CBSE 2014 ANNUAL EXAMINATION ALL INDIA
|
|
- Delphia Randall
- 5 years ago
- Views:
Transcription
1 CBSE ANNUAL EXAMINATION ALL INDIA SET Wih Complee Eplnions M Mrks : SECTION A Q If R = {(, y) : + y = 8} is relion on N, wrie he rnge of R Sol Since + y = 8 h implies, y = (8 ) R = {(, ), (, ), (6, )} So, rnge of R = {,, } Time Allowed : Hours Q If n n y, y <, hen wrie he vlue of + y + y y Sol We hve n n y n y y n y y y y+ y Q If A is squre mri such h A = A, hen wrie he vlue of 7A (I + A), where I is n ideniy mri Sol We hve, 7A (I + A) = 7A (I + A)[ (I + A) (I + A)] 7A (I A)[II IA + AI+ AA] 7A (I A)[I A+ A] { AA A A, AI A IA 7A (I A)(I A) 7A (I A A ) I [Given h A = A y z Q If y w 5, find he vlue of + y y z Sol We hve y w 5 By equliy of mrices, we hve : y, y On solving hese eqs, we ge : =, y = So, + y = Q5 If 7 8 7, find he vlue of 6 Sol Given Q6 If Sol f () sin d, hen wrie he vlue of f () We hve f () sin d Differeniing wr boh sides, Q7 Evlue : Sol We hve [log ] f () [ sin ] f () [ sin ] [sin ] sin 7 [log7 log 5] log 5 Q8 Find he vlue of p for which he vecors i ˆ j ˆ 9kˆ nd ˆ i pj ˆ k ˆ re prllel Sol As he vecors re prllel, so heir dr s mus be proporionl Therefore, 9 p p Q9 Find (b c), if ˆ ˆ ˆ ˆ ˆ ˆ = i j k, b i j k nd c = i ˆ ˆj kˆ For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
2 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) Sol As (b c) = [ b c] = So, (b c) = ( ) ( ) ( 6) Q If he cresin equion of line re y z 6, wrie he vecor equion for he line 5 7 Sol We hve y z 6 y ( ) z y y z z On compring o we hve he direcion rios s 5, 7, nd poin on he b c line s (,, ) Therefore he required vecor equion of line is r = iˆ j ˆ+ k ˆ (kˆ 5i ˆ+ 7j) ˆ Q If he funcion f : R R be given by Sol SECTION B find fog nd gof nd hence find fog () nd gof ( ) We hve And, fog () f [ g ()] f fog() 6 f () nd g : R R be given by Also, gof () g[ f ()] g( ) ( ) And, gof ( ) ( ) Q Prove h : n cos ; OR If n n, find he vlue of Sol LHS : Le Y = n Pu cos θ θ cos (i) cosθ cosθ Le Y = n cosθ sin θ cosθ cosθ = n cosθ sin θ n n θ cosθ sin θ n θ = n = n = n cosθ sin θ n θ n n θ = n n θ θ cos [By (i) = RHS OR The given equion is n n g(),, For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
3 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) n n () n n n n ( ) 6 y Q Using properies of deerminns, prove h : 5 y Sol LHS : Le y 5 y 8y 8 y 5 8y 8 [Applying R R R nd R R 8R [Epnding long C [( 5) ()( )] [ 5 ] RHS Q Find he vlue of dy θ θ θ, if e (sin θ cos θ) nd y e (sin θ cos θ) Sol On differeniing nd y wr θ boh he sides, we ge : θ θ θ θ e (sin θ cos θ) e (sin θ cos θ) e (cos θ sin θ) e sin θ dθ θ dy θ θ θ And, y e (sin θ cos θ) e (sin θ cos θ) e (cos θ sin θ) e cos θ dθ dy dy dθ θ e cosθ co θ θ dθ e sin θ dy So, co(/) θ/ b d y dy Q5 If y Pe Qe, hen prove h ( b) by b Sol We hve y Pe Qe [On dividing boh sides by b (b) e y Pe Q [On diff wr boh sides b b e y bye ( b) P( b) e b (b) e (y by) P( b) e b On muliplying boh he sides by e, we ge : e (y by) P( b) [Agin diff wr boh sides e (y by ) e (y by) e [y by y by] d y dy ie, y ( b)y by or, ( b) by HP Q6 Find he vlues of for which y [( )] is n incresing funcion OR Find he equions of ngen nd norml o he curve b e y he poin (, b) b For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
4 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) Sol dy We hve y [( )] y [ ] dy [ ][ ] For dy [ ][ ] =,, Inervl Sign of dy y is (, ) Negive Decresing (, ) Posiive Incresing (, ) Negive Decresing (, ) Posiive Incresing Since dy in (,) (, ) so, y is incresing in (,) (, ) y y dy OR Given b b dy b y [ ][ ] b ( ) b Slope of ngen (,b) b nd, Slope of norml (,b) b So, eq of ngen is : y b ( ) b y = b nd, eq of norml is : y b ( ) by = ( + b ) b sin Q7 Evlue : OR Evlue : cos 5 6 Sol Le I Using sin (i) cos f () f ( ), Adding (i) nd (ii), we ge Pu cos ( ) sin ( ) ( ) sin I cos ( ) cos (ii) sin I cos sin d Also when, = d I I d I sin I cos nd when = b I [n ] [n () n ( )] I OR Le I I 5 6 ( 5) I I [ 5 6] I 5 6 log + + C ( 5) 5 6 For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
5 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) 5 or, I 5 6 log C Q8 Find he priculr soluion of he differenil equion dy y y, given h y = when = Sol We hve dy y y dy ( )( y) dy ( ) ( y) log y C Given h y = when = so, log () C C = required soluion is : log y dy n Q9 Solve he differenil equion : ( ) y e n dy n dy e Sol Given ( ) y e y This is liner differenil equion of he form dy P()y Q() So, n e P(),Q() n Now, IF e e n n n e required soluion is : y( e ) e C n n y( e n e ) d C [Pu e d n y( e n n ) C y( e ) e C n n y C e e is he required soluion Q Show h he four poins A, B, C nd D wih posiion vecors i ˆ 5j ˆ+ kˆ, ˆj kˆ, i ˆ 9j ˆ+ kˆ nd ( ˆi ˆj + k) ˆ respecively re coplnr OR The sclr produc of he vecor = ˆi ˆj + kˆ wih uni vecor long he sum of vecors b = i ˆ j ˆ 5kˆ nd c ˆi j ˆ+ kˆ is equl o one Find he vlue of nd hence find he uni vecor long b c Sol The poins A, B, C nd D re coplnr if ABAC AD Now, AB i ˆ 6ˆj k, ˆ AC ˆi ˆj k, ˆ AD 8i ˆ ˆj kˆ 6 ABAC AD ( ) 6( ) ( ) 8 So, A, B, C nd D re coplnr b c OR Given b c b c b c (i ˆ ˆj + k)(i ˆ ˆ j ˆ 5k) ˆ (iˆ ˆj + k)( ˆ iˆ j ˆ+ k) ˆ i ˆ j ˆ 5kˆ ˆi j ˆ+ k ˆ 5 + ( ) 6 ( ) For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 5
6 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) Also, uni vecor long b c is given s : b c ( )i ˆ 6j ˆ k ˆ ( )iˆ 6j ˆ kˆ i ˆ 6ˆj kˆ b c ( ) 6 ( ) ( ) 6 ( ) 7 Q A line psses hrough (,, ) nd is perpendiculr o he lines r = (i ˆ ˆj k) ˆ (i ˆ j ˆ+ k) ˆ nd r = (i ˆ ˆj k) ˆ μ(i ˆ+ ˆj + k) ˆ Obin is equion in vecor nd Cresin form Sol Given lines re r = (i ˆ ˆj k) ˆ (i ˆ j ˆ+ k) ˆ nd r = (i ˆ ˆj k) ˆ μ(i ˆ+ ˆj + k) ˆ A line perpendiculr o he given lines will be in he direcion of ˆi ˆj kˆ (i ˆ j ˆ+ k) ˆ (i ˆ j ˆ+ k) ˆ = 6i ˆ j ˆ+ 6k ˆ or,b = i ˆ ˆj kˆ Posiion vecor of given poin (,, ) is, i ˆ ˆj kˆ Using r = b, he required vecor equion of line is : r = i ˆ ˆj k ˆ (i ˆ ˆj k) ˆ And, Cresin form of he line is : y z Q An eperimen succeeds hrice s ofen s i fils Find he probbiliy h in he ne five rils, here will be les successes Sol Le he probbiliy of success be p nd h of filure be q Therefore, p = q Since p + q = so, q + q = q, p P( les successes) P(r ) P() + P() + P(5) n r n r Using P(r) = C p q, where n = 5 r ie, P(r ) C C C So, required probbiliy 5 SECTION C Q Two schools A nd B wn o wrd heir seleced sudens on he vlues of sinceriy, ruhfulness nd helpfulness The school A wns o wrd ` ech, `y ech nd `z ech for he hree respecive vlues o is, nd sudens wih ol wrd money of `6 School B wns o spend ` o wrd is, nd sudens on he respecive vlues (by giving he sme wrd money for he hree vlues s before) If he ol moun of wrds for one prize on ech vlue is `9, using mrices, find he wrd money for ech vlue Apr from hese hree vlues, sugges one more vlue which should be considered for wrd Sol Le he wrd money spen on he vlues of sinceriy, ruhfulness nd helpfulness be ` ech, `y ech nd `z ech respecively + y + z = 6, + y + z =, + y + z = 9 6 The given siuion cn be epressed s : y z 9 6 where A, X y, B AX B X A B (i) z 9 Now, A = ( ) ( ) + ( ) = 5, so A eiss Consider A ij s he cofcors of he elemen ij of mri A For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 6
7 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) A, A, A, A, A, A, A 5, A 5, A So, A 5 dj A dj A 5 A By (i), X 5 y z By equliy of mrices, we ge :, y, z Hence he wrd money for he vlues of sinceriy, ruhfulness nd helpfulness is `, ` nd ` respecively Also, he vlue of Obedience cn be included for he wrds Q Prove h he liude of he righ circulr cone of mimum volume h cn be inscribed in sphere of rdius r is r Also show h he mimum volume of he cone is 8 of he 7 volume of he sphere Sol Le VAB be cone of mimum volume inscribed in sphere of rdius r Le OC = Then AC r = Rdius of cone, VC = OC + VO = + r = Heigh of cone Then volume of cone, V = (AC) (VC) V (r )(r ) r dv r (r ) r r A d V And, 6 r dv For poins of locl mim & locl minim, we hve ie, r r (r )(r ) r =, r d V r We shll rejec r So, r r r = So, V is mimum r r r Now, heigh of cone VC = + r r = Hence Proved r r 8 Now, volume of he cone = r r r r mimum volume of cone volume of sphere Hence Proved 7 Q5 Evlue : cos sin Sol Le I = cos sin Dividing Nr nd Dr by cos, we ge : For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 7
8 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) sec ( n ) sec I = n n ( ) I d I d Now, pu v d dv Also v I dv n C v ( ) Pu n sec d Dividing Nr & Dr by v v n So, I n C I n C n Q6 Using inegrion, find he re of he region bounded by he ringle whose verices re (, ), (, 5) nd (, ) Sol Le A(, ), B(, 5) nd C(, ) form ringle ABC Equion of side AB : y = ( 7), Equion of side BC : y = ( ), Equion of side CA : y = ( 5) So, Required re of ABC = y y y AB BC CA = ( 7) ( ) ( 5) + + = ( 7) ( ) ( 5) + + = 7 9 SqUnis Q7 Find he equion of he plne hrough he line of inersecion of he plnes + y + z = nd + y + z = 5 which is perpendiculr o he plne y + z = Also find he disnce of he plne obined bove, from he origin OR Find he disnce of he poin (,, 5) from he poin of inersecion of he line r = i ˆ j ˆ+ k ˆ (i ˆ+ j ˆ+ k) ˆ nd he plne r(i ˆ j ˆ+ k) ˆ = Sol The equion of plne hrough he line of inersecion of he plnes + y + z = nd + y + z = 5 is given s : y z + λ( + y+ z 5) = ie, (+ ) (+ )y (+ )z 5λ = (i) Since (i) is perpendiculr o he plne y + z =, So (+ )() (+ )( ) (+ )() = = Using bb cc Subsiuing he vlue of = in (i), we hve : + + y + z 5 = ie, z = For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 8
9 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) + Also, he disnce of he plne z = from origin (, ) is Unis ( ) OR Given line is r = i ˆ j ˆ+ k ˆ (i ˆ+ j ˆ+ k) ˆ so, he posiion vecor of coordines of ny rndom poin on his line is OP ( )i ˆ+ ( ) ˆj + ( )kˆ For he given line o inersec he plne r(i ˆ j ˆ+ k) ˆ =, he posiion vecor of coordines of ny rndom poin OP mus sisfy he equion of plne for some vlue of ie, ( )i ˆ+ ( ) ˆj + ( )k ˆ (iˆ j ˆ+ k) ˆ = ( ) + ( )( ) + ( ) = he poin of inersecion is OP iˆ + j ˆ kˆ ie, P(,, ) So, he required disnce of P from (,, 5) is ( ) ( ) ( 5) Unis Q8 A mnufcuring compny mkes wo ypes of eching ids A nd B of Mhemics for clss XII Ech ype of A requires 9 lbour hours of fbricing nd hour for finishing Ech ype of B requires lbour hours for fbricing nd lbour hours for finishing For fbricing nd finishing, he mimum lbour hours vilble per week re 8 nd respecively The compny mkes profi of `8 on ech piece of ype A nd ` on ech piece of ype B How mny pieces of ype A nd ype B should be mnufcured per week o ge mimum profi? Mke i s n LPP nd solve grphiclly Wh is he mimum profi per week? Sol Le he number of pieces of ype A nd ype B mnufcured per week be nd y respecively To mimize : Z ` (8 + y) Subjec o consrins :, y, 9 y 8 y 6, y Corner poins of fesible region Vlue of Z (in `) O(, ) A(, ) B(, 6) 68 C(, ) 6 Scle : smll boes on boh he es = unis So, mimum profi of `68 is obined when pieces of ype A nd 6 pieces of ype B re mnufcured by he compny per week Q9 There re hree coins One is wo-heded coin (hving hed on boh fces), noher is bised coin h comes up heds 75% of he imes nd hird is lso bised coin h comes up ils % of he imes One of he hree coins is chosen rndom nd ossed, nd i shows heds Wh is he probbiliy h i ws he wo-heded coin? OR Two numbers re seleced rndom (wihou replcemen) from he firs si posiive inegers Le X denoe he lrger of he wo numbers obined Find he probbiliy disribuion of he rndom vrible X, nd hence find he men of he disribuion Sol Le E : choosing firs (wo heded) coin, E : choosing second (bised) coin, E : choosing hird coin Also, le A : he coin showing heds 75 6 P(E ) P(E ) P(E ), P(A E ), P(A E ), P(A E ) P(A E )P(E ) By Byes' Theorem, P(E A) P(A E )P(E ) P(A E )P(E ) P(A E )P(E ) For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 9
10 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) 7 5 OR 65 Tol number of wys of selecing wo numbers form si posiive inegers Le X denoes he lrger of he Two numbers seleced So, vlues of X :,,, 5, 6 5 So, P(X ), P(X ), P(X ), P(X 5), P(X 6) disribuion cn be wrien s : 6 C 5 X 5 6 P(X) /5 /5 /5 /5 5/5 5 7 Therefore, men of he disribuion = X P(X) = NOTE: Only hose Quesions from Se nd re given here which re no in common wih Se SET Wih Complee Eplnions Q9 Evlue : e e log Sol Le I log e e Pu log d Also when d I log log log log e nd, when = e Q Find vecor of mgniude 5, mking n ngle of wih -is, wih y-is nd n cue ngle wih θ wih z-is Sol Le â be he uni vecor in he direcion of vecor Since vecor mkes n ngle of wih -is, wih y-is nd n cue ngle wih θ wih z- is herefore, cos cos cos θ [Using cos α cos β cos γ ie, cos θ cos θ θ Therefore, ˆ 5 5 cos ˆi cos ˆj cos kˆ = = 5i ˆ 5kˆ b+ c c+ + b b c Q9 Using properies of deerminns, prove h : q+ r r + p p+ q p q r y+ z z+ + y y z Sol LHS : Le b+ c c+ + b q+ r r+ p p+ q [Apply C C (C C ) y+ z z+ + y For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
11 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) c+ + b c+ + b p r + p p+ q p r + p p+ q z+ + y z+ + y c b [Tking common from C p r q [By C C C, C C C z y b c ( ) p q r [By C C y z b c p q r RHS y z dy b Q If sin ( cos ) nd y = bcos( cos), show h, Sol We hve sin ( cos ) cos ( cos ) sin d dy And, y = bcos ( cos ) bcos sin bsin ( cos ) d dy dy d bcos sin bsin ( cos ) b[cos sin sin ( cos)] d cos ( cos ) sin [ cos ( cos ) sin ] Since we know h, sin nd cos, dy b[( )] b So, Hence Proved [( ) ] Q Find he priculr soluion of he differenil equion ( y ) y(+ )dy, given h y = when = Sol We hve ( y ) y(+ )dy y dy (+ ) ( y ) ie, y dy (+ ) ( y ) log + log + y log k log + log + y log C, where log C log k + C (i) + y Given h y = when = so by (i), + C C = + Therefore, he required priculr soluion is (+ ) + y or, y + Q Find he vecor nd Cresin equions of he line pssing hrough he poin (,, ) nd perpendiculr o he lines y z nd y z 5 Sol The dr s of given lines y z nd y z 5 re respecively,, ;,, 5 Le he dr s of required line be, b, c er So by using bb cc for lines, we hve : b c nd b 5c b c 8 he dr's re, 7, for he required line For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
12 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) So, he eqion of line in Cresin form is : y z nd, he vecor form of he line is 7 given s : r = i ˆ ˆj + k ˆ (i ˆ 7ˆj + k) ˆ n co sin cos I n co sin cos cos sin sin cos d Q8 Evlue : Sol Le Pu Also cos sin sin cos I d I I sin (sin cos ) C d sin C Q8 Prove h he heigh of he cylinder of mimum volume h cn be inscribed in sphere of rdius R is R Also find he mimum volume Sol Le r nd h be he rdius nd heigh of cylinder inscribed in sphere of rdius R In BDA, BD AD AB [By using Pyhgors heorem h (r) (R) R h r (i) Now volume of he cylinder, V r h R h V h [By using (i) V R h h Differeniing wr h boh he sides : dv R h dh dv h Agin differeniing wr h boh he sides : 6h dh dv For poins of locl mim & minim, dh R h h R R h Now, R d V R dh R h = So, V is mimum R h Hence, heigh of he cylinder of mimum volume h cn be inscribed in sphere of rdius R is R For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
13 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) Also volume, Hence mimum volume is R R R V R Cubic Unis R Cubic Unis Q9 If Sol, find he vlue of 8 We hve 8 SET Wih Complee Eplnions n 8 n n 8 n n () = Q If nd b re perpendiculr vecors, b = nd = 5, find he vlue of b Sol We hve b = b = ( b)( b) = 69 b b = 69 5 b () = 69 [As b b 5 b = 69 b = b = Q9 Using properies of deerminns, prove h : Sol LHS : Le b c b c bc b b b Apply R R,R R,R R b bc bc c b c [By R R R R c c c b c b c b c bc b b b c c c [Tking common from R b c For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge
14 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) Q If Sol bc b c b b b c c c bc b c b c bc b c bc bc c b RHS cos ( cos ) nd We hve And, [By C C C, C C C [Epnding long R y sin ( sin ), find he vlue of dy cos ( cos ) cos cos 6cos sin sin = sin (cos ) dy d y sin ( sin ) sin sin cos 6 sin cos = cos ( sin ) dy dy d cos ( sin ) co d sin (cos ) dy So, co (/) = / Q Find he priculr soluion of he differenil equion y when For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge d dy log + y, given h dy dy Sol We hve log + y y y + y e e e e dy e y y e e C e + e + C 7 Given h y when so, e + e + C C y Therefore, he required priculr soluion is : e + e 7 7y z 7 7 y 5 6 z Q Find he vlue of p so h he lines l : nd l : re p p 5 perpendiculr o ech oher Also find he equions of line pssing hrough poin (,, ) nd prllel o he line l y z y 5 z 6 Sol Given lines re l : nd l p : p p p So, he dr s of hese lines re,,;,, Since l l so, by using bb cc we hve : p p ( ) () ()( 5) p 7 7 7
15 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) Also, he equion of line pssing hrough (,, ) nd prllel o l is : y z ie, y z [Since prllel lines hve proporionl dr s 7 7 Q8 If he sum of he lenghs of he hypoenuse nd side of righ ringle is given, show h he re o of he ringle is mimum, when he ngle beween hem is 6 Sol Le he lengh of he side AB of righ ringle be nd h of hypoenuse AC be y Given h + y = k (i) Are of ringle, A BC AB A y Le S A (y ) S [(k ) ] S [k k ] ds [k 6k ] d S [k k] For he locl poins of mim nd/or minim, ds k ie, [k 6k ] d S k [k k ] k/ k Th implies, S is mimum nd, hence A is lso mimum k Now, cos A y k k k cos A k cos A cos A cos ie, A 6 Q9 Evlue : sin sin cos cos Sol Le I sin sin cos cos o Hence Proved For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 5
16 CBSE Annul Em Pper (All Indi) Compiled By OP Gup ( ) sec [Dividing Nr & Dr by n n ( n )sec n n ( ) cos Pu n sec d I d [Divide Nr & Dr by I d Pu u d du Also u u I du u ( ) I u n C I n C I n C So, n I n C n Der Suden/Techer, I would urge you for lile fvour Plese noify me bou ny error(s) you noice in his (or oher Mhs) work I would be beneficil for ll he fuure lerners of Mhs like us Any consrucive criicism will be well cknowledged Plese find below my conc info when you decide o offer me your vluble suggesions I m looking forwrd for response Also I would wish if you inform your friends/sudens bou my effors for Mhs so h hey my lso benefi Le s lern Mhs wih smile :-) For ny clrificion(s), plese conc : MhsGuru OP Gup [Mhs (Hons), E & C Engg, Indir Awrd Winner] Conc Nos : Mil me : heopgup@gmilcom Officil Web-pge : For vrious suffs on Mhs, plese visi : wwwheopgupcom Pge 6
EXERCISE - 01 CHECK YOUR GRASP
UNIT # 09 PARABOLA, ELLIPSE & HYPERBOLA PARABOLA EXERCISE - 0 CHECK YOUR GRASP. Hin : Disnce beween direcri nd focus is 5. Given (, be one end of focl chord hen oher end be, lengh of focl chord 6. Focus
More informationPARABOLA. moves such that PM. = e (constant > 0) (eccentricity) then locus of P is called a conic. or conic section.
wwwskshieducioncom PARABOLA Le S be given fixed poin (focus) nd le l be given fixed line (Direcrix) Le SP nd PM be he disnce of vrible poin P o he focus nd direcrix respecively nd P SP moves such h PM
More informationSeptember 20 Homework Solutions
College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum
More informationCBSE 2013 ALL INDIA EXAMINATION [Set 1 With Solutions]
M Mrks : Q Write the principl vlue of CBSE ALL INDIA EXAMINATION [Set With Solutions] SECTION A tn ( ) cot ( ) Time Allowed : Hours Sol tn ( ) cot ( ) tn tn cot cot cot cot [Rnge of tn :,, cot : ], [ 5
More informationMotion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.
Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl
More informationPHYSICS 1210 Exam 1 University of Wyoming 14 February points
PHYSICS 1210 Em 1 Uniersiy of Wyoming 14 Februry 2013 150 poins This es is open-noe nd closed-book. Clculors re permied bu compuers re no. No collborion, consulion, or communicion wih oher people (oher
More informationVidyalankar. 1. (a) Y = a cos dy d = a 3 cos2 ( sin ) x = a sin dx d = a 3 sin2 cos slope = dy dx. dx = y. cos. sin. 3a sin cos = cot at = 4 = 1
. (). (b) Vilnkr S.Y. Diplom : Sem. III [AE/CE/CH/CM/CO/CR/CS/CW/DE/EE/EP/IF/EJ/EN/ET/EV/EX/IC/IE/IS/ ME/MU/PG/PT/PS/CD/CV/ED/EI/FE/IU/MH/MI] Applied Mhemics Prelim Quesion Pper Soluion Y cos d cos ( sin
More informationMTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
More informationPhysics 2A HW #3 Solutions
Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
More informationENGR 1990 Engineering Mathematics The Integral of a Function as a Function
ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under
More information4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
More informationCHAPTER 11 PARAMETRIC EQUATIONS AND POLAR COORDINATES
CHAPTER PARAMETRIC EQUATIONS AND POLAR COORDINATES. PARAMETRIZATIONS OF PLANE CURVES., 9, _ _ Ê.,, Ê or, Ÿ. 5, 7, _ _.,, Ÿ Ÿ Ê Ê 5 Ê ( 5) Ê ˆ Ê 6 Ê ( 5) 7 Ê Ê, Ÿ Ÿ $ 5. cos, sin, Ÿ Ÿ 6. cos ( ), sin (
More informationMATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)
MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide
More informationAvailable Online :
fo/u fopjr Hh# u] ugh vjehs de] foifr ns[ NsMs rqjr e/;e eu dj ';ea iq#" flg ldyi dj] lgrs foifr vusd] ^cu^ u NsMs /;s; ds] j?qcj j[s VsdAA jfpr% euo /ez iz.sr ln~xq# Jh j.nsmnlh egj STUDY PACKAGE Subjec
More informatione t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
More informationA 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m
PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl
More informationMagnetostatics Bar Magnet. Magnetostatics Oersted s Experiment
Mgneosics Br Mgne As fr bck s 4500 yers go, he Chinese discovered h cerin ypes of iron ore could rc ech oher nd cerin mels. Iron filings "mp" of br mgne s field Crefully suspended slivers of his mel were
More informationFM Applications of Integration 1.Centroid of Area
FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is
More informationPhysics 101 Lecture 4 Motion in 2D and 3D
Phsics 11 Lecure 4 Moion in D nd 3D Dr. Ali ÖVGÜN EMU Phsics Deprmen www.ogun.com Vecor nd is componens The componens re he legs of he righ ringle whose hpoenuse is A A A A A n ( θ ) A Acos( θ) A A A nd
More informationf t f a f x dx By Lin McMullin f x dx= f b f a. 2
Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes
More informationContraction Mapping Principle Approach to Differential Equations
epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of
More informationThree Dimensional Coordinate Geometry
HKCWCC dvned evel Pure Mhs. / -D Co-Geomer Three Dimensionl Coordine Geomer. Coordine of Poin in Spe Z XOX, YOY nd ZOZ re he oordine-es. P,, is poin on he oordine plne nd is lled ordered riple. P,, X Y
More informationMotion in a Straight Line
Moion in Srigh Line. Preei reched he mero sion nd found h he esclor ws no working. She wlked up he sionry esclor in ime. On oher dys, if she remins sionry on he moing esclor, hen he esclor kes her up in
More informationSCIENCE ENTRANCE ACADEMY PREPARATORY EXAMINATION-3 (II P.U.C) SCHEME 0F EVALUATION Marks:150 Date: duration:4hours MATHEMATICS-35
JNANASUDHA SCIENCE ENTRANCE ACADEMY PREPARATORY EXAMINATION- (II P.U.C) SCHEME F EVALUATION Mrks:5 Dte:5-9-8 durtion:4hours MATHEMATICS-5 Q.NO Answer Description Mrk(s)!6 Zero mtri. + sin 4det A, 4 R 4
More informationAll Rights Reserved Wiley India Pvt. Ltd. 1
Question numbers to carry mark each. CBSE MATHEMATICS SECTION A. If R = {(, y) : + y = 8} is a relation of N, write the range of R. R = {(, y)! + y = 8} a relation of N. y = 8 y must be Integer So Can
More informationChapter Direct Method of Interpolation
Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o
More information0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
More informationCBSE-XII-2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0
CBSE-XII- EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section
More informationSome Inequalities variations on a common theme Lecture I, UL 2007
Some Inequliies vriions on common heme Lecure I, UL 2007 Finbrr Hollnd, Deprmen of Mhemics, Universiy College Cork, fhollnd@uccie; July 2, 2007 Three Problems Problem Assume i, b i, c i, i =, 2, 3 re rel
More informationa. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines.
Mah A Final Eam Problems for onsideraion. Show all work for credi. Be sure o show wha you know. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A.
More information( ) ( ) ( ) ( ) ( ) ( y )
8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll
More informationAverage & instantaneous velocity and acceleration Motion with constant acceleration
Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission
More information2D Motion WS. A horizontally launched projectile s initial vertical velocity is zero. Solve the following problems with this information.
Nme D Moion WS The equions of moion h rele o projeciles were discussed in he Projecile Moion Anlsis Acii. ou found h projecile moes wih consn eloci in he horizonl direcion nd consn ccelerion in he ericl
More informationPhysic 231 Lecture 4. Mi it ftd l t. Main points of today s lecture: Example: addition of velocities Trajectories of objects in 2 = =
Mi i fd l Phsic 3 Lecure 4 Min poins of od s lecure: Emple: ddiion of elociies Trjecories of objecs in dimensions: dimensions: g 9.8m/s downwrds ( ) g o g g Emple: A foobll pler runs he pern gien in he
More information!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
More informationProperties of Logarithms. Solving Exponential and Logarithmic Equations. Properties of Logarithms. Properties of Logarithms. ( x)
Properies of Logrihms Solving Eponenil nd Logrihmic Equions Properies of Logrihms Produc Rule ( ) log mn = log m + log n ( ) log = log + log Properies of Logrihms Quoien Rule log m = logm logn n log7 =
More informationPhys 110. Answers to even numbered problems on Midterm Map
Phys Answers o een numbered problems on Miderm Mp. REASONING The word per indices rio, so.35 mm per dy mens.35 mm/d, which is o be epressed s re in f/cenury. These unis differ from he gien unis in boh
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
More informationSolutions to Problems from Chapter 2
Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5
More informationThe solution is often represented as a vector: 2xI + 4X2 + 2X3 + 4X4 + 2X5 = 4 2xI + 4X2 + 3X3 + 3X4 + 3X5 = 4. 3xI + 6X2 + 6X3 + 3X4 + 6X5 = 6.
[~ o o :- o o ill] i 1. Mrices, Vecors, nd Guss-Jordn Eliminion 1 x y = = - z= The soluion is ofen represened s vecor: n his exmple, he process of eliminion works very smoohly. We cn elimine ll enries
More information5.1-The Initial-Value Problems For Ordinary Differential Equations
5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil
More information(b) 10 yr. (b) 13 m. 1.6 m s, m s m s (c) 13.1 s. 32. (a) 20.0 s (b) No, the minimum distance to stop = 1.00 km. 1.
Answers o Een Numbered Problems Chper. () 7 m s, 6 m s (b) 8 5 yr 4.. m ih 6. () 5. m s (b).5 m s (c).5 m s (d) 3.33 m s (e) 8. ().3 min (b) 64 mi..3 h. ().3 s (b) 3 m 4..8 mi wes of he flgpole 6. (b)
More informationMathematics Extension 2
00 HIGHER SCHOOL CERTIFICATE EXAMINATION Mthemtics Etension Generl Instructions Reding time 5 minutes Working time hours Write using blck or blue pen Bord-pproved clcultors my be used A tble of stndrd
More informationTransformations. Ordered set of numbers: (1,2,3,4) Example: (x,y,z) coordinates of pt in space. Vectors
Trnformion Ordered e of number:,,,4 Emple:,,z coordine of p in pce. Vecor If, n i i, K, n, i uni ecor Vecor ddiion +w, +, +, + V+w w Sclr roduc,, Inner do roduc α w. w +,.,. The inner produc i SCLR!. w,.,
More informationBoard Answer Paper: October 2014
Trget Pulictions Pvt. Ltd. Bord Answer Pper: Octoer 4 Mthemtics nd Sttistics SECTION I Q.. (A) Select nd write the correct nswer from the given lterntives in ech of the following su-questions: i. (D) ii..p
More informationF.Y. Diploma : Sem. II [CE/CR/CS] Applied Mathematics
F.Y. Diplom : Sem. II [CE/CR/CS] Applied Mhemics Prelim Quesio Pper Soluio Q. Aemp y FIVE of he followig : [0] Q. () Defie Eve d odd fucios. [] As.: A fucio f() is sid o e eve fucio if f() f() A fucio
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More information1.0 Electrical Systems
. Elecricl Sysems The ypes of dynmicl sysems we will e sudying cn e modeled in erms of lgeric equions, differenil equions, or inegrl equions. We will egin y looking fmilir mhemicl models of idel resisors,
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More informationANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2
ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER Seion Eerise -: Coninuiy of he uiliy funion Le λ ( ) be he monooni uiliy funion defined in he proof of eisene of uiliy funion If his funion is oninuous y hen
More informationQuestion Details Int Vocab 1 [ ] Question Details Int Vocab 2 [ ]
/3/5 Assignmen Previewer 3 Bsic: Definie Inegrls (67795) Due: Wed Apr 5 5 9: AM MDT Quesion 3 5 6 7 8 9 3 5 6 7 8 9 3 5 6 Insrucions Red ody's Noes nd Lerning Gols. Quesion Deils In Vocb [37897] The chnge
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationMATHEMATICS PART A. 1. ABC is a triangle, right angled at A. The resultant of the forces acting along AB, AC
FIITJEE Solutions to AIEEE MATHEMATICS PART A. ABC is tringle, right ngled t A. The resultnt of the forces cting long AB, AC with mgnitudes AB nd respectively is the force long AD, where D is the AC foot
More informationForms of Energy. Mass = Energy. Page 1. SPH4U: Introduction to Work. Work & Energy. Particle Physics:
SPH4U: Inroducion o ork ork & Energy ork & Energy Discussion Definiion Do Produc ork of consn force ork/kineic energy heore ork of uliple consn forces Coens One of he os iporn conceps in physics Alernive
More informationREAL ANALYSIS I HOMEWORK 3. Chapter 1
REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs
More informationVersion 001 test-1 swinney (57010) 1. is constant at m/s.
Version 001 es-1 swinne (57010) 1 This prin-ou should hve 20 quesions. Muliple-choice quesions m coninue on he nex column or pge find ll choices before nswering. CubeUniVec1x76 001 10.0 poins Acubeis1.4fee
More informationπ = tanc 1 + tan x ...(i)
Solutions to RSPL/ π. Let, I log ( tn ) d Using f () d f ( ) d π π I log( tnc d m log( cot ) d...(ii) On dding (i) nd (ii), we get +,. Given f() + ), For continuit t lim " lim f () " ( ) \ Continuous t.
More information1. If * is the operation defined by a*b = a b for a, b N, then (2 * 3) * 2 is equal to (A) 81 (B) 512 (C) 216 (D) 64 (E) 243 ANSWER : D
. If * is the opertion defined by *b = b for, b N, then ( * ) * is equl to (A) 8 (B) 5 (C) 6 (D) 64 (E) 4. The domin of the function ( 9)/( ),if f( ) = is 6, if = (A) (0, ) (B) (-, ) (C) (-, ) (D) (, )
More informationAP Calculus BC - Parametric equations and vectors Chapter 9- AP Exam Problems solutions
AP Calculus BC - Parameric equaions and vecors Chaper 9- AP Exam Problems soluions. A 5 and 5. B A, 4 + 8. C A, 4 + 4 8 ; he poin a is (,). y + ( x ) x + 4 4. e + e D A, slope.5 6 e e e 5. A d hus d d
More information3 Motion with constant acceleration: Linear and projectile motion
3 Moion wih consn ccelerion: Liner nd projecile moion cons, In he precedin Lecure we he considered moion wih consn ccelerion lon he is: Noe h,, cn be posiie nd neie h leds o rie of behiors. Clerl similr
More informationHow to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.
How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle
More informationINTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).
INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationThink of the Relationship Between Time and Space Again
Repor nd Opinion, 1(3),009 hp://wwwsciencepubne sciencepub@gmilcom Think of he Relionship Beween Time nd Spce Agin Yng F-cheng Compny of Ruid Cenre in Xinjing 15 Hongxing Sree, Klmyi, Xingjing 834000,
More informationAP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
More informationf 5 x 3 d x x 12 x 16 x a x < 1 b x < 2 a c x e f b i c d + +
ANSWERS Chper Eercise. 0 0 c d 0 0 e f 0 [,] ],[ c ]0,] d ],0] e ],[ f ],[],[ c + c d + + c + d e - + + 0 + f + i - + ii 0 + i - + ii - + 0 {±} {±0} c Ø d {,} e {,} f {0,} - 0 0 c d 0 ],[ ],[ c ],[ + Eercise..
More informationAnswers to 1 Homework
Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:
More informationA Simple Method to Solve Quartic Equations. Key words: Polynomials, Quartics, Equations of the Fourth Degree INTRODUCTION
Ausrlin Journl of Bsic nd Applied Sciences, 6(6): -6, 0 ISSN 99-878 A Simple Mehod o Solve Quric Equions Amir Fhi, Poo Mobdersn, Rhim Fhi Deprmen of Elecricl Engineering, Urmi brnch, Islmic Ad Universi,
More informationSIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddhrh Ngr, Nrynvnm Rod 5758 QUESTION BANK (DESCRIPTIVE) Subjec wih Code :Engineering Mhemic-I (6HS6) Coure & Brnch: B.Tech Com o ll Yer & Sem:
More informationP441 Analytical Mechanics - I. Coupled Oscillators. c Alex R. Dzierba
Lecure 3 Mondy - Deceber 5, 005 Wrien or ls upded: Deceber 3, 005 P44 Anlyicl Mechnics - I oupled Oscillors c Alex R. Dzierb oupled oscillors - rix echnique In Figure we show n exple of wo coupled oscillors,
More informationChapter 2: Evaluative Feedback
Chper 2: Evluive Feedbck Evluing cions vs. insrucing by giving correc cions Pure evluive feedbck depends olly on he cion ken. Pure insrucive feedbck depends no ll on he cion ken. Supervised lerning is
More informationTax Audit and Vertical Externalities
T Audi nd Vericl Eernliies Hidey Ko Misuyoshi Yngihr Ngoy Keizi Universiy Ngoy Universiy 1. Inroducion The vericl fiscl eernliies rise when he differen levels of governmens, such s he federl nd se governmens,
More informationCBSE 2018 ANNUAL EXAMINATION DELHI
CBSE 08 ANNUAL EXAMINATION DELHI (Series SGN Code No 65/ : Delhi Region) Ma Marks : 00 Time Allowed : Hours SECTION A Q0 Find the value of tan cot ( ) Sol 5 5 tan cot ( ) tan tan cot cot 6 6 6 0 a Q0 If
More informationMinimum Squared Error
Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > for ll smples y i solve sysem of liner inequliies MSE procedure y i = i for ll smples
More informationMinimum Squared Error
Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > 0 for ll smples y i solve sysem of liner inequliies MSE procedure y i i for ll smples
More informationConvergence of Singular Integral Operators in Weighted Lebesgue Spaces
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 335-347 ISSN 1307-5543 www.ejpm.com Published by New York Business Globl Convergence of Singulr Inegrl Operors in Weighed Lebesgue
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationMH CET 2018 (QUESTION WITH ANSWER)
( P C M ) MH CET 8 (QUESTION WITH ANSWER). /.sec () + log () - log (3) + log () Ans. () - log MATHS () 3 c + c C C A cos + cos c + cosc + + cosa ( + cosc ) + + cosa c c ( + + ) c / / I tn - in sec - in
More informationLevel I MAML Olympiad 2001 Page 1 of 6 (A) 90 (B) 92 (C) 94 (D) 96 (E) 98 (A) 48 (B) 54 (C) 60 (D) 66 (E) 72 (A) 9 (B) 13 (C) 17 (D) 25 (E) 38
Level I MAML Olympid 00 Pge of 6. Si students in smll clss took n em on the scheduled dte. The verge of their grdes ws 75. The seventh student in the clss ws ill tht dy nd took the em lte. When her score
More informationMotion on a Curve and Curvature
Moion on Cue nd Cuue his uni is bsed on Secions 9. & 9.3, Chpe 9. All ssigned edings nd execises e fom he exbook Objecies: Mke cein h you cn define, nd use in conex, he ems, conceps nd fomuls lised below:
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationLog1 Contest Round 3 Theta Individual. 4 points each 1 What is the sum of the first 5 Fibonacci numbers if the first two are 1, 1?
008 009 Log1 Contest Round Thet Individul Nme: points ech 1 Wht is the sum of the first Fiboncci numbers if the first two re 1, 1? If two crds re drwn from stndrd crd deck, wht is the probbility of drwing
More information1. Consider a PSA initially at rest in the beginning of the left-hand end of a long ISS corridor. Assume xo = 0 on the left end of the ISS corridor.
In Eercise 1, use sndrd recngulr Cresin coordine sysem. Le ime be represened long he horizonl is. Assume ll ccelerions nd decelerions re consn. 1. Consider PSA iniilly res in he beginning of he lef-hnd
More informationGENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS
- TAMKANG JOURNAL OF MATHEMATICS Volume 5, Number, 7-5, June doi:5556/jkjm555 Avilble online hp://journlsmhkueduw/ - - - GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS MARCELA
More informationParametrics and Vectors (BC Only)
Paramerics and Vecors (BC Only) The following relaionships should be learned and memorized. The paricle s posiion vecor is r() x(), y(). The velociy vecor is v(),. The speed is he magniude of he velociy
More informationLecture 10: Wave equation, solution by spherical means
Lecure : Wave equaion, soluion by spherical means Physical modeling eample: Elasodynamics u (; ) displacemen vecor in elasic body occupying a domain U R n, U, The posiion of he maerial poin siing a U in
More informationA Time Truncated Improved Group Sampling Plans for Rayleigh and Log - Logistic Distributions
ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 A Time Trunced Improved Group Smpling Plns for Ryleigh nd og - ogisic Disribuions P.Kvipriy, A.R. Sudmni Rmswmy
More information15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )
- TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the
More informationPART - III : MATHEMATICS
JEE(Advnced) 4 Finl Em/Pper-/Code-8 PART - III : SECTION : (One or More Thn One Options Correct Type) This section contins multiple choice questions. Ech question hs four choices (A), (B), (C) nd (D) out
More informationSection P.1 Notes Page 1 Section P.1 Precalculus and Trigonometry Review
Secion P Noe Pge Secion P Preclculu nd Trigonomer Review ALGEBRA AND PRECALCULUS Eponen Lw: Emple: 8 Emple: Emple: Emple: b b Emple: 9 EXAMPLE: Simplif: nd wrie wi poiive eponen Fir I will flip e frcion
More informationProbability, Estimators, and Stationarity
Chper Probbiliy, Esimors, nd Sionriy Consider signl genered by dynmicl process, R, R. Considering s funcion of ime, we re opering in he ime domin. A fundmenl wy o chrcerize he dynmics using he ime domin
More information3. Renewal Limit Theorems
Virul Lborories > 14. Renewl Processes > 1 2 3 3. Renewl Limi Theorems In he inroducion o renewl processes, we noed h he rrivl ime process nd he couning process re inverses, in sens The rrivl ime process
More informationBEng (Hons) Telecommunications. Examinations for / Semester 2
BEng (Hons) Telecommunicaions Cohor: BTEL/14/FT Examinaions for 2015-2016 / Semeser 2 MODULE: ELECTROMAGNETIC THEORY MODULE CODE: ASE2103 Duraion: 2 ½ Hours Insrucions o Candidaes: 1. Answer ALL 4 (FOUR)
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationHow to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
More informationKINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationADDITIONAL MATHEMATICS PAPER 1
000-CE A MATH PAPER HONG KONG EXAMINATIONS AUTHORITY HONG KONG CERTIFICATE OF EDUCATION EXAMINATION 000 ADDITIONAL MATHEMATICS PAPER 8.0 am 0.0 am ( hours This paper mus be answered in English. Answer
More informationPhysics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
More informationA Kalman filtering simulation
A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer
More information