in e th e an d ra d Fin d o u t a ) in in g so iu is 6 s c m , th e v o lu m e o a c o n ic f th c re m a c a a v ity o f h e ig h t in in s o lid id ) cTn 7 ' c m c m d c m w re 3 4 ) - re m 3 603 is p h e m is 2 7 53 6 - e ts h e ig h t is \ 1 c o n e It is fil e d c m IR is h o o w f a n in o u m he a r t io n h e ig h t is 8 se c m a r ea o n r th c \ o v o e rm Is n u m d ic a p o c o n s u r fa c ! c m in th e fo ic h is d iu ra se th e t o t a 603 n s - lid a m m c m (A a rea n s : A tm w f th e b a s e e te r o m o u n te co n e s u r fa c e a r e a a n ( 18 ) F ro ( 9 ) A wra rm ith w w c m to th e r im a te r u p ppeo \n to a n d th e W n en ra d iu s o f its to p h o O t t n e w a t e r \0 W 5 le a d s h o ts Ea c f 2 w u H s F in d th e a s p n u m ber Z ) Cm o i r& u S U n e re f ead o s h o ts T W 20 v e th a (i) c o s (ii) (1 (i ) (3 ) V e t h a i (i) O i) ta n s in A + co t 4 A + s in A a re d ro n D p p e d in to th e v e sse 4 A )7 A + 2 ta i A + = (1 = = se c 2 co s co t sec 4 A 2 c o se c = A IA n s u u \ 440) A l A A )? A = 2 sec c o sec 2 = A 2 2 A A co sec 1+ co s A co s l 5 1+ co s A s in 1+ co s A 1 M e v e sse G O N O M E T R IC A L ID I# ll aH (l) P r o \ e t h e i t y t a n A + C O t A (2 ) p r o , (c o s e c A + cot A A )? Scanned by ( iil) (4 ) P r o v e th a i CO S A c o t 1 s in A A se c v e th a t : v e ta n A A c 1 (i) th a t s (8 ) K . ' + ' c o s e c A c o t A A ' A + ta s e c se c s e c ta n A A n A A ta n 1 1 + s in A A + 1 s e c A A c o s s in 2 A + ta n A 1 = cas B s ta n z + m A \n A A + ta n (i) s in t A o 1 + s tn A (ii) ( 1 (7 1 If ta n A A c o se c _ (ii) (6 1 P r o + : c o s e c (S ) P n A an , A ) + tn A c o s co s A ( + + in z s in A - s d ta n A s in - b - A . in B . d y A A P ro n . v e - A s th a t . Ia n in m A + A G 41 sh . . M aE l 1l ii 2 H E IG H T S A N D D IS T A N c E s ( 1) T h e le n g th o f th e sha e le v a t io n o n t h e s u N (2 ) T h e an g le o f e le v a tio h o r iz o n ta p la n e is fo do w o fa v e r tic a to w er is tim e s h s h e ig h t F in d M e an g e o f 30) lA n s n o f th e to p un d to b e 3 0 o f a to w er at a d is ta i c e o F in d th e h e ig h t o f th e f 12 0 m fro m to w e r IA ns its fo o to n a 69 28 m ) Scanned by (3 ) A g u a r d le v e b se o To b e (i) C a c u w o Pe f e e 70 m o s o m e t im e tw o d e c im a (6 ) A n o d e p re s h ip s bse rv if th e lp ac p sh O n th e o p o s ite i p s des 3 (8 ) T h e an e s id e o o in g d ir e c tl to y to 4 5 an g le o g le o ho , w f n an th e , b se o f 29 o r v a t io n to w e r a t a h e iB h t f 18 o o a m b v e se a d is ta l IA w a rds n o n s th a t t h e e g e s f th e b o n c e o 45 m 13 6 6 m c o f 25 w e r as v e r t ic a bo f d e p re o id e l to is 2 o o FA to w e r a a t s s io n a n pe o in d 5o p le w er o n to 3 o fr o a t th e fo o t o m f th e b se o r v a t io n c n s a s tr a re s TA e v fr o m o c lifi F f th e o m an a bo d 3o fa m c if (A n s : If it ta k e s 12 w fle c tio ) m ig h t in e w e o t o f th e o b s e r v a t io n ; ( ii) 4 I 5 9 it h it m th e a n g e s e a s u re y F th e h e ig h t f th e o to w e r is 9 1 3 8m ) g r o d in u n a lc u la te th e v e th e s e a re s p e c le v tiv e y ill th e n (i) 14 6 4 n to w e r o m m ; b se r v in u te s fo c a r re a c s ta t io n a r y c lo u re fo c re a s e s h e ig h t o f th e b y o m to w e r w , he n th e c o rre c t to e l , o bse rv es th e a n F in d th e d is ta n c e g le s o b e tw f e e n th e , l o b s e r v a tio fte r th is f its 50 it th e ( i) 3 2 5 p e c t iv n s e m ) ifr 2 o o ip s to b e 4 5 f th e so o n a f ? p r e s s io d e e n th e tw o fa a v e r tic a f e le v a tio n th e la k e le v e m a re (ii) co m o fa ip s to p w th e to p o th e s a m o n th e b e tw n s o rv e an f th e to o IA o n n a n b se h a n g e s fr o (i) (7 ) A o o n th e s a m do es f th e tw sh it is n ce sha su n c e r m s s io n o , f th e o f th e o , e s ta n d in p g f n d th e d is t a h itu d e f d e p r e s s io o a lc u la te th e n e w C (5 ) T h e e n th g a fr o , to th e n e a r e s t m e te r t io n o f th e t o v a , a t e r w to w e r w o y bo e n e m g le a t a n an a te (ii) A fte r (4 ) T rv e s a n r h th e (ii) 5 4 6 4 e s a car m o v m ) in g a a u n ifo r u spe e th e a n le o f d s e p r e s s io n to c han ge o b s e r v a tio d fr o m p o in t 2 5 rn n tlo w e r ? lA n s d fom 16 3 9 ) abo v e a la k e is 3 o a n d th e in th e la k e is 6 o M [h a t is th e h i e g h t o f th e c lo u d a b o v e Scanned by t9 ) F r o be m a h th a s \K ng th e g r o u n d o n i ta n m v at i ge n t \e r t ic a \ po e f th t op e m e an n w e g a f e le v o lk in g 5 o of th e to a t io n o f th e to p t o w a r d s th e m w e r i s f o u nd w e F in d th e (2 ) F in d th e f th e to a r ith m m e an e o to p m w e r a r e o n th e s a o f th e , o to w e r f t h e to S M E D M , ) o f 5 pe rs o n s o w (in k i ig h to F in d th e h e ig h t is 3 0 w o n (M E A N (l ) T h e O a v e r t ic a d an e v a t io th e a n g le o f e o is th e of Th m . be e \e v v e r t ic a o fa w c r l lo th e ta n i nd the he i i frund m w gen t o f \h e gh t of the n w r, Io \ 2 0 m l n s 1 0 ) A t e \e or k lA p o in t tic m m o f th e ir w e is 6 o an e g r o d th e u n a n if th e h e ig h t w e r g \e o F ro d o m th e tl p o f th e p o le f d e p r e s s io o n f th e p o \e iq 2 1) m f th e (A n s # f)m ) N D M O D E ) Q u A R 1 L E S A a re ig h ts 3 7 3 5 , 7 1 57 IA n s 6 \ k g ) , , an d 45 f lA n s 6 85) (3 ) F d th ie m e a l o c a s s in l rv a F r e q lA n s M e an t & . ' f 10 O o L 20 10 6 20 8 30 1 40 30 \2 \ L 50 40 5 41 ' Scanned by F (4 ) F m d m e a n t h e fo l o w in g d is t r ib u t io o Y lA n s (5 ) k la e an e an m 1 30 1 F re q u e n c y lA n s M e a n (6 ) If th e g shortut e th o m V 45 6 f th e fo o w in g d is tr ib u t io o I te r v a u s in d Y q 46) - F in d th e in 40 7 m ss 40 25 n 30 T 10 40 40 1 50 50 1 8 60 60 f th e ? lo o in g d is tr ib u t io w n is 7 5 , fin d th e u e n c to t a ln be u m 5 0 F in d th e ass (8 ) F in d th e (9 ) T h e 1W 1 U m c a in w e m d ia n ig h ts n e a re s t u c e (10 ) F in d th e m m f45 kg) ! e n lc u la te th e o y n s m e , , n in . 60 1 / 1 2 , 15 6 2 0 8 4 , h ild r e 1 , a c la s s w 7 w d ia n fo 5 lA l4 0 24 ) ? in g d is tr ib u t io T a n d l2 w e re an d 10 record n s b le is 1 2 o a 60 M e d ia n n d th e ir l8 0 I 32 IA ta n m is e an 0 0 f, - t o th e n e a r e s t 9) kg as , fo llo w s e d ia n e c in th e fo lo l2 0 I f o a tio n s f m is s in g fie u e n c i es q 20 = 9 Y f o bs erv v a lu e s o 10 1 17 tF r e q u e n c y r o 80 \ (? ) T h e 70 5 is s in g fr e q m 70 l 12 49 6) m m n r 6 d ia n is 2 6 ) 8 e ig h t r th e fo llo w in g lA 1 ns M e d ia n ' 12 w e \ + 55 2 ig h t disbution 7 10 [ 8 \ 5 Scanned by ? ( ]i) T Ii e . A1a ks b y 2 0 0 s t u d e n ts in b ta in e d U s in g a g w p h p a p e r ) d r a w (i) th e m e d ia n O g iv e fo an th e a b o r in a tio a n ex a m v e d is tr ib t tio (ii) th e n u m b e r o f s tu d e n ts w h o o b ta in e d m o re th a n 8 0 % (iii) th e n u m b e r o f s tu d e n ts w h o d id n o t p a s s i th e p a s s (A M e d ia n n s: T h e d a ily (12 ) w 2c e as s e th e s c a n a ees R - w a m 2c m - 57 ;n o o ge s o n s: 20 s tu I O 1 0 s d e n ts o a g iv fo e m o r e th a n 30 r lo th e 1 7 13 7 , , C 1 (A ns: Low (15 ) U s in g o M N lA o o ns er (A e d ia n n an d 35 w as 130 20 ;n o = g iv a re 4 0 40 o f s tu d e n t en be o w : 50 l5 0 60 no t pa s s ii 7o s 45 o re 34 ) - da y a th a n R s 4 5 a da y 30% ) , , - - 10 10 l 3 36 ;N 15 o o ta b le , 20 a r tile - fin d (iii) In te r q u a r tile 15 l 4 15 5 ; U p p e r q u - L o io lm a - in a tio e x a m w o rkers g r a p h p a p e r , d ra w b ta in e d in th e G e n e ra l K M 80% - a f s tu d e n ts e to e s tim a te q u a r tile , u p p e r q u a r tile a n d in te r q u a r tile r a n g e fo r th e d a ta : 9 1 1 1 5 , 19 , 9 ; u p p e r q u a r tile n s : L o w e r q u a r tile 1 7 ; in te r q u a r tile r a n e 8) g q u a r ti e a rk s O g iv w er 5 : Th e a b o v e d is tr ib u tio n (14 ) F r o m th e fo llo w in g fr e q u e n c y d is tr ib u tio n (i) L o w e r q u a r tile e O i) U p p e r q u a r ti F re q u e n m u r w is p e r c e n ta g e o f th e w o r k e r s w h o e a m m o re th a n R 3 5 ; th e p e r c e n t o f w o r k e r s w h o e a m m m e d ia n (13 ) F in d ge b e lo yo se in th e p e r c e n ta b u i d in g p r o je c t l 20 an in d e n d o th e r a x th e o n s c o re w o r k e rs f 16 0 , (A m a r k s O ri o n e a x is a n f s tu d e n ts U s in g th e g r a p h p a p e r d r a w u s e y o u r o g iv e to e s tim a te (i) th e m e d ia n w a g e o f th e (ii) th e 10 g iv U n m a rk s , U n a re 20 1 25 ra n ge 1 2 5 1 9 2 5 5 ; In te r q u a rtile ra n 30 7 \ - f s tu d e n ts s c o r in g m a rk s a bo v e 65 = 35 10 ) g iv e fo r th e fo llo w in g d is tr ib u t io n w h ic h s h o w n o w le d g e p a p e r b y 1 0 0 s tu d e n ts 2o l 3 o 1 3 o 4 o 1 4 o 5 o 6 o 10 20 25 15 1 12 an o 30 1 s th e m a rk s 1 6o 7o 1 8 1 l E l= a 8) Scanned by (1) A b a g s iz e th e in c o n ta M o b a l d r a w b a c k ba l s a h it ta k e is n (i) re d b a ll a a re b a ll f r o s o u t a n b rg d ba l? 3 (2 ) In a s in e th r o g w o f (3 ! F r o (4 ) In n w a th a n 2 c e c ard ) en b a l ) f n d th e p r o b a b i ity o f g e tt in g a i i l ( ) e s s a n o r e q u a l to 2 (iii) ) = ; (l' ' " h a n ll s h u f e d d e c k o f 5 2 ill (i) b e a fa c e c a r d P (fa n s P {g r e d ie a a w e m d ra w lA P (g r e a te r n s a w 3 (i) g r e a te r th a n 2 lA d t h e b a g r e e n b a ll a \s a r e id e n t ic a in s h a e a n d p ith o u t o o k in g in to it w h a t is th c p r o b a b ilt th a t y ? ( ii) b a lc k b (iii) g r e e n b a l ? th e m i = P (n ; c a rd s fa qu a o n e c a rd , (ii) o t a o r e n o - n u m r t g r a te r th a ; l '" 2 ) is a be n o a w n p ? ' F in d th " e n 2 p r o b a b ility th a e c e c a rd ) n m a tc h b e t w e e n R a e s h a n d J o s e p h , t h e p r o b a b ilit o f w in j y F in d th e p r o b a b ilit o f y t w in in g o f R a e s h j (ii) w in in g o f J o s e p h P (n o t w in in g o f R a je s h ) = 0 4 2 ; P (w in in g o f J o s e p h ) = 0 4 2 ) is 0 5 8 lA (5 ) In a s in (A (6 ) n s g le th r o A d ie is t h r o lA an n s o P (g e tt in g n s: (i) w o P dd la n w n n u m o f a n u m o n ce be dd d ie a r n u m fin d th e p r o b a b ility o f g e tt in (i) 7 g b e r 7 ) O ; P (n u m b e r le s s th a n 7 ) 1) , (ii) a n in g n u m fR o j a e sh b e r le ss th a n 7 = F in d t h e p r o b a b ility (ii) a n u m b e r g r e a te be r) t th e c a r d t b e a fa c e c a r d b a d m in to (i) n o ) a '" h a n 2 )= l = 2 , P in o o r f g e tt in g : th a n 4 (iii) G r e a te r th a n 4 ) = a n u m P b e r b e tw in o B e tw ee n een 2 a n d 6 2 & 6) 1 = Scanned by bq /I s T \ \ d ic e a th m s itt t u \\ n lta ( i\ th e d iw s \o w n th e \ li th e H r s t d ic e s h L \ m 6 \ iii) tl to ta w u rlr ) o th e n u {iv ) d ic e s h o n s P (h t a l n o P (to P {fu ca n n u m n d is d r a w be r w o n h ic h is ho s du o n th e fio a d ic e a re ro e Is lle d 6 be r n u m d ic th e o n a t e r th a n f 1o o o 9 9 IA a re n s d to g e th e r (ii) e x a c t ly tw (v ) 9) c a rd s n u m b e re d 1 W n at o he a d s O i) h e a ds ) = s im u lta n e o u s p in d p r o b a b ilit y 1o o to P o f d r a w in g a ) (v i) a ll ta ils (e x a c t y tw h e a d s) o (a t le a s t o n (iv ) P (a ll ta ils ) ; 2 P (iv ) ; 18 is th e p r o b a b ility o f o b ta in in g (iii) e x a c t ly o n e h e a d a t le a s t tw o = = = he a ds ; he ad ) is 8 ) e q u ir e d p r o b a b itity re a r e to s s e o 9 1 is 9 ) = p e r fe c t s q u o n e n ' 6 = ly F in d th e p ro b a b ility a m u ltip le o f3 he ad) e 3 = 8 = 8 ' ) 8 o f d a m u ltip le o f3 o n th e o th e r d ie a s th e s u m (i) r e q u ir e d p r o b a b ility (iii) r e q u ir e d an p ro b a b ility - 6 ; O i) a ll o (iii) g e tt in g n s t ti 1 ) b ta in in g a to ta o f a t le a s t 9 O i) g e tt in g a m u ltip le o f 2 o n o n e d ie lA a \ d ic e is \ pac k a m e a s t tw (v ) P (a t (i) d ic f th e c t o (iii) P (e x a c tly w o n u m e n u n lb e r 6)= w s (i) P ta ll h e a d s ) n s ( 10 ) T tm o n th e ta tl p r o ta th e s w d e n t ic a l c o in s (9 ) T h r e e (i) a ll h e a d s (iv ) a t le a s t o n e h e a d lA n d th e p r o b a b ilit th a y be r o H fic s t d ic e (8 ) A sa m e ly m b e r s Q n t h e d ic e is 9 f h t pn du ct e n u m b e t o n th e d i c e is 8 to ta o f th n u n b e r s o i t h e d ic e is B r e a te r th a t ite \ ) th e A n ru s r e q u ire d p ro b a b ility - 36 ' - Scanned by (1 1 ) A bo x c o n b a l is tw ( 12 ) F r o A (Ii) n s a fa a n f ft h s o e f b la c k b a lls a w c e a n d 30 a \ c W n u m be re d re d c a rd h ile b a lls n u m be r o If th e p r o b a b ility o f d r a w in B f b la c k b a lls in t h e b o x IA n s a - a rd s w n at c a rd e v e n w h ile b a ll fin d th e p a c k o f 5 2 p la y in g c a r d s is n o w d r a w n a t r a n d o m a c a rd (i) (A m ta in s s o m o h o s e n u m b e r a r e m u \t i le s o f 3 p is th e p r o b a b ilit y t h a t th e c a r d d ra w a re n re m b la c k \2 ) o v e d s ? (i) X (T X X X X X X B E S R) Scanned by

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