OMPLEX NUMBERS AND DEMOIVRES THEOREM. Geel om o omp umes + y whee s Rel pt d y s Imgy pt.. Sum o oot o uty s zeo. Poduct o oot o uty ( ). ue oots o uty e, ω, ω 5. + ω + ω, ω, 6. Ag z t pcp vlue o θ s π θ π 7. Ag o + y s y θ t o evey > 0, y > 0 8. y Ag o y s θt o evey > 0, y > 0 9. y Ag o + y s θπt o evey > 0, y > 0 0. Ag o y s θπ+ t o evey > 0, y > 0.. 5. 6. 7. ω, ω. Ag. z z z Agz z Agz Ag Ag Ag + zz z Agz Ag +,, +,( + ),( ) + + +, + z + z z + z ; z z z z whee ( + ) + ( ) os + π + π ( + ) + ( ) os I ee comp umes Z, Z, Z e colle e z z z z z z 8. Ae o tg omed y Z, IZ, Z + Z s Z 9. Ae o tg omed y Z, ωz, Z + ωz s Z 0. I Z ZZ + Z e og, Z, Z oms equltel tg. I Z, Z, Z oms equltel tg d Z 0 s ccum cete e Z + Z + Z Z, 0 AIEEE Memtcs Quck Revew + y. I Z, Z, Z oms equltel tg d Z 0 s ccum cete e Z + Z + Z Z Z + Z Z + Z Z + z + z z z ;. Dstce etwee two vetces Z, Z s.z z. z z 0 s cc w dus p d cete z 0 5. zz + zα+ zα+β 0 Repesets cc W dus α β whee α s oel comp d β s cost t 6. I z z k (k ) epesets cc w z z kz ± z eds o dmete k± I k e locus o z epesets le o pepedcul secto. 7. z z + z z k,k > z z e locus o z epesets Ellpse d k< zz t s ss, e t epesets hypeol 8. A(z ),B(z ),(z ), d θ s g etwee z z AB AB, A e e θ z z A 9. e θ osθ + Sθ osθ, e π, π π e,log 0. (osθ + Sθ) osθ + Sθ. osθ+sθsθ, sα. sβs (α+ β), sβ s( α+β ) sβ. I osθ+sθ e osθ Sθ + os S α α + osα Sα. I Σosα ΣSα Σosα ΣSα Σos α ΣS α, Σos α ΣS α / Σosα os(α + β + γ), ΣSα S(α + β + γ) Σos(α β γ), ΣS(α β γ),. + + c c ( + + c) ( + ω + cω ) ( + ω + cω) Qudtc Epessos. Stdd om o Qudtc equto s + +c Sum o oots, poduct o oots c, dscmte c I α, β e oots e Qudtc equto s (α + β) + αβ. I e oots o + + c e, c e + + c. I e oots o + + c e to m : e m (m + ) c. I oe oot o + + c s sque o e oe e c + c + c 5. I > 0 e e st vlue o + s 6. I,,..., e postve e e st vlue o www.sksheducto.com
www.sksheducto.com + +... + + +... + ( ) s 7. I + + c K e ge o K + c + c s,k 8. I e two oots e egtve, e,, c wll hve sme sg 9. I e two oots e postve, e e sg o, c wll hve deet sg o '' 0. () s polyoml e e equto whose oots e ecpocl o e oots o () s cesed y 'K' s ( K), multpled y K s (/K). Fo,, h R e oots o ( ) ( ) h e el d uequl. Fo,, c R e oots o ( ) ( ) + ( ) ( c) + ( c) ( ) e el d uequl. Thee oots o cucl equto e A.P, ey e tke s d,, + d. Fou oots A.P, d, d, +d, +d 5. I ee oots e G.P,, e tke s oots 6. I ou oots e G.P,,, e tke s oots 7. Fo + + c + d () Σα β (αβ + βγ + γα) (α + β + γ) αβ γ s s s () α +β +γ s s () (v) α +β +γ s s s + ss + s α +β +γ s ss + s (v) I + + c... to elmte secod tem oots e dmshed y Boml Theoem Ad Ptl Fctos. Nume o tems e epso ( + ) s +. Nume o tems e epso + ( ) + +... +. s I T+ + +, T p. Fo + q depedet tem s p p+ q + 5. I ove, e tem cotg s s 6. ( + ) s dvs y d ( + ) s dvs y. 7. oecet o (+) (+)...(+) 8. oecet o (+) (+)...(+) s ( + ) 9. oecet o ove s ( + )( )( + ) 0. I () ( + y) e sum o coecets s equl to (). Sum o coecets o eve tems s equl to (). Sum o coecets o odd tems s equl to. I () + + e A.P ( ) +. Fo (+y), s eve e oly oe mdd tem t s + tem. 5. Fo ( + y), s odd ee e two mdd tems t s + + tem d tem. 6. I e epso ( + y) s eve getest coecet s 7. I e epso ( + y) s odd getest coecets e, + s odd 8. Fo epso o (+ ) Geel otto 0 o,, 9. Sum o oml coecets o + + +... + 0. Sum o eve oml coecets o + + +.... Sum o odd oml coecets + + +... 5 MATRIES. A sque mt whch evey emet s equl to '0', ecept ose o pcpl dgol o mt s cld s dgol mt. A sque mt s sd to e scl mt ll e emets e pcpl dgol e equl d Oe emets e zeo's. A dgol mt A whch ll e emets e pcpl dgol e d e est '0' s cld ut mt. A sque mt A s sd to e Idem-potet mt A A, 5. A sque mt A s sd to e Ivolu-ty mt A I 6. A sque mt A s sd to e Symm-etc mt A A T A sque mt A s sd to e Skew symmetc mt A-A T 7. A sque mt A s sd to e Nlpotet mt I e ests postve tege such t A '' s e de o Nlpotet mt 8. I 'A' s gve mt, evey sque mt- c e epessed s sum o symme-tc d skew symmetc mt whee T Symmetc pt A+ A T A+ A usymmetc pt 9. A sque mt 'A' s cld oo-gol mt AA T I o A T A - 0. A sque mt 'A' s sd to e sgul mt det A. A sque mt 'A' s sd to e o sgul mt det A 0. I 'A' s sque mt e det Adet A T. I AB I BA e A d B e cld veses o ech oe. (A - ) - A, (AB) - B - A - 5. I A d A T e vet e (A T ) - (A - ) T 6. I A s o sgul o ode, A AdjA s vet, e A det A d 7. I A A d-c 0 c d d c c 8. (A - ) - A, (AB) - B - A -, (A T ) - (A - ) T (AB) - - B - p s + p+ q www.sksheducto.com
www.sksheducto.com A -. I A s o- sgul mt, e ) A(AdjA) A I ) Adj A A A - c) (Adj A) - Adj (A - ) d) Adj AT (Adj A) T e) Det (A - ) ( Det A) - ) Adj A A - g) ladj (Adj A ) l A ( - ) h) Fo y scl 'k' Adj (ka) k - Adj A 9. I A d B e two o-sgul mtces o e sme type e () Adj (AB) (Adj B) (Adj A) () Adj (AB) Adj A Adj B Adj B Adj A 0. To deteme k d soluto st co-vet mt to Echolo om.e. A Echolo om oa 0 y z 0 0 0 k l No o o zeo ows Rk o mt I e system o equtos AXB s cosstet e coe mt A d ugmeted mt K e o sme k Let AX B e system o equtos o '' ukows d ks o coe mt d k o ugmeted mt I, e AX B s cosstt,.e. t hs o soluto I e AXB s cosstt, t hs uque soluto I < e AXB s cosstt d t hs tely my ume o solutos Rdom Vs- Dstutos & Sttstcs. Fo polty dstuto w ge (,, ----) d P( ) e e poltes e me µ Σ P(- ) Vce σ Σ p -µ Stdd devto vce. I e postve tege p e el ume such t 0 P dom v X w ge (0,,,----) s sd to ollows oml dstuto. Fo Boml dstuto o (q+p) ) polty o occuece p ) polty o o occuece q ) p + q v) polty o '' successes P ( ) q p v) Me µ p v) Vce pq v) Stdd devto pq. I ume o tls e lge d po-lty o success s vey smll e posso dstuto s used d gve s k e P( k) λ λ k. ) I,,,... e vlues o vt, e ts Ametc Me ) Fo dvdul sees I A s ssumed vege e A.M ( A) A+ ) Fo dscete equecy dstuto: d A whee d A + N v) Med F l+ whee l Lowe lmt o Med clss equecy N Σ Wd o Med clss F umultve equecy o clss just pecedg to med clss v) Fst o lowe Qut devto N F Q l +. whee equecy o st qu clss F cumultve equecy o e clss just pecedg to st qut clss v) uppequtdevto m v) Mode Z l+. whee m l lowe lmt o modl clss w mmum equecy equecy pecedg modl clss equecy successve modl clss equecy o modl clss v) Mode Med - Me Q ) Qut devto Q ) coecet o qut devto N F Q l+. Q Q Q + Q ) coecet o Rge Rge Mmum + Mmum VETORS. A system o vectos,,... e sd to e lely depedet e ests scls,.... Such t + +... +... 0. Ay ee o copl vectos e le-ly depedet A system o vectos,,... e sd to e lely depedet ee + +... + tst oe o 0,,,. Ad detemt. Ay two colle vectos, y ee copl vectos e lely depedet. Ay set o vectos cotg ull vectos s lely depedet. I ABDEF s egul hego w cete 'G' e AB + A + AD + AE + AF AD 6AG. 5. Vecto equto o sphee w cete t c d dus s c o. c+ c 6., e eds o dmete e equto o sphee (. ) 7. I, e ut vectos e ut vecto log secto o AOB s ( + + ) o + ± + 8. Vecto log tel gul secto s ± λ + 9. I 'I' s cete o AB e, www.sksheducto.com
www.sksheducto.com 0. I 'S' s ccum cete o AB e, SA + SB + S SO. I 'S' s ccum cete, 'O' s oocete o AB e, OA + OB + O OS. I (,, ) & es e otted ough ) - s (, cosα + s α, cosα + s(90 α) ) y - s ( cos( 90 + α) + s ( 90 + α),,( cos α + s α)) ) z - s ( cosα + s α, ( cos( 90 + α) + s ( 90 + α), )) I 'O' s ccumcete o AB e Σ OAs A OA + OB + O (osde equltel ).. cosθ whee 0 θ 80 ). > 0 0< θ < 90 θ s cute ). < 0 90 < θ < 80 θ s otuse ). θ 90 two vectos e to ech oe.. I ght gd AB, AB s e hypoteuse d AB P e AB. B + B. A + A. AB P 5. AB s equltel tg o sde '' e AB. B AB. B + B. A + A. AB 6. B IA + A IB + AB I. +. j + k. ; + j + k 7. Vecto equto. o le pssg ough e pot A w P.V. d pll to '' s + t 8. Vecto equto o le pssg ough A (), B () s (-t) +t 9. Vecto equto. o le pssg ough & to c, + t( c) 0. Vecto equto. o p pssg ough pt A () d- pll to o-colle vectos & c s + s+ tc. s,t R d lso gve s c c c. Vecto equto. o p pssg ough ee o-colle Pots. A (), B (), c () s AB A AP.e + s + t c ( s t) + s+ sc,, c. Vecto equto. o p pssg ough pts A () B() d pll to () c s AP AB. Vecto equto o p, t dstce p (p >0) om og d to s. p. Pepedcul dstce om og to p pssg ough,,c c c + c + 5. P pssg ough d pll to,c s [ -, - c] d [ c] [c] 6. Vecto equto o p pssg ough A,B, w posto vectos,,c s [ -, -, c-] 0 d.[ c + c + ] c 7. Let, 0 e two vectos. The ) The compoet o o s. ) The pojecto o o s (. ). 8. ) The compoet o o s (. ) ) e pojecto o o s ) e pojecto o o vecto pepe-dcul to' ' e p geeted y (. ), s 9. I, e two ozeo vectos e. cos (, ) 0. I, e ot pll e s pepedcul to o o e vectos,.. I, e ot pll e., om ght hded system.. I, e ot pll e s (. ) d hece. I s y vecto e. I, e two vectos e -. 5. - s cld tcommuttve lw. 6. I, e two ozeo vectos, e s (, ) 7. I AB s tg such t AB, A e e vecto e o AB s d scl e s [ ] 8. I,,c e e posto vectos o e vetces o tg, e e vecto e o e tg + c+ c 9. I ABD s pllogm AB, B d e e vecto e o ABD s l l 0. The g o e pojecto o o vecto pepedcul to e p geeted y, s. The pepedcul dstce om pot P to e le jog e pots A,B s AP AB AB. Toque: The toque o vecto momet o momet vecto M o oce F out pot P s deed s M F whee s e vecto om e pot P to y pot A o e le o cto L o F..,,c e copl e [c]0. Volume o pllopped [c] w,, c s cotemus edges. 5. The volume o e teedo ABD s ± AB A AD 6 6. I,,c e ee cotemous edges o teedo e e volume o e teedo ± [ ] 6 c 7. The ou pots A,B,,D e copl AB A AD 8. The shotest dstce etwee e skew les +s d c+ td s [ c, d] d 9. I,j,k e ut vectos e [ j k] 50. I,,c e vectos e [+, +c, c+] [c] www.sksheducto.com
www.sksheducto.com 5. [, c, c ] (c) 5. Σ ( ) 5. +.. c. d. 5. ( )(. c d) c. d. 55. I A,B,,D e ou pots, d AB D + B AD + A BD ( AB ) c c 56.,, c [ c] [ c] [ c] e cld ecpocl system o vectos 57. I,,c e ee vectos e [ c] [ c ] [c ] -[ c] -[c ] -[ c ] 58.Thee vectos e copl det I + j + k, + j + k, + j + ck whee c e copl e + + ) c ) + + c c Pepto Tps - Memtcs Memozg ld mk poms (emem-eg stdd omu, cocepts so t you c pply em dectly) d eg stog metl clcultos e essetl (Neve use e clculto dug you ete AIEEE pepto. Ty to do st d sec-od vel o clcultos metlly You e gog to ppe o AIEEE s ye, you must e vey codet, do't p-c,t s ot dcult d tough. You eed to some specl tps d tcks to solve e AIEEE questos to get e top k. Do't ty to tke up ew topcs s ey co-sume tme, you wll lso lose you code-ce o e topcs t you hve ledy pe-ped. Do't ty to ttempt 00% o e ppe ul-ess you e 00% codet: It s ot ece-ssy to ttempt e ete questo ppe, Do't ty you e ot sue d codet s ee s egtve mkg. I you e codet out 60% o e questos, t wll e eough to get good k. Neve swe questos ldly. Be wse, peplg s vey mpott. Thee e mly ee dculty vels, s-mp, tough d vege. Fst ty to sh ll e smp questos to oost you o-dece. Do't oget to solve questo ppes o pevous yes AIEEE eoe e emt-o. As you pepe o e od emt-o, you should lso pepe d solve e lst ye questo ppes o AIEEE. You lso eed to set e hous tme o ech d evey pevous ye ppe, t wll help you to judge yousel, d s wll t you kow you wek d stog es. You wll gdully ecome codet. You eed to cove you ete syllus ut do't ty to touch y ew topc e e-mto s close y. Most o e questos AIEEE e ot d-cult. They e just deet & ey equ-e deet ppoch d deet m-dset. Ech questo hs emet o su-pse t & studet who s dept tck-lg 'supse questos' s most lkely to sl ough successully. It s vey mpott to udestd wht you hve to ttempt d wht you hve to omt. Thee s lmt to whch you c mpove you speed d stke te eyod whch wht ecomes vey mpott s you sec-to o questo. So success depeds upo how judcously oe s to sect e questos. To optmze you peomce you should quckly sc o esy questos d come ck to e dcult oes lte. Rememe t cut-o most o e e-ms moves etwee 60 to 70%. So you o-cus o esy d vege questo.e. 85% o e questos, you c esly scoe 70% mks wout eve ttemptg dcult qu-estos. Ty to esue t e tl hous o e ppe e ocus should e c-ly o esy d vege questos, Ate hous you c decde whee you wt to move to dcult questos o evse e oes ttempted to esue hgh stke te. Topc-wse tps Tgoomety: I tgoomety, studets usully d t d-cult to memoze e vst ume o omul-e. Udestd how to deve omu d e pply em to solvg poms.the mo-e you pctce, e moe ged you - ese omu wll e, elg you to e-cll em y stuto. Dect questos om tgoomety e usully ss ume, ut e use o tgoometc cocepts oo-dte Geomety & lculus s vey pouse. oodte Geomety: Ths secto s usully cosdeed ese tgoomety. Thee e my commo coc-epts d omu (such s equtos o tg-et d oml to cuve) coc sectos (cc, pol, ellpse, hypeol). Py tt-eto to Locus d elted topcs, s e udestdg o ese mkes coodte Geome-ty esy. lculus: lculus cludes cocept-sed poms whch eque lytcl sklls. Fuctos e e ckoe o s secto. Be oough w popetes o ll types o uctos, such s tgoometc, lgec, vese tgoom-etc, logmc, epoetl, d sgum. Appomtg sketches d gphcl tep-ettos wll help you solve poms ste. Pctcl pplcto o devtves s vey vst e, ut you udestd e sc cocepts volved, t s vey esy to scoe. Alge: Do't use omu to solve poms top-cs whch e logc-oeted, such s pemut-tos d comtos, polty, locto o oots o qudtc, geometcl pplct-os o comp umes, vectos, d D-geomety. AIEEE 009 Memtcs Secto Alyss o BSE syllus O ll e ee sectos e AIEEE 009 ppe, e Memtcs secto ws e toughest. Questos wee eqully dvded etwee e syll o lss XI d XII. My cddtes stuggd w e lculus d oodte Geomety potos. lss XI Syllus Topc No. o Questos Tgoomety Alge (XI) 6 oodte Geomety 5 Sttstcs -D (XI) lss XII Syllus Topc No. o Questos lculus 8 Alge (XII) Polty -D (XII) Vectos www.sksheducto.com