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VECTOR ALGEBRA D Weghtge 7 Ms SYLLABUS: VECTOR ALGEBRA Vetos sls, mgtue eto of veto Deto oses eto tos of veto Tpes of vetos (equl, ut, eo, pllel olle vetos, posto veto of pot, egtve of veto, ompoets of veto, to of vetos, multplto of veto sl, posto veto of pot vg le segmet gve to Sl (ot pout of vetos, poeto of veto o le Veto (oss pout of vetos Sl tple pout of vetos SUMMERY OF QUESTIONS (YEAR WISE YEAR VSA ( M SA ( Ms LA ( Ms Totl Ms 9 7 PREVIOUS YEARS QUESTIONS 9 F poeto of o f 8 Wte ut veto the eto of F the vlue of p fo whh 9 p e pllel vetos If, show tht s pllel to Wte veto of mgtue 5 uts the eto of veto Wht s the ose of the gle whh the veto mes wth -s F the posto veto of the pot R whh ves os of pots P Q whose posto vetos e : tell Also show tht P s the m pot of RQ Fo wht vlue of the vetos 8 e olle F the eto oses of the veto 5 F the ut veto pepeul to eh of the vetos whee Wte veto of mgtue 5 uts the eto of veto Wht s the ose of the gle whh the veto mes wth -s? F the posto veto of the pot R whh ves os of pots P Q whose posto vetos e espetvel etell the to : etell Also show tht P s the m pot of RQ

If ut veto mes gles wth, wth ute gle wth, f the vlue of If e two equl vtos, the wte vlue of + + If e two vetos suh tht, the pove tht s pepeul to veto Top Wse Dstuto of questos Top 9 Itouto Sl Pout Veto Pout Tple Sl Pout ----------- ----------- ----------- ----------- ----------- Impott esults Fo pot A (,,, OA = Fo pots A,, B,, ; AB = + + If, the = Two vetos e olle f = Pots A, B C e olle f AB = AC 5 ; = mob OA mob OA OB OA OP (Itell ; OP (etell; OP (m pot m m 7 Fo OP veto ompoets e, sl ompoets e, 8 Fo veto,, e eto tos Fo ut veto l m l, m, e eto oses l m 9 Fo eto tos,, eto oses e lulte s l, m DOT (SCALAR PRODUCT: ; Also ( ; Fo two pepeul vetos, ; ; Poeto of o = ; Poeto veto of o = 5 Fo vetos ; To f gle etwee two vetos os ; Coto fo two vetos to e pepeul

Coto fo two vetos to e pllel OR 7 ; VECTOR (CROSS PRODUCT = s, wth e the gle etwee the two vetos the ut veto pepeul to oth the vetos ; Also (, Mgtue of oss-pout s Ut veto pepeul to the ple otg vetos, Vetos of mgtue pepeul to s gve 5 Fo ; Geometll veto pout epesets the e of the pllelogm whose ses e epesete the two vetos e Ae of pllelogm wth oseutve ses epesete ; Ae = 7 Ae of tgle wth ses ; Ae = 8 Ae of pllelogm wth gols epesete, ; Ae = 9 Lgge s Iett SCALAR TRIPLE PRODUCT: Fo vetos,, sl tple pout s OR s eote [ ] Popetes: Fo plleleppe hvg ts otemous eges epesete vetos, Volume of plleleppe = [ ] Thee vetos, e opl ff [ ] = Pots A, B, C D e opl f [ AD AC AB ] = Fo vetos [ ] = 5 [ ] = - [ ], MLM

ONE MARKER Fo the followg pots A, B f the veto AB s well s the mgtues A (, -,, B (, -5, F the ut veto log the vetos F so tht s ut veto F the veto log the eto of the veto of mgtue 5 5 F the sl veto ompoets of the veto AB wth A(,,, B(, -, F the eto tos eto oses of the le pssg though the pots (,, (-5,, 7 F the sl pout of the vetos 5 8 If, f ut veto pllel to the veto 9 F the posto veto of the pot whh ves os of pots : tell etell usg vetos F the posto veto of the pot vg the o of A ( -,,, B (, -, ( : tell ( : etell usg vetos F so tht 9 s ( pepeul ( pllel to F, f 8 8 Fo, ; f Show tht 5 Evlute F the gle etwee the vetos f 7 F the gle etwee the vetos 8 FOUR MARKERS 8 Show tht the pots A (, -,, B (, -5, C (-,, 9 e olle 9 Show tht pots A, B C wth pv 5 foms ght tgle If,, e mutull pepeul ut vetos, show tht If,, e mutull pepeul vetos of equl mgtue, show tht,, e equll le to ( If,, f so tht s pepeul to

F poeto poeto veto of o fo 5, 8 Fo, epess the fom of two vetos, suh tht s pllel to pepeul to 5 If,, 5, 7, f the gle etwee Fo vetos,, f,,,, F 7 F veto of mgtue 9 pepeul to oth the vetos 8 F the e of ( pllelogm ( tgle whose et ses e epesete 9 Gve, ; f Let, 7 F veto whh s pepeul to oth 5 If, show tht s pllel to F the volume of the plleleppe whose ses e gve 7 5, 5 7 7 5 Show tht pots (,-7,,(,-9,-, (,,- (,5, e opl F so tht pots, 5 e opl 5 Show tht Show tht Lst of mpott questos emples (NCERT EXAMPLES: 8,9,,,,5,9,,,8,9, E:,,5,8,,,,,5,,7 E :,,5,,,,,5, E:,,5,7,9,, MISC: 5,,7,8,9,,,,,8,9 5

THREE DIMENSION GEOMETRY SYLLABUS Deto oses eto tos of le og two pots Ctes veto equto of le, opl sew les, shotest ste etwee two les Ctes veto equto of ple Agle etwee ( two les, ( two ples ( le ple Dste of pot fom ple SUMMERY OF QUESTIONS (YEAR WISE YEAR VSA ( M SA ( Ms LA ( Ms Totl Ms 9 ------- PREVIOUS YEARS QUESTIONS 9 7 7 7 5 F p so tht les e pepeul ; p p 5 F the equto of the ple pssg though the pot (-,, pepeul to eh of the ples + + = 5 + + = 5 Wte veto equto of the le 7 F the Ctes equto of the ple pssg though the pots A(,,, B (, -, pllel to le 7 The pots A (, 5,, B(,, C (,,- e thee vetes of pllelogm ABCD F the equtos of the ses AB BC lso f the o-otes of pot D Wte the teept ut off the ple + = 5 o -s F the gle etwee the followg p of les 8 5 7 he whethe the les e pllel o pepeul F the equto of ple whh ots the les of teseto of the ples ( ; ( 5 whh s pepeul to the ple (5 8 Wte the veto equto of the followg le: 5 7 F the Ctes equto of the ple pssg though the pots A(,, B(,, pllel to the le 7

The pots A(, 5,, B(,, C(,, e thee vetes of pllelogm ABCD F the veto equtos of the ses AB BC lso f the ootes of pot D F the Ctes equto of the le whh psses though the pot (-,, -5 s pllel 8 to the le 5 F the ootes of the pot, whee the le teset the ple + 5 = Also f the gle etwee the le ple OR F the equto of ple whh ots the les of teseto of the ples ( ; ( 5 whh s pepeul to the ple (5 8 F the veto equto of the ple pssg though the pots wth posto vetos ; Also f the pot of teseto of ths ple the le ( Top Wse Dstuto of questos Top 9 St Le 7 7 5 Ple 7 St le Ple ------------ ------------ ------------ ------------ ------------ Impott esults STRAIGHT LINE Le pssg though pot A (PV pllel to m s m Ctes fom: A (,,,, e D R of veto pllel to le A geel pot o the le s P (,, Le pssg though two pots A (PV B (PV s Ctes fom: Fo pots A (,, B (,, Fo les hvg gle etwee them m, m OR ; m os = m m m Fo pepeul les m m Fo pllel les m m ; 5 Sew les spe e les whh e ethe pllel o tesetg Shotest ste etwee two sew les Fo two les m ; m 7

( ( m m S D = OR SD = m m Coto fo two les to teset m m Fo pllel les ( ( m; m S D = ( m PLANE Geel equto of ple s + + + = Hee,, s the of oml (pepeul to ple Veto Fom, hee s oml (pepeul to ple Ple though oe pot A (,, wth pv oml to veto (Deto tos,, ; OR Equto of ple though thee pots A ; B ; C,, m,, OR Equto of ple hvg teepts,, o the es 5 Ple pepeul to ut veto t ste of p fom og p Ctes Fom: l + m + = p, hee l, m, e oses of oml to ple Equto of ple though pot A (,, ( pepeul to ples + + + = ; + + + = o ( pllel to two vetos wth eto tos o ( pllel to two les hvg eto tos,,,, 7 Equto f ple though two pots A (,,, B (,, ( pepeul to ple + + + = OR ( pllel to oe veto o ( pllel to le hvg DR s,, 8

9 8 Equto of ple tht ots the les s Two les e opl f 9 Equto of ple tht ots pllel les s Equto of ple though the le of teseto of ples : : s Dste of pot fom ple: Fom pot A (pv Fom pot A (,, ple the ple + + + = p = p = Agle etwee the ples: If e the gle etwee the ples OR os = Fo pepeul ples ; Fo pllel ples ; LINE AND PLANE Agle etwee le ple: Fo le ple m OR Fo le Ple m m S Fo le to e pllel to ple m Fo le to e pepeul to ple ; m

Equto of le pepeul to ple + + + = pssg though the pot (,, s IMPORTANT QUESTIONS (THREE DIMENSIONS F the equto of le tes veto fom pssg though pot pllel to 5 F equto of le though (, -, pllel to the le whose equto s 7 F veto Ctes equto of le pssg though the pots (,, ( -,, F the equto of s 5 F the pot whee le 5 meets ple + = Show tht les teset ( ; ( 7 F the pot of teseto of the les ( ; 5 ( 8 F the gle etwee the les ( ; 5 9 ( 9 F p so tht les e pepeul 7 p ; 5 5 7 7 p F p so tht les e pllel ( ; p 9 ( F the ste to the le 7 7 fom the pot (,, F the mge the le 8 ( of the pot (, -, 5 F the shotest ste etwee the les ( ; ( F the equto of ple Ctes fom ( 5 F the equto of the ple veto fom + = F the equto of ple pssg though pots (,, -, (,,, (7,, 7 F the equto of ple pssg though the pot (,, - pepeul to the ples + - =, = 8 F the ple pssg though the pot (,, pllel to the vetos

9 F the ple pssg though the pot (,, pllel to the les 7 F the equto of ple pssg though the pots ( -,,, (,, 7 pepeul to the ple + + = F the equto of ple t ste of fom og oml to F the ple pssg though the pot (,, oml to veto F the ple though (, -, pllel to ple + + = The foot of pepeul fom og to ple s (, 5, 7 F the equto of ple 5 F the eto oses of oml to the ple + = 7 F the equto of ple hvg teepts, o the es 7 F p so tht ples e pepeul ( ; 9 7 ( p 8 F p so tht ples e pllel ( p p ; 5 ( 9 F the gle etwee the le ( ple ( F m so tht le ( s pllel ple ( m Show tht le s lg the ple = F the ste of the pot fom the ple 9 ( F the ste etwee the ples + + = ; + = F p so tht ste of pot (,, p fom ple + + = s 5 F the ple otg the les ( ; ( Show tht les e opl ( ; ( 7 ( F the equto of ple though the le of teseto of ples 7 + = 5 + + = the pot (-,, ( F the equto of ple though the le of teseto of ples + + = + - + 5 = pepeul to the ple 5 + + 8 = ( F the equto of ple though teseto of ples ( ; ( pllel to the le ( 8 F the legth o-otes of foot of pepeul fom the pot (7,, 5 to the ple + = Lst of mpott questos emples (NCERT EXAMPLES:,7,,,7,9,,,5,,7,8, E:, E :,,,,8,,,,5,7 E:,(,5(,(,,,,(,(e,(,( MISC:,,7,9,,,,,5,,7,8,9,,,

LINEAR PROGRAMMING Weghtge Ms SYLLABUS Itouto, elte temolog suh s ostts, oetve futo, optmto, ffeet tpes of le pogmmg (LP polems, mthemtl fomulto of LP polems, gphl metho of soluto fo polems two vles, fesle fesle egos, fesle fesle solutos, optml fesle solutos (up to thee o-tvl ostts SUMMERY OF QUESTIONS (YEAR WISE YEAR VSA ( M SA ( Ms LA ( Ms Totl Ms 9 PREVIOUS YEARS QUESTIONS 9 A et s to ot t lest 8 uts of Vtm A uts of mels Two foos F F e vlle Foo F osts Rs pe ut foo F osts Rs pe ut Oe ut of foo F osts uts of Vtm A uts of mels Oe ut of foo F ots uts of Vtm A uts of mels Fomulte ths s le pogmmg polem f gphll the mmum ost fo the et tht ossts of mtue of these foos lso meets the mmum uttol equemets Oe of e eques g of floo 5 g of ft, othe of e eques 5 g of floo g of ft F the mmum ume of es whh e me fom 75 g of floo g of ft, ssumg tht thee s o shotge of geets use mg the es Me t LPP solve ot gphll A fto mes tes ets et ts A tes et tes 5 hous of mhe tme hous of ftsm s tme ts mg whle et t tes hous of mhe tme hou of ftsm s tme I, the fto h the vllt of ot moe th hous of mhe tme hous of ftsm s tme If the poft o et t s Rs Rs espetvel, f the ume of tes ets et ts tht fto must muftue to e the mmum poft Me t s LPP solve t gphll A smll fm muftues gol gs hs The totl ume of gs hs muftue pe s t most It tes hou to me g mutes to me h The mmum ume of hous vlle pe s If the poft o g s Rs tht o h s Rs 9, f the ume of gs hs tht shoul e muftue pe, so s to e the mmum poft Me t s LPP solve t gphll A oopetve soet of fmes hs 5 hete of l to gow two ops X Y The poft fom ops X Y pe hete e estmte s Rs,5 Rs 9, espetvel To otol wees, lqu hee hs to e use fo ops X Y t tes of ltes ltes pe hete Futhe, o moe th 8 ltes of hee shoul e use oe to potet fsh wl lfe usg po

whh ollets ge fom ths l How muh l shoul e llote to eh op so s to mmse the totl poft of the soet? Top Wse Dstuto of questos Top 9 Muftug Polem ------------ ------------ ------------ Det Polem ------------ ------------ ------------ ------------ Alloto Polems ------------ ------------ ------------ Lst of mpott questos emples (NCERT EXAMPLES:,,7,, E, 8 E :,,5,,7,8,9, E5:,,,,7,8,9 MISC: 5,7,8,

PROBABILITY Weghtge Ms Cotol polt, multplto theoem o polt epeet evets, totl polt, Be's theoem, Rom vle ts polt stuto, me ve of om vle Repete epeet (Beoull tls Boml stuto SUMMERY OF QUESTIONS (YEAR WISE YEAR VSA ( M SA ( Ms LA ( Ms Totl Ms 9 9 A e s thow g g utl thee ses e ote F the polt of otg the th s the sth thow of the e Thee gs ots lls s show the tle elow Bg Nume of whte lls Nume of Bl lls Nume of Re lls I II III A g s hose t om two lls e w fom t The hppe to e whte e Wht s the polt tht the e fom th g O multple hoe emto wth thee possle swes (out of whh ol oe s oet fo eh of fve questos, wht s the polt tht te woul get fou o moe oet swes ust guessg? Oe fom p of 5 s s lost Fom the emg s of the p, two s e w t om e fou to e oth lus F the polt tht lost eg of lus OR Out of lot of uls whh lues efetves, smple of uls s w t om F the polt of the ume of efetve uls Poltes of solvg spef polem epeetl A B e espetvel If oth t solve the polem epeetl, f the polt tht ( the polem s solve ( etl oe of them solve the polem Suppose tht 5% of me 5% of wome hve ge h A ge he peso s selete t om Wht s the polt of ths peso eg mle? Assume tht thee e equl ume of mles femles A fml hs hle F the polt tht oth e os, f t s ow tht ( t lest oe of the hle s o ( the ele hl s o A g ots lls Two lls e w t om, e fou to e whte Wht s the polt tht ll lls e whte?

A spes tuth % of ses, whle B 9% of the ses I wht peetge of ses e the lel to ott eh othe sttg the sme ft? I the ses of otto o ou th, the sttemet of B wll moe weght s he spes tuth moe ume of ses th A? Assume tht the hes of ptet hvg het tt s % It s lso ssume tht metto og ouse eue the s of het tt % pespto of et ug eues ts hes 5% At tme ptet hoose oe of the two optos wth equl poltes It s gve tht fte gog though oe of the two optos the ptet selete t om suffes het tt F the polt tht the ptet followe ouse of metto og? Top Wse Dstuto of questos Top 9 Cotol Polt -------- -------- -------- -------- Multplto Th, Iepeet Evets -------- -------- -------- Bs Th Polt Dstuto -------- -------- -------- -------- -------- Boml stuto -------- -------- -------- Impott Results Bes theoem:- Fo the smple spe S ssote wth om epemet Let E, E,, E e mutull elusve ehustve evets ssote wth smple spe S If A s PEI PA/ E evet of S the P( E / A P( E P( A/ E I Cotol Polt: If A B e two evets of smple spe S ssote wth om epemet, the polt of ouee of A pove B hs le oue s lle otol P( A B polt s eote P(A/B ts vlue s gve P( B Polt stuto:-if om vle X tes vlues,,, wth espetve poltes p, p,, p the followg ptte s lle polt stuto X : P(X : p p p p Me E(X = p Ve V(X = E( X E( X SD = Ve Boml stuto:- f epemet s epete fte ume of tmes,evets ssote e epeet polt of suess o flue s ostt fo ll tls the polt of om vle X hvg vlue s gve P ( X C q p Hee : Totl ume of tls of epemets 5

PROBLEM SOLVING ALGORITHM PROBABILITY Polems volvg De (Up to tosses o Cos (Up to tosses E ( Fom Smple Spe use the esult PE ( S ( Q: A o s tosse two tmes F the polt of gettg t lest oe tl Sol: S ={HH, HT, TH, TT} E: Gettg t lest oe tl = {HT, TH, TT} E ( PE ( S ( Polems volvg Cs, Mles, Blls, Bos/ Gls et Ol oe oet s selete ( E o of fvoule oets Use PE ( ( S Totl o of oets Q: If oe s selete fom e of plg s, f the polt tht t s e g oof e g s Sol P( e g totl oof s 5 Note: C use esults of lge of evets P( E o F P( E P( F P( E F P E / ( P( E P E F P E F P E F / / / ( ( ( Moe th ol oe oet s selete Oets e selete oe oe wth Replemet Evets e epeet Use vul ses wth otto o use Boml Dstuto Q: If two lls e selete wth eplemet fom o hvg e, 5 gee l lls, f the polt of gettg oe l lls oe e lls Sol P( BRoRB P( BR P( RB P( B P( R P( R P( B 5 Oets e selete t om wthout Replemet Evets e epeet Use omto(c metho o otol polt Q: If two lls e selete fom o hvg e, 5 gee l lls, f the polt of gettg oe l ll oe e ll Sol P( oee oel ll C C = C 5 5 Lst of mpott questos emples (NCERT EXAMPLES:,,,,7,9,,8,9,,,,,,8,9,,,,5, E:,5,7,8,9,,, E :,,5,7,8,9,,,, E:,,,5,,7,9,,, E:,,5,,7,,,5 E5:,,5,,7,8,, MISC:,5,,7,8,,,

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