12 STD BUSINESS MATHEMATICS FORMULAE

Transcription

1 STD BUSINSS MATHMATICS FORMULA CHAPTR APPLICATIONS OF MATRICS AND DTRMINANTS T Adjoit o mtri A is AdjA A c whr A c is coctor mtri) Ivrs o mtri A is A AdjA A Rslts: i) AAdjA AdjA) A AI ii) Adj AB) AdjB ) AdjA iii) AB B A iv) AA A A I v) A A 4 Th rk o zro mtri irrspctiv o its ordr) is 5 Coditios or cosistc o Simltos Lir qtios No homogos): i) I A, B) A), th th qtios r cosistt d hs iq soltio ii) I A, B) A), th th qtios r cosistt d hs iiitl m soltios iii) I A, B) A), th th qtios r icosistt d hs o soltio 6 Coditios or cosistc o Simltos Lir qtios Homogos): i) I A, B) A), OR) I A th th qtios hv trivil soltios ol ii) I A, B) A), OR) I A th th qtios hv o trivil soltios lso z 7 Crmr s rl: ; ; z 8 Tcholog mtri B 9 Otpt mtri X I B D PAA PAB Trsitio Proilit Mtri P T OR) T PBA P BB P dpds o th m o th prodcts A, B or P, Q) For idig qilirim shr o mrkt A + B = OR) P + Q = This stp crris mrk d it is complsor) PP QP P P PQ QQ wwwklvisolicom K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

2 wwwklvisolicom CHAPTR ANALYTICAL GOMTRY SP PM cctricit o prol = cctricit o llips < 4 cctricit o hprol > 5 cctricit o rctglr hprol 6 Prol: = 4 ops rightwrd) = -4 ops ltwrd) = 4 ops pwrd) = - 4 ops dowwrd) Vrt,),),),) Focs,,,, Dirctri Ltsrctm Ais 7 llips:,, Ctr,),) cctricit OR) OR) Vrtics,,,,,, Dirctri Ltsrctm Foci,,,,,, K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

3 wwwklvisolicom 8 Hprol: Ctr,),) cctricit OR) OR) Vrtics,,,,,, Dirctri Ltsrctm Foci,,,,,, 9 Th grl qtio o Rctglr Hprol RH) is = c whr c sl or ojctivs) Th cctricit o Rctglr Hprol RH) is K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

4 wwwklvisolicom CHAPTR APPLICATIONS OF DIFFRNTIATION I C ) k Avrg cost AC) = or) ) Avrg vril cost AVC) = k Avrg id cost AFC) = dc 4 Mrgil cost MC) = d 5 Mrgil vrg cost MAC) = dac d 6 Totl rv R = p R 7 Avrg rv AR) = Avrg rv = Dmd ctio i, AR = p) dr 8 Mrgil vrg rv MR) = d p d 9 I = p) is dmd ctio, th lsticit o dmd d dp Whr qtit dmdd ; p pric) p dq Not : For dmd ctio q = p), d q dp p d I = p) is sppl ctio, th lsticit o sppl s dp Whr qtit spplid ; p pric) Rltio tw MR d lsticit o dmd is MR p d At qilirim lvl, Q d = Q s qtio o tgt is m m 4 qtio o orml is K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

5 wwwklvisolicom CHAPTR 4 APPLICATIONS OF DIFFRNTIATION II lr s thorm : I is homogos ctio o d with dgr th, or z c sd i th plc o dpds o th m o th ctio) Prtil lsticitis q p p q d q p q p p q q p RC coomic ordr qtit q ) C whr R Rqirmt ; C ordrig cost ; C crrig cost) % itpric 4 I it pric d prctg o ivtor r giv th crrig cost C q 5 Tim tw two cosctiv ordrs t ) R R 6 Nmr o ordrs = q 7 Miimm vrg vril cost = RC C R 8 Totl ordrig cost = C q q 9 Totl crrig cost = C K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

6 wwwklvisolicom CHAPTR 5 APPLICATIONS OF INTGRAL CALCULUS Proprtis o Diit itgrls: ) d ) d I ) is odd ctio, i, i -) = -) th ) d I ) is v ctio, i, i -) = ) th ) d ) d 4 ) d ) d 5 ) d ) d 6 Th r dr th crv ), th -is d th ordits t d is Ar d 7 Th r dr th crv = g ), th -is d th lis = c d = d is d Ar d c 8 I MC is th mrgil cost ctio th totl cost ctio is giv C MC d k 9 I MR is th mrgil rv ctio th totl rv ctio is giv R MR d k Th prodcrs srpls or th sppl ctio p g ) or th qtit d pric p is P S p g ) d Th cosmrs srpls or th dmd ctio p = ) or th qtit d pric p is C S ) d p K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

7 wwwklvisolicom CHAPTR 6 DIFFRNTIAL QUATIONS Th Grl orm o Homogos dirtil qtios is d d Workig rl or idig th soltio o lir dirtil qtios i) trct P d Q ii) Fid P d iii) Fid Itgrtig Fctor IF) = P d g,, d Th soltio to lir dirtil qtios o tp P Q Whr P d Q r ctios o d ol) is I F QI F Pd Pd d C OR) Q d C d 4 Th soltio to lir dirtil qtios o tp P Q Whr P d Q r ctios o d ol) is I F QI F Pd Pd d C OR) Q d C 5 Scod ordr lir dirtil qtios I m d m r th roots o th Aillir qtio is o th tp + + c = Qdrtic qtio) m i) I th roots m d m r rl d distict, CF = m A B m ii) I th roots m d m r rl d qlm = m ), CF = A B iii) I th roots m d m r rl, i, i CF Complmtr Fctio) m i, CF = Acos Bsi K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

8 CHAPTR 7 INTRPOLATION Forwrd oprtor dlt) ) or) ) ) )) h Bckwrd oprtor l) ) or) ) ) )) h h Th Shitig oprtor ), ), ) d so o 4 Th rltio tw orwrd oprtor dlt) d shitig oprtor is or) 5 For missig trm prolms) ) ) c) Grgor Nwto s orwrd orml :!!!! Whr h d h ql itrvl tw th - vls mr o trms i th orml dpds o th mr o trms i th prolm) 7 Grgor Nwto s ckwrd orml :!!!! Whr h d h ql itrvl tw th - vls mr o trms i th orml dpds o th mr o trms i th prolm) 8 Lgrg s orml: dpds o th mr o trms giv i th prolm) K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH : wwwklvisolicom

9 wwwklvisolicom 9 Li O Bst Fit: Norml qtios r Th li o st it is = + CHAPTR 8 PROBABILITY DISTRIBUTION I X is cotios rdom vril, th P X ) ) d For discrt rdom vril X, M X) i pi X ) Vr X ) X i p For cotios rdom vril X, i ) M X ) ) d X ) Vr X ) X ) d ) X ) X ) 4 I th discrt rdom vril X ollows Biomil distritio th P X ) C p q,,,, 5 Rslts rltd to Biomil distritio: M = p ; Vric = pq ; d p + q = 6 I th discrt rdom vril X ollows Poisso distritio th P X ),,,,! 7 Rslts rltd to Poisso distritio: p ; Vric = M I Poisso distritio M = Vric K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

10 wwwklvisolicom 8 I th cotios rdom X ollows Norml distritio, th its pd is giv, 9 To covrt Norml vrit X to stdrd Norml vrit z w s, z X CHAPTR 9 SAMPLING DISTRIBUTION Nottios: ) N Popltio siz ) Smpl siz c) d) X M o th smpl M o th popltio ) s - Stdrd dvitio SD) o smpl ) - Stdrd dvitio SD) o popltio Coidc limits or = X Z s c I N is ot giv) s N X c I N is giv) N = Z Coidc itrvls or proportio = p Z Not : For 95% coidc itrvl Z c = 96 pq c I N is ot giv) pq N p c I N is giv) N = Z For 99% coidc itrvl Z c = 58 4 Tstig o Hpothsis Forml: X Tst sttistic Z p P Tst sttistic Z pq K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

11 wwwklvisolicom 5 For 5% lvl o sigiicc : Accptc rgio Z 96 Criticl rgio Z 96 6 For % lvl o sigiicc : Accptc rgio Z 58 Criticl rgio Z 58 CHAPTR APPLID STATISTICS Corrltio coicit orml: N XY ) r X, Y) N X X I X, Y XY NY Y r itgrs or o-itgrs) ) r, ) Whr X X d Y Y I X, Y r itgrs) d X Y X d Y c) r X, Y) N d N dd dd d Nd d I X, Y r itgrs or o - itgrs) Whr d X A d d Y B A, B r ritrr vls o X d Y) Rgrssio Forml: Not: Corrltio coicit shold li tw - d ) ) Rgrssio li o X o Y is X X ) Y Y) ) Rgrssio li o Y o X is Y Y) X X ) Whr X X Y d Y Whr X Y N XY Y d Y I X, Y N XY X X r itgrs or o - itgrs) X Y Whr d I X, Y r itgrs) K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :

12 wwwklvisolicom Not: Rgrssio lis will itrsct t X,Y ) Qtrl vrg Ssol Id = Grd vrg 4 Id Nmrs: ) Lspr s pric Id mr P L p q ) Psch s pric id mr P P p p q q c) Fishr s pric id mr P F p q p q pq pq F L P OR) P P P p q d) Cost o Livig Id mrs: pq i) Aggrgt pditr mthod CLI) p q PV ii) Fmil Bdgt mthod CLI) V 5 Sttisticl Qlit Cotrol SQC) Forml: Rg chrt R Chrt): CL = R R UCL = LCL = p Whr P d V p q p R D 4 R D X X Chrt : CL = X UCL = X A R Whr LCL = X A R CL Ctrl Li ; UCL - Uppr Cotrol Li ; LCL - Lowr Cotrol Li) K MANIMARAN MSc,Bd ; PG Asst - GOLDN GATS MAT HR SC SCHOOL, SALM 8 PH :