CLASS 9 FORMULAE MATHS - … · Rational numbers – All the numbers which can be written in the...

CLASS9FORMULAE MATHS

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NUMBERSYSTEMSNaturalnumbersare-1,2,3,………………denotedbyN.Wholenumbersare–0,1,2,3,…………..denotedbyW.Integers-……………….-3,-2,-1,0,1,2,3,…………denotedbyZ.Rationalnumbers–Allthenumberswhichcanbewrittenintheformp/q,q≠0arecalledrationalnumberswherepandqareintegers.Irrationalnumbers–Anumbersiscalledirrational,ifitcannotbewrittenintheformp/qwherepandqareintegersandq≠0Thedecimalexpansionofarationalnumberiseitherterminatingornonterminatingrecurring.Thuswesaythatanumberwhosedecimalexpansioniseitherterminatingrnonterminatingrecurringisarationalnumber.Thedecimalexpansionofairrationalnumberisnonterminatingnonrecurring.Alltherationalnumbersandirrationalnumberstakentogether.Makeacollectionofrealnumber.Arealnoiseitherrationalorirrational.Ifrisrationalandsisirrationalthenr+s,r-sarealwaysirrationalnumbersbutr/smayberationalorirrational.EveryirrationalnumbercanberepresentedonanumberlineusingPythagorastheorem.Rationalizationmeanstoremovesquarerootfromthedenominator.Integersareacombinationofwholenumbersandnegativenaturalnumbers

POLYNOMIALSConstants:Asymbolhavingafixednumericalvalueiscalledaconstant.Variables:Asymbolwhichmaybeassigneddifferentnumericalvaluesisknownavailable.AlgebraicExpressions-Acombinationofconstantsandvariablesconnectedbyoperators+,-,xand/isknownasalgebraicexpression.Polynomials:Analgebraicexpressioninwhichthevariablesinvolvedhaveonlynonnegativeintegralpowersiscalledapolynomial.Coefficient-AnumericalorconstantquantityplacedbeforeandmultiplyingthevariableinanalgebraicexpressionDegreeofapolynomial–Degreeofapolynomialinonevariableisthehighestpowerofthevariableintheexpression.ClassificationofPolynomialsbasedonDegree: Degree Polynomial (a) 1 Linear (b) 2 Quadratic (c) 3 Cubic (d) 4 Biquadratic


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ClassificationofPolynomialsbasedonNumberofTerms: No.ofterms Polynomial(i) 1 Monomial(ii) 2 Binomial–(iii) 3 Trinomial–

Constantpolynomial:Apolynomialcontainingonetermonly,consistingaconstantermiscalledaconstantpolynomialthedegreeofnon-zeroconstantpolynomialiszero.

Zeropolynomial:Apolynomialconsistingofoneterm,namelyzeroonlyiscalledazeropolynomial.Thedegreeofzeropolynomialisnotdefined.Zeroesofapolynomial:Letp(x).beapolynomial.Ifp(α)=0,thenwesaythatiszeroofthepolynomialofp(x)Findingthezeroesofpolynomialp(x)meanssolvingtheequationp(x)=0.RemainderTheorem:Letf(x)beapolynomialofdegree≥1andletabeanyrealnumber.Whenf(x)isdivided(x-a)bythentheremainderisf(a)

Factortheorem:Letf(x)beapolynomialofdegreen>1andletabeanyrealnumber.(i)Iff(a)=0then(x-a)isfactoroff(x)(ii)If(x-a)isafactoroff(x)thenf(a)=0Factor:Apolynomialp(x)iscalledfactorofq(x)dividesq(x)exactly.Factorization:Toexpressagivenpolynomialastheproductofpolynomialseachofdegreelessthanthatofthegivenpolynomialsuchthatnosuchafactorhasafactoroflowerdegree,iscalledfactorization.

ModesofFactorizationFactorizationbytakingoutthecommonfactorFactorizationofquadratictrinomialsbymiddletermsplittingmethod.Identity:Identityisaequation(trigonometric,algebraic)whichistrueforeveryvalueofvariable.Somealgebraicidentitiesusefulinfactorization:(i)(x+y)2=x2+2xy+y2(ii)(x-y)2=x2-2xy+y2(iii)x2–y2=(x-y)(x+y)(iv)(x+a)(x+b)=x2+(a+b)x+ab(v)(x+y+z)2=x2+y2+z2+2xy+2yz+2zx(vi)(x+y)3=x3+y3+3xy(x+y)(vii)(x–y)3=x3-y3-3xy(x-y)(viii)x3+y3+z3–3xyz=(x+y+z)(x+y+z)(x2+y2+z2–xy–yz-zx) x3+y3+z3=3xyz ifx+y+z=0


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COORDINATEGEOMETRY• Thehorizontalreallineiscalledx-axisandtheverticallineiscalledy-axis• ThepointofintersectionofthesetwoaxesiscalledtheoriginanditisdenotedbyO.• TheX-Co-ordinateisalsocalledabscissaandthey-co-ordinateisalsocalledordinate.• Theaxesdividetheplaneintofourregionscalledquadrants.• Thecoordinatesoftheoriginare(0,0).• Thepositivenumbersappeartotherightoftheoriginonthex-axisandabovetheoriginonthey-axis.• Thenegativenumbersappeartotheleftoftheoriginonthex-axisandbelowtheoriginonthey-axis.• Inanorderedpairthefirstnumberdenotedthex-axisandthesecondnumberdenotedthey-axis.• Thecoordinatesofapointonthex-axisareoftheform(x,0).• Thecoordinatesofapointonthey-axisareoftheform(x,y).• Infirstquadrantbotharepositive;secondquadrantyispositiveandxisnegative;thethirdquadrantyis

negativeandxisnegative;thefourthquadrantxispositiveyisnegative.

LINEAREQUATIONSIN2VARIABLELinearEquationInTwoVariables Anequationoftheformax+by+c=0wherea,b,carerealnumbersandx,yarevariables,iscalledalinearequationintwovariables.Here‘a’iscalledcoefficientofx,‘b’iscalledcoefficientsofyandciscalledconstantterm.SolutionsOfLinearEquationThevalueofthevariablewhichwhensubstitutedforthevariableintheequationsatisfiestheequationi.e.L.H.S.andR.H.S.oftheequationbecomesequal,iscalledthesolutionorrootoftheequation.RulesForSolvingAnEquation(i)Samequantitycanbeaddedtobothsidesofanequationwithoutchangingtheequality(ii)Samequantitycanbesubtractedfrombothsidesofanequationwithoutchangingtheequality(iii)Bothsidesofanequationmaybemultipliedbyasamenon-zeronumberwithoutchangingtheequality(iv)Bothsidesofanequationmaybedividedbyasamenon-zeronumberwithoutchangingtheequality.• ALinearEquationin2Variablescanhaveinfinitelymanysolutions.GraphOfLinearEquationsThegraphofanequationinxandyisthesetofallpointswhosecoordinatessatisfytheequation:Inordertodrawthegraphofalinearequationax+by+c=0mayfollowthefollowingalgorithm.Step1:Obtainthelinearequationax+by+c=0Step2:Expressyintermsofxi.e.y=-((ax+b)/c))Step3:Putanytwoorthreevaluesforxandcalculatethecorrespondingvaluesofyfromthe

expressionvaluesofyfromtheexpressionobtainedinstep2.Letwegetpointsas(α1,β1),(α2,β2),(α3,β3)

Step4:Plotpoints(α1,β1),(α2,β2),(α3,β3)ongraphpaper.Step5:Jointhepointsmarkedinstep4toobtain.Thelineobtainedisthegraphoftheequation

ax+by+c=0


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Note:(i)Thereasonthatadegreeonepolynomialequationax+by+c=0iscalledalinearequationisthatitsgeometricalrepresentationisastraightline

(ii)Thegraphoftheequationoftheformy=kxisalinewhichalwayspassesthroughtheoriginEquationsOfLinesParallelToTheX—AxisAndY—AxisEverypointonthex–axisisoftheform(x,0).Theequationofthex–axisisgivenbyy=0.Similarly,observethattheequationofthey–axisisgivenbyx=0.Considertheequationx-2=0.Ifthisistreatedasanequationinonevariablexonly,thenithastheuniquesolutionx=2,whichisapointonthenumberline.Anequationintwovariables,x-2=0isrepresentedbythelineABinthegraphinbelowgivenfig.

Note:(i)Thegraphofx=aisastraightlineparalleltothey–axis(ii)Thegraphofy=aisstraightlineparalleltothex–axis

INTRODUCTIONTOEUCLID’SGEOMETRYTheGreeksdevelopedgeometryisasystematicmannerEuclid(300B.C.)agreekmathematician,fatherofgeometryintroducedthemethodofprovingmathematicalresultsbyusingdeductivelogicalreasoningandthepreviouslyprovedresult.TheGeometryofplanefigureisknowas“EuclideanGeometry”Axioms:ThebasicfactswhicharetakenforgrantedwithoutproofarecalledaxiomssomeEuclid’saxiomsare(i)Thingswhichareequaltothesamethingareequaltooneanother.i.e. a=b, b=c ⇒ a=c(ii)Ifequalsareaddedtoequals,thewholesareequali.e.a=b⇒a+c=b+c(iii)Ifequalsaresubtractedfromequals,theremaindersareequali.e. a=b⇒a-c=b–c(iv)Thingswhichcoincidewithoneanotherareequaltooneanother.(v)Thewholeisgreaterthanthepart.Postulates:Axiomsarethegeneralstatements,postulatesaretheaxiomsrelatingtoaparticular

field.


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Educlid’sfivepostulatesare.(i)Astraightlinemaybedrawnanyonepointtoanyotherpoint(ii)Aterminatedlinecanbeproducedindefinitely(iii)Acirclecanbedrawnwithanycentreandanyradius(iv)Allrightanglesareequaltooneanother(v)Ifastraightlinefallingontwostraightlinesmakestheinterioranglesonthesamesideofittakentogetherlessthantworightangles,thenthetwostraightlines,ifproducedindefinitelymeetonthatsideon

whichtheanglesarelessthantworightangles.Statements:Asentencewhichiseithertrueorfalsebutnotboth,iscalledastatement.Theorems:Astatementthatrequiresaproofiscalledatheorem.Corollary–Resultdeducedfromatheoremiscalleditscorollary

LINESANDANGLES(1)Point–Weoftenrepresentapointbyafinedotmadewithafinesharpenedpencilonapieceofpaper

(2)Line–Alineiscompletelyknownifwearegivenanytwodistinctpoints.LineABisrepresentedbyas Alineorastraightlineextendsindefinitelyinboththedirections.

(3)Linesegment–Apart(orportion)ofalinewithtwoendpointsiscalledalinesegment.(4)Ray–Apartoflinewithoneendpointsiscalledaray (5)Collinearpoints–Ifthreeormorepointslieonthesameline,theyarecalledcollinearpointsotherwisetheyarecallednon-collinearpoints.TypesofAngles–(1)Acuteangle–Anacuteanglemeasurebetween00and900(2)Rightangle–Arightangleisexactlyequalto900(3)Obtuseangle–Ananglegreaterthan900butlessthan1800(4)Straightangle–Astraightangleisequalto1800(5)Reflexangle–Ananglewhichisgreaterthan1800butlessthan3600iscalledareflexangle(6)Complementaryangles–Twoangleswhosesumis900arecalledcomplementaryangles(7)Supplementary–Twoangleswhosesumis1800arecalledsupplementaryangles(8)Adjacentangles–Twoanglesareadjacent,iftheyhaveacommonvertex,acommonarmandtheirnoncommonarmsareondifferentsidesofcommonarm.(9)Linearpair–Twoanglesformalinearpair,iftheirnon-commonarmsformaline(10)Verticallyoppositeangles–Verticallyoppositeanglesareformedwhentwolinesintersecteachotheratapoint.


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LinesAndATransversal

Whenatransversalintersects2lines,followinganglesareformed(referthefigureabove)-(a)Correspondingangles: (i)∠1and∠5 (ii)∠2and∠6 (iii)∠4and∠8 (iv)∠3and∠7(b)Alternateinteriorangles: (i)∠4and∠6 (ii)∠3and∠5(c)Alternateexteriorangles: (i)∠1and∠7 (ii)∠2and∠8(d)Interioranglesonthesamesideofthetransversal: (i)∠4and∠5 (ii)∠3and∠6Ifatransversalintersectstwoparallellines,then(i)eachpairofcorrespondinganglesisequal.(ii)eachpairofalternateinterioranglesisequal.(iii)eachpairofinteriorangleonthesamesideofthetransversalissupplementary.Ifatransversalinteractstwolinessuchthat,either(i)anyonepairofcorrespondinganglesisequal,or(ii)anyonepairofofalternateinterioranglesisequalor(iii)anyonepairofinterioranglesonthesamesideofthetransversalissupplementarythenthelinesareparallel.Lineswhichareparalleltoagivenlineareparalleltoeachother.Thesumofthethreeanglesofatriangleis180°.Ifasideofatriangleisproduced,theexterioranglesoformedisequaltothesumoftwointerioroppositeangles.• Triangle–Aclosedfigureformedbythreeintersectinglinesiscalledatriangle.Atrianglehasthreesides,

threeanglesandthreevertices.• Congruentfigures–Congruentmeansequalinallrespectsorfigureswhoseshapesandsizesareboththe

sameforexample,twocirclesofthesameradiiarecongruent.Alsotwosquaresofthesamesidesarecongruent.

• CongruentTriangles–twotrianglesarecongruentifandonlyifoneofthemcanbemadetosuperposeontheother,soastocoveritexactly.

• IftwotrianglesABCandPQRarecongruentunderthecorrespondenceA↔P,B↔QandC↔Rthensymbolically,itisexpressedas∆ABC≅∆PQR


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• Incongruenttrianglescorrespondingpartsareequalandwewrite‘CPCT’forcorrespondingpartsof

congruenttriangles.• SAScongruencyrule–Twotrianglesarecongruentiftwosidesandtheincludedangleofonetriangleare

equaltothetwosidesandtheincludedangleoftheothertriangle.Forexample∆ABCand∆PQRasshowninthefiguresatisfySAScongruentcriterion.

• ASACongruenceRule–Twotrianglesarecongruentiftwoanglesandtheincludedsideofonetriangleare

equaltotwoanglesandtheincludedsideofothertriangle.Forexamples∆ABCand∆DEFshownbelowsatisfyASAcongruencecritertion.

• AASCongruenceRule–Twotrianglearecongruentifanytwopairsofanglesandonepairofcorresponding

sidesareequalforexample∆ABCand∆DEFshownbelowsatisfyAAScongruencecriterion.

• AAScriterionforcongruenceoftrianglesisaparticularcaseofASAcriterion.• IsoscelesTriangle–Atriangleinwhichtwosidesareequaliscalledanisoscelestriangle.ForexampleABC

shownbelowisanisoscelestrianglewithAB=AC.


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• Angleoppositetoequalsidesofatriangleareequal.• Sidesoppositetoequalanglesofatriangleareequal.• Eachangleofanequilateraltriangleis60°.• SSSCongruenceRule–Ifthreesidesofonetriangleareequaltothethreesidesofanothertrianglethenthe

twotrianglesarecongruentforexample∆ABCand∆DEFasshowninthefiguresatisfySSScongruencecriterion.

• RHSCongruenceRule–Ifintworighttrianglethehypotenuseandonesideofonetriangleareequaltothe

hypotenuseandonesideoftheothertrianglethenthetwotrianglearecongruent.Forexample∆ABCand∆PQRshownbelowsatisfyRHScongruencecriterion.

RHSstandsforrightangle–Hypotenuseside.• Apointequidistantfromtwogivenpointsliesontheperpendicularbisectorofthelinesegmentjoiningthe

twopointsanditsconverse.• Apointequidistantfromtwointersectinglinesliesonthebisectorsoftheanglesformedbythetwolines.• Inatriangle,angleoppositetothelongersideislarger(greater)• Inatriangle,sideoppositetothelarger(greater)angleislonger.• Sumofanytwosidesofatriangleisgreaterthanthethirdside.Aquadrilateralisaclosedfigureobtainedbyjoiningfourpoint(withnothreepointscollinear)inanorder.Everyquadrilateralhas:(i)Fourvertices,(ii)Foursides,(iii)Fouranglesand(iv)Twodiagonals.

SUMOFTHEANGLESOFAQUADRILATERALStatement:Thesumoftheanglesofaquadrilateralis360°.

TypesOfQuadrilaterals1.Trapezium:Itisquadrilateralinwhichonepairofoppositesidesareparallel.2.Parallelogram:Itisaquadrilateralinwhichboththepairsofoppositesidesareparallel.3.Rectangle:Itisaquadrilateralwhoseeachangleis90°.ABCDisarectangle.(i)∠A+∠B=90°+90°=180°⇔AD||BC(ii)∠B+∠C=900+900=180°⇔AB||DC


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RectangleABCDisaparallelogramalso.4.Rhombus:Itisaquadrilateralwhoseallthesidesareequal.5.Square:Itisaquadrilateralwhoseallthesidesareequalandeachangleis90°.6.Kite:Itisaquadrilateralinwhichtwopairsofadjacentsidesareequal.Note:• Square,rectangleandrhombusareallparallelograms.• Kiteandtrapeziumarenotparallelograms.• Asquareisarectangle.• Asquareisarhombus.• Aparallelogramisatrapezium.

PARALLELOGRAM:Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.Itisdenotedby

PROPERTIESOFPARALLELOGRAM:1.Adiagonalofaparallelogramdividesitintotwocongruenttriangles.2.Theoppositesidesofaparallelogramareequal.Theorem:Ifeachpairofoppositesidesofaquadrilateralisequal,thenitisaparallelogram.3.Theoppositeanglesofaparallelogramareequal.Theorem:Ifinaquadrilateral,eachpairofoppositeanglesisequal,thenitisaparallelogram.4.Thediagonalsofaparallelogrambisecteachother.Theorem:Ifthediagonalsofaquadrilateralbisecteachother,thenitisaparallelogram.

MIDPOINTTHEOREM(BASICPROPORTIONALITYTHEOREM)Statement1:Thelinesegmentjoiningthemid-pointsofanytwosidesofatriangleisparalleltothethirdside


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Statement2:Thelinedrawnthorughthemid–pointofonesideofatriangle,parallelsidebisectsthethirdside.• Areaofafigureisaunitassociatedwiththepartoftheplaneenclosedbythatfigure.• Twocongruentfigureshaveequalareabutthefigureshavingequalareasneednotbecongruent.• IfaplanarregionformedbyafigureXismadeupofoftwonon–overlappingplanarregionsformedbyfigures

AandB,thenar(X)=ar(A)+ar(B),wherear(P)denotestheareaoffigureP.• Twofiguresaresaidtobeonthesamebaseandbetweenthesameparallels,iftheyhaveacommonbaseand

theverticesoppositetothecommonbaseofeachfigurelieonalineparalleltothebase.• Parallelogramsonthesamebaseandbetweenthesameparallelsareequalinarea.• Areaofaparallelogramistheproductofitsbaseandthecorrespondingaltitude.• Parallelogramsonthesamebaseandhavingequalareasliebetweenthesameparallels.• Ifaparallelogramandatriangleareonthesamebaseandbetweenthesameparallels,thenareaofthe

triangleishalftheareaoftheparallelogram.• Trianglesonthesamebaseandbetweenhesameparallelsaeequalinarea.• Areaoftriangleishalftheproductofitsbaseandthecorrespondingaltitude.• Trianglesonthesamebaseandhavingequalareasliebetweenthesameparallels.• Amedianofatriangledividesitintotwotrianglesofequalareas.Thecollectionofallthepointsinaplane,whichareatafixeddistancefromafixedpointintheplane,iscalledacircle.Acircleisaclosedcurveallofwhosepointslieinthesameplaneandareatthesamedistancefromthecentre.Thefixedpointiscalledthecentreofthecircleandthefixeddistanceiscalledtheraidusofthecircle.Note:Thelinesegmentjoiningthecentreandanypointonthecircleisalsocalledaradiusofthecircle.

INTERIORANDEXTERIOROFACIRCLEAcircledividestheplaneonwhichitleisintothreeparts.Theyare(i)insidethecircle,whichislaoscalledtheinteriorofthecircle.(ii)thecircleand(iii)outsidethecircle,whichisalsocalledtheexteriorofthecircle

CHORDAchordofacircleisalinejoiningtwopointsofthecircumference.Achordpassesthroughthecentreiscalleddiameter.Note:Diameteristhelongestchordandalldiametershavethesamelength,whichisequaltotwotimestheradius.Apieceofacirclebetweentwopointsiscalledanarc.Inacircleequalchordshaveequalares.WhenPandQareendsofadiameter,thenbotharcsareequalandeachiscalledasemicircle.


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Thelengthofthecompletecircleiscalleditscircumference.Theregionbetweenachordandeitherofitsarcsiscalledsegmentofthecircle.

Theminorarecorrespondstotheminorsectorandthemajorarecorrespondstothemajorsector.Whentwoarcsareequal,thatis,eachisasemicircle,thenbothsegmentsandbothsectorsbecomethesameandeachisknownasasemicircularregion.

Theorem1:Equalchordsofacirclesubtendequalanglesatthecentre.Theorem1:Iftheanglessubtendedbythechordsofacircleatthecentreareequal,thenthechordsareequal,

PERPENDICULARFROMTHECENTRETOACHORDTheorem3:Theperpendicularfromthecentreofacircletoachordbisectsthechord.Theorem4:Thelinedrawnthroughthecentreofacircletobisectachordisperpendiculartothechord.Theorem5:Thereisoneandonlyonecirclepassingthroughthreegivennon–chollinearpoints.Remark:IfABCisatriangle,thenbyabovegivenTheoremthereisauniquecirclepassingthroughthethreeverticesA,BandCofthetriangle.ThiscircleiscalledthecircumcircleoftheAABC.Itscentreandradiusarecalledrespectivelythecircumcentreandthecircumradiusofthetriangle.

EQUALCHORDSANDTHEIRDISTANCESFROMTHECENTRETheorem6:Equalchordsofacircle(orofcongruentcircles)areequidistantfromthecentre(orcentres).Theorem7:Chordsequidistantfromthecentreofacircleareequalinlength.

ANGLESUBTENDEDBYANARCOFACIRCLEResult:Congruentarcs(orequalarcs)ofacirclesubtendequalanglesattthecentre.Theorem8:Theanglesubtendedbyanarcatthecentreisdoubletheanglesubtendedbyitatanypointontheremainingpartofthecircle.Note:Theoremgivestherelationshipbetweentheanglessubtendedbyanareatthecentreandatapointonthecircle.

ANGLEFORMEDINTHESEGMENTTheorem9:Anglesinthesamesegmentofacircleareequal.Note:Angleinasemicircleisarightangle.


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Theorem10:Ifalinesegmentjoiningtwopointssubtendsequalanglesattwootherpointslyingonthesamesideofthelinecontainingthelinesegment,thefourpointslieonacircle(i.e.theyareconcyclic).

CYCLICQUADRILATERAL:AquadrilateralABCDiscalledcyclicifallthefourverticesofitlieonacircle.

Theorem11:Thesumofeitherpairofoppositeanglesofacyclicquadrilateralis180°.Theorem12:Ifthesumofapairofoppositeanglesofaquadrilateralis180°,thequadrilateraliscyclic.

HERON’SFORMULA*Trianglewithbase‘b’andaltitude‘h’is

𝐴𝑟𝑒𝑎 =12×(𝑏×ℎ)

*Trianglewithsidesa,bandc(i)Semiperimeteroftriangle𝑠 = !!!!!

!

(ii)Area= 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)squareunits.


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*Equilateraltrianglewithside‘a’

𝐴𝑟𝑒𝑎 = !!𝑎!squareunits

SURFACEAREASANDVOLUMES

SURFACEAREAOFACUBOID:Letusconsideracuboidoflength=1unitsBreadth=bunitsandheight=hunitsThenwehave:(i)Totalsurfaceareaofthecuboid=2(1*b+b*h+h*1)sq.units(ii)Lateralsurfaceareaofthecuboid=[2(1+b)*h]sq.units(iii)Areaoffourwallsofaroom=[2(1+b)*h]sq.units.=(Perimeterofthebase*height)sq.units(iv)Surfaceareaoffourwallsandceilingofaroom=lateralsurfaceareaoftheroom+surfaceareaofceiling=2(1+b)*h+1*b(v)Diagonalofthecuboid=√12+b2+h2SURFACEAREAOFACUBE:Consideracubeofedgeaunit.(i)TheTotalsurfaceareaofthecube=6a2sq.units(ii)Lateralsurfaceareaofthecube=4a2sq.units.(iii)Thediagonalofthecube=√3aunits.


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SURFACEAREAOFTHERIGHTCIRCULARCYLINDERCylinder:Solidslikecircularpillars,circularpipes,circularpencils,roadrollersandgascylindersetc.aresaidtobeincylindricalshapes.Curvedsurfaceareaofthecylinder=Areaoftherectangularsheet=length*breadth=Perimeterofthebaseofthecylinder*height=2πr*hTherefore,curvedsurfaceareaofacylinder=2πrhTotalsurfaceareaofthecylinder=2πrh+2πr2Sototalareaofthecylinder=2πr(r+h)Ifacylinderisahollowcylinderwhoseinnerradiusisr1andouterradiusr2andheighththenTotalsurfaceareaofthecylinder

= 2𝜋𝑟!ℎ + 2𝜋𝑟!ℎ + 2𝜋 𝑟!! − 𝑟!! = 2𝜋 𝑟! + 𝑟! ℎ + 2𝜋 𝑟! + 𝑟! 𝑟! − 𝑟!

= 2𝜋 𝑟! + 𝑟! ℎ + 𝑟! − 𝑟!

SURFACEAREAOFARIGHTCIRCULARCONE:curvedsurfaceareaofacone=l/2*1*2π=πlwhererisbaseradiusandlitsslantheightTotalsurfaceareaoftherightcircularcone.=curvedsurfacearea+Areaofthebase=πrl+πr2=πr(l+r)Note:l2=r2+h2ByapplyingPythagorasTheorem,herehistheheightofthecone.

Thus𝑙 = 𝑟!+ ℎ! 𝑎𝑛𝑑 𝑟 = 𝑙

!− ℎ!

ℎ = 𝑙!+ 𝑟!


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SURFACEAREAOAASPHERESphere:Asphereisathreedimensionalfigure(solidfigure)whichismadeupofallpointsinthespacewhichlieataconstantdistancecalledtheradius,fromafixedpointcalledthecentreofthesphere.Note:Asphereislikethesurfaceofaball.Thewordsolidsphereisusedforthesolidwhosesurfaceisasphere.Surfaceareaofasphere:Thesurfaceareaofasphereofradiusr=4×areaofacircleofradiusr=4*πr2=4πr2Surfaceareaofahemisphere=2πr2

Totalsurfaceareaofahemisphere=2πr2+πr2=3πr2Totalsurfaceareaofahollowhemispherewithinnerandouterradiusr1andr2respectively

= 2πr!! + 2πr!! + 𝜋 𝑟!! − 𝑟!! = 2𝜋 𝑟!! + 𝑟!! + 𝜋(𝑟!! − 𝑟!!)

Volumeofacuboid:Volumeofacuboid=Areaofthebase*heightV=l*b*hSo,volumeofacuboid=basearea*height=length*breadth*heightVolumeofacube:Volumeofacube=edge*edge*edge–a3wherea=edgeofthecubeVOLUMEOFACYLINDERVolumeofacylinder=πr2hvolumeofthehollowcylinder𝜋𝑟!!ℎ − 𝜋𝑟!!ℎ= 𝜋 𝑟!! − 𝑟!! ℎVOLUMEOFARIGHTCIRCULARCONEVolumeofacone=1/3πr2h,whereristhebaseradiusandhistheheightofthecone.VOLUMEOFASPHERE Volumeofaspherethesphere=4/3πr3,whereristheradiusofthesphere.Volumeofahemisphere=2/3πr3


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APPLICATION:Volumeofthematerialofahollowspherewithinnerandouterradiir1andr2respectively

=43

𝜋𝑟!! −43𝜋𝑟!! =

43𝜋(𝑟!! − 𝑟!!)

Volumeofthematerialofahemispherewithinnerandouterradiusr1andr2respectively=!!𝜋(𝑟!! − 𝑟!!)

STATISTICSPrimarydata:Datawhichcollectedforthefirsttimebythestatisticalinvestigatororwiththehelpofhisworkersiscalledprimarydata.Secondarydata:Thesearethedataalreadycollectedbyapersonorasocietyandthesemaybeinpublishedorunpublishedform.ThesedatashouldbecarefullyusedPRESENTATIONOFDATARawdata:Whenthedataiscompiledinthesameformandorderinwhichitiscollected,itisknownasRawdataThedifferenceofthehighestandthelowestvaluesinthedataiscalledtherangeofthedataFrequency:Itisanumber,whichtellsthathowmanytimesdoesaparticulardataappearinagivensetofdataFrequencyDistribution:Atabulararrangementofdatashowingtheircorrespondingfrequenciesiscalledafrequencydistribution.ThetableshowingdatawiththeircorrespondingfrequenciesiscalledafrequencytableClassMark:Themid-pointofaclassiscalledtheclassmarkoftheclassThus,

𝐶𝑙𝑎𝑠𝑠 −𝑚𝑎𝑟𝑘 = !""#$ !"#$$ !"#"$!!"#$% !"#$$ !"#"$!

MEASURESOFCENTRALTENDENCY(AVERAGES)Mean:Themean(oraverage)ofanumberofobservationsisthesumofthevaluesofalltheobservationsdividedbythetotalnumberofobservations.Itisdenotedbythesymbolxas‘xbar’

Ingeneral,fornobservalas𝑥 = !!!!!!!

Forangroupedfrequencydistribution,

𝑥 = !!!!!!!!

!!!!!!

wherefiisthefrequencycorrespondingtotheobservationxiNote: 𝑓!𝑥!!

!!! = 𝑓!𝑥! + 𝑓!𝑥! + 𝑓!𝑥! +⋯+ 𝑓!𝑥!MedianWhenthedataisarrangedinascending(ordescending)orderthemedianofungroupeddataiscalculatedasfollows:


©BYJU’S

Whenthenumberofobservation(n)isodd,themedianisthevalueofthe((n+1)/2)thobservation.MODE:Themodeisthatvalueoftheobservationwhichoccursmostfrequently,i.e.,anobservationwiththemaximumfrequencyiscalledthemode.

PROBABILITYPROBABILITY–ANEXPERIMENTAL(EMPIRICAL)APPROACH.Atrialisanactionwhichresultsinoneorseveraloutcomes.Aneventforanexperimentisthecollectionofsomeoutcomesoftheexperiment.Letnbethetotalnumberoftrials.TheempiricalprobabilityP(E)ofaneventEhappeningisgivenby

𝑃 𝐸 = !"#$%& !" 𝑡𝑟𝑖𝑎𝑙𝑠 !" !!!"! !!! !"!#$ !!""#$#%!!! !"!#$ !"#$%& !" !"#$%&

NumberoftrialsinwhichtheeventhappenedNote:Theempiricalprobabilitydependsonthenumberoftrials.Theprobabilityofaneventliesbetween0and1,i.e.,Itcanbeanyfractionfrom0to1.Thesumoftheprobabilitiesofallthepossibleoutcomesofatrialis1.Probabilityoftheoccurrenceofanevent+Probabilityofthenon-occurrenceofthatevent=1

CLASS 9 FORMULAE MATHS - … · Rational numbers – All the numbers which can be written in the...

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