Sets,Relationsandfunctions

PHYSICSSYLLABUS

SYLLABUS:Class-XI

UnitI:PhysicalWorldandMeasurement Physics-scopeandexcitement;natureofphysicallaws;Physics,technologyandsociety. Needformeasurement:Unitsofmeasurement;systemsofunits;SIunits,fundamentaland derivedunits.Length,massandtimemeasurements;accuracyandprecisionofmeasuringinstrum ents;errorsinmeasurement;significantfigures. Dimensionsofphysicalquantities,dimensionalanalysisanditsapplications. UnitII:Kinematics

Frameofreference,Motioninastraightline:Position-timegraph,speedandvelocity.

Uniformandnon-uniformmotion,averagespeedandinstantaneousvelocity.Uniformly acceleratedmotion,velocitytimeandposition-timegraphs. Relationsforuniformlyacceleratedmotion(graphicaltreatment). Scalarandvectorquantities;Positionanddisplacementvectors,equalityof vectors, multiplicationofvectorsbyarealnumber;additionandsubtractionofvectors.Relativevelocity.Uni tvector;Resolutionofavectorinaplane-rectangularcomponents.Scalarand Motioninaplane.Casesofuniformvelocityanduniformacceleration-projectilemotion. Uniformcircularmotion. UnitIII:LawsofMotion Intuitiveconceptofforce.Inertia,Newton'sfirstlawofmotion;momentumandNewton's secondlawofmotion;impulse;Newton'sthirdlawofmotion. Lawofconservationoflinearmomentumanditsapplications.

Equilibriumofconcurrentforces.Staticandkineticfriction,lawsoffriction,lubrication.

Dynamicsofuniformcircularmotion:Centripetalforce,examplesofcircularmotion(vehicle onalevelcircularroad,vehicleonbankedroad). UnitIV:Work,EnergyandPower Workdonebyaconstantforceandavariableforce;kineticenergy,work-energytheorem, power. Notionofpotentialenergy,potentialenergyofaspring,conservativeforces:conservation ofmechanicalenergy(kineticandpotentialenergies);non- conservativeforces:motioninaverticalcircle. UnitV:MotionofSystemofParticlesandRigidBody Centreofmassofatwo-particlesystem,momentumconservationandcentreofmass Centreofmassofarigidbody;centreofmassofauniformrod. Momentofaforce,torque,angularmomentum,lawsofconservationofangularmomentum anditsapplications. Equilibriumof rigidbodies,rigidbodyrotationandequationsof rotationalmotion, comparisonoflinearandrotationalmotions. Momentofinertia,radiusofgyration.Valuesofmomentsofinertia,forsimplegeometrical objects(noderivation).Statementofparallelandperpendicularaxestheoremsandtheirapplicatio ns. UnitVI:Gravitation Keplar'slawsofplanetarymotion.Theuniversallawofgravitation. Accelerationduetogravityanditsvariationwithaltitudeanddepth. Gravitationalpotentialenergyand gravitationalpotential.Escapevelocity.Orbitalvelocityof asatellite.Geo-stationarysatellites. UnitVII:PropertiesofBulkMatter Elasticbehaviour,Stress-strainrelationship,Hooke'slaw,Young'smodulus,bulkmodulus, shearmodulusofrigidity,Poisson'sratio;elasticenergy. Pressureduetoafluidcolumn;Pascal'slawanditsapplications.Effectofgravityonfluid Viscosity, Stokes' law, terminal velocity, streamline and turbulent flow, critical Surfaceenergyandsurfacevelocity.Bernoulli'stheoremtension,anditsapplications.angleofcontact,excessofpressureacrossacurved surface,applicationofsurfacetensionideastodrops,bubblesandcapillaryrise. Heat,temperature,thermalexpansion;thermalexpansionof solids,liquidsandgases, anomalousexpansion ofwater;specificheatcapacity;Cp, Cv-calorimetry;changeofstate- latentheatcapacity. Heattransfer-conduction,convectionandradiation,thermalconductivity,Qualitativeideas ofBlackbodyradiation,Wein'sdisplacementLaw,Stefan'slaw,Greenhouseeffect. UnitVIII:Thermodynamics Thermalequilibriumanddefinitionoftemperature(zerothlawofthermodynamics).Heat, workandinternalenergy.Firstlawofthermodynamics.Isothermalandadiabaticprocesses. Secondlawofthermodynamics:reversibleandirreversibleprocesses.Heatengineand refrigerator. UnitIX:BehaviourofPerfectGasesandKineticTheoryofGases Equationofstateofaperfectgas,workdoneincompressingagas. Kinetictheory of gases- assumptions,concept of pressure.Kineticinterpretationof temperature;rmsspeedofgasmolecules;degreesoffreedom,lawofequi- partitionofenergy(statementonly)andapplicationtospecificheatcapacitiesofgases;conceptof meanfreepath,Avogadro'snumber. UnitX:OscillationsandWaves Periodicmotion-timeperiod,frequency,displacementasafunctionoftime.Periodic functions. Simpleharmonicmotion(S.H.M)anditsequation;phase;oscillationsofaspring-restoring forceandforceconstant;energyinS.H.M.Kineticandpotentialenergies;simplependulumderivati onofexpressionforitstimeperiod. Free,forcedanddampedoscillations(qualitativeideasonly),resonance. Wavemotion.Transverseandlongitudinalwaves,speedofwavemotion.Displacement relationforaprogressivewave.Principleofsuperpositionofwaves,reflectionofwaves,standingwa vesinstringsandorganpipes,fundamentalmodeandharmonics,Beats,Doppler

CLASSXII-PHYSICS UnitI:Electrostatics ElectricCharges;Conservationofcharge,Coulomb’slaw-forcebetweentwopointcharges,forcesbetween multiplecharges;superpositionprincipleandcontinuouschargedistribution. Electricfield,electricfieldduetoapointcharge,electricfieldlines,electricdipole,electricfielddue toa dipole, torqueonadipoleinuniformelectricfleld. Electricflux,statementofGauss’stheoremanditsapplicationstofindfieldduetoinfinitelylongstraightwire, uniformlychargedinfiniteplanesheetanduniformlychargedthinsphericalshell(fieldinsideandoutside). Electricpotential,potentialdifference,electricpotentialduetoapointcharge,adipoleandsystemofcharges; equipotentialsurfaces,electricalpotentialenergyofasystemoftwopointchargesandofelectricdipoleinanelectrostati cfield. Conductorsand insulators,freechargesand bound chargesinsidea conductor.Dielectricsandelectric polarisation,capacitorsandcapacitance,combinationofcapacitorsinseriesandinparallel,capacitanceofaparallelplat ecapacitorwithandwithoutdielectricmediumbetweentheplates,energystoredinacapacitor.VandeGraaffgenerator . UnitII:CurrentElectricity Electriccurrent,flowofelectricchargesinametallicconductor,driftvelocity,mobilityandtheirrelationwith electriccurrent;Ohm’slaw,electricalresistance,V-Icharacteristics(linearandnon- linear),electricalenergyandpower,electricalresistivityandconductivity.temperaturedependenceofresistance. Internalresistanceofacell,potentialdifferenceandemfofacell,combinationofcellsinseriesandinparallel. Kirchhoff’slawsandsimpleapplications.Wheatstonebridge,metrebridge. Potentiometer-principleanditsapplicationstomeasurepotentialdifferenceandforcomparingemfoftwo cells;measurementofinternalresistanceofacell. UnitIII:MagneticEffectsofCurrentandMagnetism Conceptofmagneticfield,Oersted’sexperiment. Biot-Savartlawanditsapplicationtocurrentcarryingcircularloop. Ampere’slawanditsapplicationstoinfinitelylongstraightwire.Straightandtoroidalsolenoids,Forceona movingchargeinuniformmagneticandelectricfields. Forceonacurrent-carryingconductorinauniformmagneticfield.Force betweentwoparallelcurrent-carrying conductors- definitionofampere.Torqueexperiencedbyacurrentloopinuniformmagneticfield;movingcoilgalvanometer- Currentloopasamagneticdipoleanditsmagneticdipolemoment.Magneticdipolemomentofarevolving electron.Magneticfieldintensityduetoamagneticdipole(barmagnet)alongitsaxisandperpendiculartoitsaxis.Torqu eonamagneticdipole(barmagnet)inauniformmagneticfield;barmagnetasanequivalentsolenoid,magneticfieldline s;Earth’smagneticfieldandmagneticelements.Para-,dia-andferro-magnetic substances,withexamples.Electromagnetsandfactorsaffectingtheirstrengths.Permanentmagnets. UnitIV:ElectromagneticInductionandAlternatingCurrents Electromagneticinduction;Faraday’slaws,inducedemfandcurrent;Lenz’sLaw,Selfandmutualinduction. Alternatingcurrents,peakand rmsvalue of alternatingcurrent/voltage;reactanceand impedance;LC oscillations(qualitativetreatmentonly),LCRseriescircuit,resonance;powerinACcircuits,wattlesscurrent. ACgeneratorandtransformer. UnitV:Electromagneticwaves Electromagneticwavesandtheircharacteristics(qualitativeideasonly).Transversenatureofelectromagnetic waves. Electromagneticspectrum(radiowaves,microwaves,infrared,visible,ultraviolet,X-rays,gammarays)including elementaryfactsabouttheiruses. UnitVI:Optics Reflectionof light,sphericalmirrors,mirrorformula.Refractionof light,total internalreflectionand its applications,opticalfibres,refractionatsphericalsurfaces,lenses,thinlensformula,lensmaker’sformula.Magnificatio n,powerofalens,combinationofthinlensesincontactcombinationofalensandamirror.Refractionanddispersionoflig htthroughaprism.

Scatteringoflight-bluecolourofskyandreddishapprearanceofthesunatsunriseandsunset.

Opticalinstruments:Microscopesandastronomicaltelescopes(reflectingandrefracting)andtheirmagnifying powers. Waveoptics:WavefrontandHuygen'sprinciple,reflectionandrefractionofplanewaveataplanesurface usingwavefronts.ProofoflawsofreflectionandrefractionusingHuygen'sprinciple.InterferenceYoung'sdoubleslitex perimentandexpressionforfringewidth,coherentsourcesandsustainedinterferenceoflight.Diffractionduetoasingl eslit,widthofcentralmaximum,microscopesandastronomicaltelescopes.Polarisation,planepolarisedlightBrewster' slaw,usesofplanepolarisedlightandPolaroids. UnitVII:DualNatureofMatterandRadiation Dualnatureofradiation.Photoelectriceffect,Einstein’sphotoelectricequation-particlenatureoflight. Matterwaves-wavenatureofparticles,deBroglierelation.Davisson-Germerexperiment(experimentaldetails shouldbeomitted;onlyconclusionshouldbeexplained). UnitVIII:Atoms&Nuclei Alpha-particlescatteringexperiment;Rutherford’smodelofatom;Bohrmodel,energylevels,hydrogen Radioactivityalpha,betaandgammaparticles/raysandtheirproperties;radioactivedecaylaw.Mass-energy relation,massdefect;bindingenergypernucleonanditsvariationwithmassnumber;nuclearfission,nuclearfusion.

UnitIX:ElectronicDevices Energybandsinsolids(Qualitativeideasonly)conductors,insulatorandsemiconductors;semiconductordiode – I-Vcharacteristicsin forwardandreverse bias,diode asa rectifier;I-VcharacteristicsofLED,photodiode, solarcell,andZenerdiode;Zenerdiodeas avoltageregulator.Junctiontransistor,transistoraction,characteristicsofatransistor,transistorasanamplifier(comm onemitterconfiguration)andoscillator.Logicgates(OR,AND,NOT,NANDandNOR).Transistorasaswitch. UnitX:CommunicationSystems Elementsofacommunicationsystem(blockdiagramonly);bandwidthofsignals(speech,TVanddigitaldata); bandwidthoftransmissionmedium.Propagationofelectromagneticwavesintheatmosphere,skyandspacewavepro pagation.Needformodulation.Productionanddetectionofanamplitude-modulatedwave. Chemistry SYLLABUS:Class-XI&XII Unit-1AtomicStructure Contents CONCEPT IntroductiontoStructureofAtom Dalton’satomictheory Thomsonmodel Rutherfordmodel Atomicmodels Bohrmodel DualbehaviorofMatter Conceptoforbitals Heisenberg’s QuantumMechanicalModel uncertaintyprinciple Quantumnumbers Shapeofs,panddorbitals ShapesofAtomicOrbitals Nodeandnodalsurface Shieldingeffect Aufbauprinciple RulesforFillingElectronsinOrbitals Pauli’sexclusionprinciple Hund’sruleElectronicconfigurationofatoms StabilityofCompletelyFilleda ndhalf-filledOrbitals Unit-2ChemicalBonding TypesofChemical Ionicbond Covalentbond Bonds Polarcovalentbond Hybridization ValenceBondTheory VSEPRtheory Resonance Magnetic characteristics MolecularOrbitalTheory Bondorder Intermolecularhydrogenbonding HydrogenBond Intramolecularhydrogenbonding Unit-3StatesofMatter:GasesandLiquids Typesofintermolecularforces IntermolecularForces Natureofintermolecularforces Boyle’slaw Charleslaw LawsGoverningGaseousState Gay-lussac Avogadrolaw Idealgasequation IdealBehaviour Dalton’slawofpartialpressure Kinetictheoryofgasespressure Compressibilityfactor DeviationfromIdealBehaviour Boyle’sTemperature LiquefactionofGases Criticaltemperature,criticalpressureandcriticalvolume Vapourpressure LiquidState Viscosity Surfacetension Unit-4Thermodynamics Conceptsof: system,surrounding typesofsystem ThermodynamicTerms stateofasystem statefunctionandpathfunction extensiveandintensiveproperties reversibleandirreversibleprocess Work ThermodynamicQuantities Heat InternalEnergy Enthalpy FirstLawofThermodynamics Heatcapacity Measurementof U Measurementof H Enthalpychangeinachemicalreaction Endothermicand Exothermicreactions Standardenthalpyofreactions Thermochemistry Enthalpychangesduringphasetransformations Standardenthalpyofformation Thermochemicalequations Hess'sLawofConstantHeatSummation Enthalpiesfordifferenttypesofreactions Entropy SecondlawofThermodynamics Spontaneity Gibb'senergychangeforspontaneousandnon-spontaneousprocesses Criteriaforequilibrium ThirdLawofThermodynamics Unit-5ChemicalEquilibrium IntroductiontoEquilibrium Dynamicnatureofequilibrium Solid-liquidequilibrium Liquid-vapourequilibrium EquilibriuminPhysicalProcesses Solidvapourequilibrium Equilibriuminvolvingdissolutionofsolidandgasesinliquids Dynamicnatureofchemicalequilibrium EquilibriuminChemicalProcesses lawofchemicalequilibrium Equilibriumconstant HomogenousEquilibria TypesofChemicalEquilibria HeterogeneousEquilibria Predictingtheextentofareaction ApplicationsofEquilibriumConstan Predictingthedirectionofthereaction t CalculatingEquilibriumConcentrations FactorsAffectingEquilibria LeChatelier’sprinciple Strongandweakelectrolytes IonicEquilibriuminSolution Acids,basesandsalts IonicproductofWater pHscale IonizationofAcidsandBases Ionizationconstantofweakacidsandbases Factorsaffectingacidstrength Commonioneffect BufferSolutions Bufferactionandrelevantexamples SolubilityEquilibriaofSparin Solubilityproduct glySolubleSalts Commonioneffectofsolubilityofionicsalts Unit-6SolidState IntroductiontoSolidState CharacteristicsofSolidState Chemistry ClassificationofSolidsontheBasis CrystallineandamorphousSolids ofOrderintheArrangement PrimitiveandCentredUnitCells CrystalLatticesandUnitCells NumberofatomsinperunitCellinacubicunitcell PackinginSolids Voids ClosePackinginsolids PackingEfficiency CalculationofDensityofunitcell TypesofPointDefects StoichiometricandNon-StoichiometricDefects ImperfectionsinSolids MetalExcessDefect MetalDeficiencyDefect ImpurityDefects Conductors,semiconductorsandinsulators ElectricalProperties Bandtheoryofsolids n&ptypesemiconductors Paramagnetic Diamagnetic MagneticProperties Ferromagnetic Antiferromagnetic Ferrimagnetic Unit-7Solutions Solute Introductiontosolutions Solvent Solution GaseousSolutions TypesofSolutions LiquidSolutions Solidsolutions Variousquantitiesusedtoexpressconcentrationofasolution ExpressingtheConcentrationo MoleFraction fSolutionsofSolidsinLiquids Molarity Molality Solubilityofsolidinliquid Solubility Solubilityofgasinliquid Henry’sLaw Solutionoftwovolatileliquids VapourPressureofLiquidSolutions Solutioncontainingnon-volatilesolute Raoult’sLaw Idealsolutions ClassificationofLiquid- NonIdealsolutions LiquidSolutionsonthebasisof Positivedeviation Raoult’sLaw Negativedeviation Relativeloweringofvapourpressure Elevationofboilingpoint ColligativeProperties Depressionoffreezingpoint Osmoticpressure Determinationofmolecularmassesusingcolligativeproperties AbnormalMolecularMass van’tHoffFactor–Numericalsbasedontheabove Unit-8RedoxreactionsandElectrochemistry OxidationandReductionReactions RedoxReactionsinTermsofElectro Mechanismofredoxreactionsbyelectrontransferprocess n Evolutionoftheelectrochemicalseries. OxidationNumberTransferReactions Calculationofoxidationnumber TypesofRedoxReactions Oxidationnumbermethod BalancingofRedoxReactions Halfreaction Method TypesofElectrochemicalCells Electrolyticcells Galvaniccells Electrode Electrolysis Signconventionsatanodeandcathode Lawsofelectrolysis Metallicandelectrolyticconductance Typesofelectrolytes Conductance ConductanceinElectrolyticSolutio Resistance ns Molarconductivity Variationofconductivitywithconcentration Kohlrausch’slaw EMFofacell Standardelectrodepotential GalvanicCells Nernstequationanditsapplicationtochemicalcells RelationbetweenGibbsenergychangeandemfofacell Corrosion Conceptandmechanismofcorrosioninrelationtoemf Unit-9s-Block&p-BlockElementsandmetallurgy Electronicconfiguration PhysicalProperties Chemicalproperties Positionofhydrogenintheperiodictable S-BlockElementsGroup1Elements Diagonalrelationship &Group2Elements Biologicalimportance Waterandhydrogenperoxide SomeAlkalimetalcompounds SomeAlkalineearthmetalcompounds Electronicconfiguration P-BlockElementsGroup13,14,15, OccurrenceInertpaireffectReactivity 16,17and18Elements SomecompoundsofGroup13to18elements Unit-10dandf-BlockElementsandCoordinationCompounds Generalpropertiesof3delements. Electronicconfiguration VariablevalencyconceptColor d-Blockelements Magneticproperties Catalyticproperties Compounds Electronicconfiguration F-BlockElements Oxidationstates Lanthanidecontraction Generalcomposition Coordinationnumber Coordination Compounds Typesofligands Wernertheory IUPACNomenclatureofCoordinati on IUPACrules ValenceBondTheoryCompounds asAppliedto Valencebondtheory Coordination Compounds Crystalfieldtheory Analyticalapplications ImportanceofCoordinat Industrialapplications ionCompounds Biologicalapplications Unit-11SurfaceChemistry Physisorption AdsorptiononaSurface Chemisorption Factorsaffectingtheadsorptionofgasesonsolids Homogenousandheterogeneouscatalysis Catalysis Shapeselectivecatalysis Enzymecatalysis Distinctionbetweentruesolution,colloidandsuspension Classificationofcolloids Colloids Propertiesofcolloids:Mechanical,Optical,Electrical Hardy-Schulzerule applicationofcolloids Unit-12ChemicalKinetics Averagerateofreaction RateofChemicalReaction Instantaneousrateofreaction Concentrationofreactants,temperature,catalyst,natureofreactants,pressure(gases),presenceoflight,surf aceareaofthereactants FactorsAffectingRateofaReaction RateLawandSpecificRateConstant OrderAndMolecularity Zeroorderreactions IntegratedRateEquationsandH Firstorderreactions alflife PseudoFirstorderreaction Activation TemperatureDependenceofRate Energy ofReaction ArrheniusEquation CollisionTheory Unit-13Hydrocarbons,HaloalkanesandHaloarenes Typesofhybridizationincarboncompounds TypesofHybridizationofCarbon Shapesoforganicmolecules 2Dand3Dstructuralrepresentationoforganiccompounds basedonfunctionalgroups ClassificationofOrganicCompound basedonstructure s Priorityorderoffunctionalgroups IUPACNomenclatureofOrga Prefixesandsuffixesforfunctionalgroups nicCompounds DerivationofstructuralformulafromagivenIUPACnameandvice-versa Structuralisomerism Stereochemistryandstereoisomerism Projectionformulae Interconversionofprojectionformulas StereochemistryandIsomerism Conformationsandtheirrelativestabilities(ethaneandbutane) Geometricalisomerism(cisandtrans) Opticalisomerism Absoluteandrelativenomenclatureofopticalisomers carbocation HomolyticandHeterolyticFission carbanion ofaCovalentBond freeradical Electrophilicandnucleophilicreagents BasicsofOrganicReaction Typesoforganicreactions inductiveeffect ElectronicDisplacementsi electromericeffect naCovalentBond resonance hyperconjugation Stabilityofaromaticcompounds Aromaticity Huckel’srule Methodsofpreparation(Reduction,Wurtzreaction,Kolbe'selectrolysis) Physicalproperties Alkanes(Upto5CarbonAtoms) Chemicalreactions(Halogenation,Isomerisation,Oxidation,Aromatization,Combustion,Pyrolysis)

Methodsofpreparation(Partialreduction,dehydrohalogenati on,dehydration,dehalogenation) Alkenes(Upto5CarbonAtoms) Physicalproperties Chemicalreactions(AdditionofH2,X2,Markovnikov’sandanti-Markovnikov’srule) AdditionofHX,andH2O,ozonolysis,oxidationandpolymerization Methodsofpreparation(Hydrolysisofcalciumcarbide,dehydrohalo-genation) Alkynes(Upto5CarbonAtoms) Physicalproperties Chemicalreactions(AdditionofH2,X2,HX,andH2Oandpolymerization) Nomenclature,resonanceandstabilityofbenzene,orientationeffectofsubstituentsinbenzene,pre Arenes parationphysicalandchemicalpropertiesofbenzene Structure Classification Structureof1 ,2 and3 haloalkanesandhaloarenes Haloalkanesandhaloarenes ⁰ ⁰ ⁰ Nomenclature Isomerism Preparationandproperties Unit-14OxygencontainingOrganiccompounds Structureofalcohols,phenolsandethers Structure Classification Preparationofalcohols(hydrationofalkenes,hydroboration- PreparationofAlcoholsandPhenols oxidation,reductionofcarbonylcompounds,fromGrignard'sreagent) PreparationofPhenols(fromchlorobenzene,benzeneandcumene) PhysicalPropertiesofAlcohols,PhenolsandEthers PropertiesofAlcohols,Phenolsan ChemicalPropertiesofAlcohols(withmetals,esterification,esterification,withHX,dehydration) dEthers ChemicalPropertiesof Phenols(halogenation,nitrationandsulphonation, KolbesReimer- Tiemann,deoxygenationandoxidation) Preparationfromalcohols PreparationofEthers&chemic Williamsonsethersynthesis alProperties EthercleavagebyHX halogenation,nitrationandFriedelcraftsreaction StructureofAldehydes,Ketonesa ndCarboxylicAcids Fromalcohols Fromalkenes Fromalkynes PreparationofAldehydesa Fromaromatichydrocarbons ndKetones Gattermann-Koch Fromacidchlorides Fromnitriles PhysicalPropertiesofaldehydesandketones Physical,ChemicalPropertiesa ChemicalPropertiesofAldehydesandKetones(nucleophilicadditionreactions,nucleophilicaddition- ndUsesofAldehydesandKeton eliminationreactions,reduction,oxidation,Aldolcondensation,Cannizzarroreaction,electrophiclics es ubstitutioninaromaticaldehydes) Structureofcarboxylicacid Preparationofcarboxylicacids(byoxidation,hydrolysis,fromGrignardreagents) Carboxylicacids Physicalpropertiesofcarboxylicacids Chemicalpropertiesofcarboxylicacids Unit-15NitrogencontainingOrganiccompounds Structure PreparationofAmines Byreductionofnitrocompounds,nitrilesandamides Ammonolysisofalkylhalides PhysicalandChemicalPropertiesof PhysicalPropertiesofAmines Amines ChemicalPropertiesofAmines Nomenclature Structure Methodsof DiazoniumSalts Preparation Physicalproperties ChemicalProperties Structureandimportanceofazodyesandexamples Unit-16Bio-MoleculesandPolymers Carbohydrates Aminoacidsandproteins Biomolecules Nucleicacids Vitamins Classification Polymers Methodsofpolymerization PreparationofSomepolymers Unit-17Chemistryineverydaylife antacids,antihistamines,tranquilizers,analgesics,antimicrobials(antibiotics,antisepticsand disinfectants),antifertilitydrugsandchemotherapy ChemicalsinMedicines,Foodan foodadditives,artificialsweeteningagents,preservativesandantioxidants dHygiene(SoapsandDetergent saponification,Soaps&cleansingproperty s) detergentsandbio-degradabledetergents Unit-18EnvironmentalChemistry Environmental Environmentalpollution Pollution Conservationofnaturalresources Typesofwaterpollutants WaterPollution Treatmentofwaterpollution BOD Industrialandagricultural chemicalsthat IndustrialPollution causeenvironmentaldegradation Industrialwastemanagement GreenChemistry Mathematics CLASSXI Sets,RelationsandFunctions

Unit1:Sets

Setsandtheirrepresentations,Emptyset,Finite&Infinitesets.Equivalent andequal sets.Subsets.Subsetsofasetofrealnumbersespecially intervals(with notations). Powerset,Universalset. Venndiagrams.UnionandIntersectionofsets.Differenceofsets.Complem entofaset.Propertiesofcomplementsets.Practicalproblemsonunionandi ntersectionofsets.

Contents LearningOutcomes:

1.1 Setsandtheirrepresent identifysetsaswelldefinedcollections.representsetsi -tations nrosterandsetbuilder form. identifythesymbols and and understandthedifferencebetweenthetwo. Conversionfromsetbuilder formtorosterformandviceversa. EmptySet 1.2 identifyemptysets(nullsets). SingletonSet 1.3 Identifysingletonsetandframeexamples.

1.4 Finiteand identifyfiniteandinfinitesets;andtheirrespectiverepresentations. infiniteSets understandmeaning 1.5 Equivale ofequalandequivalentsets.differentiatebetweenequalandequi nt andEqualSets valentsets.determinewhetherthegivenpairofsetsisequalornot. Subsets 1.6 identifythesubsetsofagivensetanditssymbol( ) understand thateverysethastwotrivialsubsets-nullsetandthesetitself. understandthedifferencebetweenasubsetandpropersubset. PowerSet 1.7 identifypowersetassetofsubsets.

UniversalSet 1.8 identifyuniversalsetanditssymbol( )

1.9 Complementof findthecomplementofasubsetofagivenset,withinagivenuniverse. a Set Intervalsas 1.10 closedinterval,openinterval,righthalfopeninterval,lefthalf SubsetsofR open interval.

1.11 Venn representsetsusingvenndiagrams. diagrams

1.12 Unionand findtheintersection of setsandunion ofsets. Intersectionof Sets showtheintersectionandunionofsetsusing Venndiagrams.identifydisjointsetsanditsrepresentationusing venn 1.13 Differenceof diagram. sets findthedifferenceofsetsandtheirrepresentationusing venndiagram. 1.14 Lawsof applythefollowinglawsofalgebra onsets: Operationson Sets - Lawsofunionofsets(commutativelaw,a ssociativelaw,idempotentlaw,identity law) - lawsofintersectionofsets - distributivelaws 1.15 PropertiesofComplem De entSets applypropertiesofcomplementsets. 1.16 Practical Problemsonun ionandInterse solvepracticalproblemsonunionandintersectionofsets. ctionofSets applyresultsandsolveproblemsonnumberofelementsof setsusingpropertieslike 1) 2)

Orderedpairs,Cartesianproductofsets.NumberofelementsintheCartesianpro Unit2:Relationsan ductoftwofinitesets.Cartesianproductofthe d Functions setofallrealswithitself(uptoRxRxR).Definitionofrelation,pictorialdiagrams, domain,co- domainandrangeofarelation.Functionasaspecialkindofrelationfrom onesetto another.Pictorialrepresentationofafunction,domain,co- domain&rangeof afunction.Realvaluedfunctions,domainandrangeofthefunctions:constant,id entity,linear&quadraticpolynomial,rational,modulus,signumandgreatestint egerfunctionswiththeirgraphs.Sum,difference,productandquotientsof functions,Evenandoddfunction Contents LearningOutcomes

2.1 OrderedPairs Studentswillbeableto: identifyanorderedpair. identifytheequalityoftwoorderedpairs. 2.2 Cartesian identifyacartesianproductoftwo ProductofSets nonemptysets.identifythetwosetsgiventheir cartesianproduct.findtheunionandintersectiononcartesian products.findorderedtriplets(R R R). identifythenumberofelementsinthecartesianproductoftwofinitesets. identifycartesianproductofsetofallrealnumberswithitself. understandrelationoftwosetsasasubsetoftheircartesianproduct. 2.3 DefinitionofRelatio n 2.4 ArrowDiagram pictorialrepresentationofarelationbetween twosets.

2.5 Domain,Co- identifydomain,co-domainandrangeofarelation. domainandR angeofaRela tion 2.6 identifyfunctionasaspecialkindofrelationfromonesettoanother. FunctionasaSpe cialKindofRelati determinewhen arelationis afunction. onfromoneSetto describeandwritefunctionalrelationshipsforgivenproblemsituations. another understandthatf R

2.7 AXA.representfunctionsusinggrap Pictorial representation hs. ofaFunction tounderstandthat everygraphdoes notrepresentafunction. 2.8 Domain,Co- identifydomain,co- domainandR domainandrangeofafunction.findingdomainandrangeofa angeofaFunc tion givenfunction.identifyevenand odd functions. findspecificfunctionvalues findthealgebraoffunctionscovering:(fg) (x)=f(x) g(x)=g(x) f(x) (fg)(x)=f(x).g(x) , g(x) 0

recognisethefollowingrealvaluedfunctionso 2.9 Realvaluedfu constantfunction nctionsandthe o irgraphs i d e n t i t y f u n Unit3:Trigonome tricFunctions c t i o n o

l i n e a r f u n c t i o n o quadraticfun c ction t o i polyn o omial n fun o greatestintegerfunction on Studentisexpectedtodrawthegraphsoftheabovementionedrealvalued functions ration Positiveandnegativeangles.Measuringanglesinradiansandindegreesandconv al ersionfromonemeasuretoanother.Definitionoftrigonometricfunctionswitht ti hehelpofunitcircle.Truth oftheidentitysin²x+cos²x=1,forallx.Signsoftrigonometricfunctions.Domain o andrangeoftrigonometricfunctionsandtheirgraphs.Trigonometricfunctionsa modusperiodicfunctions,theiramplitude,argumentperiod&graph.Expressingsin(x lus +y)andcos (x+y)intermsofsinx,siny,cosx&cosy.Deducingidentitieslikethefollowing : ction

signu mfun tanx tan(x+y)= 1 tanxtany y cotxcoty cot(x+y)= x x x sinx+siny=2six cos ,cosx+cosy=2cos .cos 2 2 2

2 x x x sinx-siny=2cos .sin , cosx cosy=2sin .sin 2 2 2

2 Identitiesrelatedtosin2x,cos2x,tan2x,sin3x,cos3x andtan3x.General solution oftrigonometricequationsofthetypesin= sin,cos =cosandtan=tan. Proofandsimpleapplicationofsineandcosinerulesonly,lawofsine,lawofcosine and theirapplications. Contents LearningOutcomes

Studentswillbeableto: 3.1 Positiveandnegativean identifypositiveandnegativeangles. gles

3.2 Measuring measureanglesinbothdegreesandin anglesinradian radians,andconvertbetweenthesemeasures sandindegreesa ndconversionfr omonemeasure toanother

3.3 Definitionoftrigonomet ricfunctionswiththehel pofunitcircle definetrigonometricfunctions withthehelpofunitcircle. 3.4 Signoftr igonometricfunctions

identifythechangeofsignsoftrigonometricfunctionsindifferentquadra nts.

developandapplythevalueoftrigonometricfunctionsat0, /6, /4, /3, /2radiansandtheirmultiples*. usethereciprocal andco- 3.5 Domainandrangeoftr functionrelationshipstofindthevaluesofthesecant,cosecantandcotang igonometricfunctions ent0,/6, /4, /3, /2radians valueoftrigonometricfunctionsatn ,wheren 3.6 Trigono isapositiveinteger metricfunction sasperiodicfunc identifythedomainandrangeoftrigonometricfunctions. tions,theirampl itude,argument ,periodandgrap h identifytrigonometricfunctionsas periodic functions with 3.7 Trigon sineandcosinefunctionshavingaperiodof2,tangentandcotangentfunctions ometricfuncti havingaperiodof , secantandcosecantfunctionshavingaperiodof2 . onsofsumand differenceof constructthegraphsoftrigonometricfunctionsanddescribetheirbehavi our,includingperiodicity,amplitude,zerosandsymmetry.

expresssin(x±y)andcos(x±y)intermsofsinx,siny,cosxandcosy. tan(x 3.8 Expresssuman sinx cos ddifferenceof 2 2 T- cosx cos Functionsast 2 2 heproductofT -ratios sinx sin 2 2 cosx sin 2 2 usetheaboveidentitiestosimplifytrigonometricequations.

3.10 General solutionoftrigo nometricequat ionsofthe typesin

=sin ,cos= cos andtan = tan .

solveforanunknownsideorangle,usingthelawofsinesorthelawofcosine.

determinetheareaof atriangleorparallelogram,giventhemeasureoftwosidesandtheinclude dangle.

Algebra

UNIT4:Principle Processofthe proof byinduction,motivatingtheapplicationof the ofMathematicalI methodbylookingatnaturalnumbersastheleastinductivesubsetofrealnumbers.The nduction: principle ofmathematical inductionandsimpleapplications.

Contents LearningOutcomes Students willbeableto: 4.1 Processof apply theprincipleofmathematicalinductiontoestablishthe proofbyinductio validityofageneralresultinvolvingnaturalnumbers. n 4.2 Principleofma applicationoftheprincipleofmathematicalinductioninsolving problems thematicalind uctionanditsa pplications UNIT5:ComplexN umbers andQuadraticEqu Need forcomplexnumbers,especially ,tobe motivatedbyinabilityto ations solvesomeofthequadraticequations,standardformofacomplex numberAlgebraicpropertiesofcomplex numbers,Argand plane,themodulusandconjugateofacomplex numberandpolar representationofcomplexnumbersStatementofFundamentalTheoremofAlgebra, solutionofquadraticequationinthecomplexnumbersystem. Squarerootofacomplex number.Cube roots ofunity and theirproperties. Contents

Le ar ni ng O ut co m es St ud en ts wi llb ea bl et o: Need for yiotatobemotivatedbyinabilitytosolvesomeofquadraticeq complex uation. number, 5.2 Standard especiall formofcomplexnumber u r tand theconcept n s ofiotaanditsapplication d 5.3 M e od ul r us s an t dc on a ju n definea complexnumber (z = a+ib) and identify itsreal and ga d imaginaryparts te t concept ofpurelyrealand of co h purelyimaginarycomplex m e numbergetfamiliarwithequalityofcomplexnumbers pl n understandthe addition and subtraction of ex complexnumbers and itsproperties n e u e identifytheconjugate of a complex number and familiarizedwith itsproperties m d be identifythe modulus of a complex number r o andfamiliarizedwith itsproperties 5.4 f understandthemultiplicationofcomplexnumbersandits ultipl I propertiesunderstandthedivisionofcomplex ica- m numbersanditspropertiesidentify tiona nddi a themultiplicativeinverseorreciprocal of visio g acomplexnumber nofc i mpl xn n mber a s r y Q u a n t i t i e s u n d e 5.5 Polarrepr understandthepolarortrigonometricalformof esent- acomplexnumberfindthemodulusofacomplex number tationofco mplexnu findtheargument of acomplex number mber ArgandPlane 5.6 geometricalrepresentationofacomplexnumber understanddifferentpropertiesofcomplex numbersanditsrepresentationonargandplane solvedifferentmathematicalproblemsusing argand plane 5.7 Statem getfamiliarwithfundamentaltheore entofFu mofalgebra ndamen taltheo remofa lgebra findthesquareroot ofacomplex 5.8 Squarerootofac number omplexnumber 5.9 Solutionof quadra solvethequadraticequationsinthecomplex numbersystem ticequa tionsint hecom plexnu mbersy stem

UNIT6:Li Linearinequalities.Algebraicsolutionsoflinearinequalitiesinon nearandQ evariableandtheirrepresentationonthenumberline. uadraticIn Graphicalsolutionof linearinequalityintwovariables. equalities Graphicalsolutionofsystemoflinearinequalitiesintwovariables. Inequalitiesinvolvingmodulusfunction.Practicalproblemsonlin earinequality,algebraicsolutionofquadraticinequality. Contents LearningOutcomes

6.1 Linear understandlinearinequalities inequati ons findalgebraicsolutionsoflinearinequalities inonevariable 6.2 Algebraicsoluti representthesolutionoflinear inequalitiesinonevariableona onsoflinearineq numberline uationsinoneva riable simultaneoussolutionoftwolinearinequalitiesalgebraicallyas wellasonnumberline 6.3 Algebraicsol findalgebraicsolutionsoflinearinequalitiesintwovariables utionsoflinea r 6.4 Graphicalsolutionofli Solutionoflinearinequalityintwovariablesandthe graphofitssolutionset nearinequationsintw ovariables Solutionofsystemof linearinequalitiesintwovariablesandthegraphofits solutionset

6.5 Inequationssolving inequalitiesinvolvingmodulusfunction. modulusfunctions

understandwavycurve methodfor2nddegree and higher UNIT 7: degreepolynomialsexpressedintheform (x+a)(x+b) ...... (thenumber of suchterms Permutation correspondingtothedegree ofthepolynomial) andCombination Fundamental principle of counting. Factorial n. (n!)Permutationsandcombinations. Propertiesof combination,derivation offormulae and theiconnections,simpleapplications.

Contents LearningOutcomes

Students willbeableto: 7.1 Fundamentalprincipl knowthefundamentaladditionprinciplesofcountingandapply ittofindout esofcounting numberofwaysparticular eventcanoccur know thefundamentalmultiplicationprinciplesofcountingandapply ittofindoutnumberofwaysparticular eventcanoccur Factorialn(n!) 7.2 knowthe meaningoffactorial anditssymbol knowhowtocompute factorial knowhowtorepresentproductofconsecutivenumbersinfactorialknowhowtor epresentproductofconsecutivenumbersinfactorial Permutation 7.3 familiaritywith themeaningofpermutation

derive theformulaeofalinearpermutationnP

use the formula of permutation to find other results p(n,n)=n!0! =1 Combinations 7.4 familiarwiththemeaningofcombination distinctionbetweencombinationandpermutation derivetheformulasofcombinationnC r n

7.5 Derivationofpropertie familiarityanduseofproperties ofcombination s of For nC combination 0 wehave , r 7.6 Typesof linearpermutationscircular permutations permutationrestrictedperm utation permutationswhen particularthing istobe includedeverytimepermutationswhen particularthing isnever tobeincludedpermutationofobjects are notalldifferent Permutationwith repetition

UNIT8:Binomial Theorem: positive integral indices. General and middle termin binomialexpansion,simpleapplications.

Students willbeableto:

8.2History, knowthebinomialtheoremsforpositiveintegralindicesandtheirproofexpandanexpre statement andproofofthe ssionusingbinomial theorem binomialtheor emsforpositive integralindices

8.4Applicationofbino mialtheorem UNIT Sequencesand 9:SequencesandSeri Series,ArithmeticProgression(A.P.),ArithmeticMean(A.M.),GeometricProgression(G.P es: .),generaltermofaG.P.,sumofntermsofaG.P.,infiniteG.P.anditssum,geometric mean (G.M.), relationbetweenA.M.andG.M., Sumto n terms of thespecial series , and

ArithmeticGeometricSeries. HarmonicProgression. Contents

L e a r ni n g O u tc o m es S t u d e n ts w ill b e a bl et o: 9.1 Arithmeti Progression,GeometricProgression c msofanA. identifyana rithmeticsequencefindtheformula P. rithmeticor forthenthtermofageometricsequence. geometrics 9.5 Sumtonter equence. prove a givensequencefromanarithmeticprogressionor a msofaG.P. geometricprogression f determineaspecifiedtermofanarithmeticseque 9 9.6 Infinite i G.P.anditssum ncedetermine . n 2 aspecifiedtermofageometricsequence. 9.7 Relation d A between generateorconstructsequences r A.M.and t i G.M. h fromgivenrecursiverelationshipsfindthearithmeticme t e an. h 9.8 Sumton f m terms i arithmeticmeans between2givennumbers ofspecials e o findthe t eries r geome i m c tricme m u e 9.9 Arithme- l an.i a ticogeome a findthesumoffinitetermsofanarithmeticprogression. n tricseries f 9 9.10Harmonic o findthesum offiniteterms . Progressi r on 3 t ofageometricprogression.findthesum G e h o e ofaninfinite geometricprogression. m n e t identifyand applythe relationbetweenarithmeticmean t andgeometricmean. r h i t c e findthesumtonterms ofthespecialseries m e r a m findthesum of n termsofa series givenits nthtermusing results of n o . f 9.4 S identifyarithmetico-geometricseries. u m a t n o identifyharmonicprogression. a n findthesumtontermsofharmonicprogression. t e r Co-ordinate Geometry

Unit10:StraightLinesBriefrecalloftwodimensionalgeometryfromearlierclasses,Shiftingoforigin,Sl opeofalineandanglebetweentwolines.Variousformsofequationsofa line:paralleltoaxes,pointslope form,slope interceptform,two-point form,interceptformandnormalform.Generalequationofaline.Equation offamilyoflinespassingthroughthepointofintersectionoftwolines.Distan ceofapointfromaline,distancebetweenparallellines.

Studentswillbeableto:

10.1 Briefrecalloftwodimensio distancebetweentwopoints nalgeometryfromearlierc lasses areaoftrianglewhoseverticesaregiven

co-ordinatesofapointdividesthejoinoftwogivenco- ordinatesintheparticularratio

co-ordinatesofmidpointofalinesegmentjoiningtwoco- ordinates co-ordinatesofcentroidandincenterofatriangle

Shiftingof origin 10.2 comprehendthechangeinequationonshiftingthepoint oforigin

Slopeofaline 10.3 findtheslopeofalinewhenangleofinclinationisgiven

identifytheslopeofalineintermsofco-ordinatesofanytwopointsonit

familiarwithconditionofparallellinesandperpendicularlinesinterms ofslope

use slopesoflinesto investigate geometricrelationships,includingparallellines,perpendicularlines .

10.4 Angle between haveafamiliaritywith thetheoremthat anglebetweentwo two lines lineshavingslopem andm isgivenby tan 1 2

equationoflinesparalleltotheco-ordinateaxis 10.5 Variousformsofe quationofaline:p arallelto axis,pointslopfor formtheequationoflinewhenco-ordinatesofpoint m, slop- throughwhichlinepassesandslopeisgiven(point-slopeform). interceptform,tw formtheequationoflinewhenco- opointform,interc eptform,andnorm ordinatesoftwopointsthroughwhichlinepassesaregiven(twopointform alform ). familiarwithinterceptsofalineontheaxes.

formtheequationofline makingslopemandmakinganintercept cony/xaxis(slopeinterceptform).

formtheequationoflinewhenalinecutsof finterceptsa&brespectivelyonxandyaxis (interceptform).

formtheequationoflinewhenthelengthof theperpendicularonitandangleofthatper pendicularisgiven(normalformofline). usedifferent formsofalinetofindoutmissingparamet ersofalineinsymmetricform.

10.6 Generaleq identifygeneralequationandtransformitindif uationofal ferentstandardforms. ine

10.7 Equation findthepointofintersectionoftwolines. of familyofli understandtheconceptoffamilyoflines nespassin passingthrough gthrough intersectionoflinesl andl intermsofl +kl =0. the l 2 1 2 thepointo fintersect givetheequationoflinespassingthroughthepoin ionoftwol tofintersectionoftwolinesundergivencondition ines s.

10.8 Distanceo computethedistance ofapointfromaline. fapointfro maline

10.9 Distanceb computethedistance between parallellines. etweenpa rallelline s

Unit11:ConicSection

Sectionsofacone:circle,ellipse,parabola,hyp erbola,apoint,straightlineandpairofintersect inglinesasadegeneratedcaseoaconicsection. Standardequationofacircle;Generalequatio noacircle,generalequationofconicsectionsw henitsfocus,directrian d

eccentricity are

given,

standardequationsan d

simplpropertiesofpar abola,ellipseandhype rbola.

Contents

LearningOutcomes

Stu den ts illb e le

11.1 Introductionto

identifythecircle,parabola, ellipseand hyperbolaascross sectionofacone sectionsofadoublenappedconebyaplane.

11.2 Circle(Standardf identifytheequationofacircleinstandardformhavingtheCentre(h,k) orm) and radiusr. equationofacirclehavingcentreatoriginandradius r. equationofacirclewhentheendpointsofadiameteraregiven.

11.3 Circle(ge generalequationofacirclewithcentreat(-g,-f)andradius neral form)

find the equationofthecircleusinggivenconditions.

find theconditionforalinetobeatangenttoacircle

11.4 Parabola identifythestandardparabola(righthanded,lefthanded,upwardan (standardform) ddownwardparabola)

findtheaxis,vertex,focus,directrixandthelatusrectumofthestandard parabola

11.5 Parabola identifythegeneralequationof aparabola (generalform) reductionofgeneralformofparabolatothestandardform.

findtheaxis,vertex,focus,directoryandthelatusrectumfromthegene ralequationoftheparabola.

findtheequationofparabolaundergivencondition.

11.6 Ellipse identifytheverticalandhorizontalellipse. (standardform) horizontal&ver findthevertices,majorandminoraxis,foci,directrix,centre,eccentricity ticalellipse andlatusrectumoftheverticalandhorizontalellipse.

11.7 Ellipse(generalfo identifythegeneralformofanellipse(vertical&horizontal)reductionofg rm) eneralformofellipsetothestandardform.

findthevertices,majorandminoraxis,foci,directrix,centre,eccentricit yandlatusrectumfromthegeneralfromofellipse.

findtheequationofanellipseundergivenconditions.

11.8 Hyperbole identifythehyperbolain standardform(alsoconjugatehyperbola) (standardform) findthecentre,vertices,foci,directrix,transverseandconjugate axes,eccentricityandlengthoflatus rectum.

11.9 Hyperbole identifythegeneralformofhyperbole. (generalform) reductionofgeneralformofhyperbolatostandardform.

findthecentre,vertices,focidirectrix,transverseandconjugateaxes ,eccentricity&latusrectumfromthegeneralequationofhyperbola

findtheequationofhyperboleundergivencondition

11.10 Applicationof applytheconceptsofparabola,ellipseandhyperbolainthegivenp conicsection roblems.

Unit12:Introductionto Co-ordinate axes andco-ordinateplanesinthreedimensions.Co- ThreedimensionalGeo ordinatesofapointinspace. metry Distance betweentwopoints and section

formula,directioncosinesofaline,directionratiosofline,anglebetweentwo lines.

12.1 Co- ordinateaxesand identifyco- co- ordinateplanesin ordinateplanesinthreedimensions.findco- threedimensions ordinatesofapointinspace.

12.2 Distance finddistancebetweentwopoints.applyse betweentwopoi ntsandsectionfo ctionformula. rmula

12.3 Someresultsonli directioncosinesofalinedirectionrati ne inspace osofaline

anglebetweentwolines.

Calculus difference, product and quotient of function, Derivative ofpolynomialsandtrigonometricfunction.

Contents

LearningOutcomesStude ntswillbeableto:

13.1 Limitof understandthemeaningofx function

13.2 Fundam-ental aunderstandthelimitoffunctionatapointappl theoremonlimits yfundamentaltheoremsonlimits

1) limf x x

2) limf x x limf x.gx .limg x 3) x

x 4) lim x a ,provided x x g x limc.f x x 5) x

if f x 6)

13.3 Standard n wherea 0 resultsonli lim x mitsandthe x

irapplicati x on lim e x

a lim x

log(1 ) limx x

lim1 x

13.4 Trigono- sinx metric lim x limits limtanx wherexisinradius x Probability Contents

14.1 Random learntheconceptofrandomexperiment,outcomesofrandomexperimen experiment:outc tandsamplespaces omes,samplespa ces(setrepresent listthesamplespacesofarandomexperiment ation). 14.2 Events: occurrenceofeven understandthetermeventasasubsetofsamplespacewriteevents/s ts,'or','and', & 'not'events amplespaceforagivenexperimentrecognise'or','and'&'not'event s 14.3 Exhaustive events,mutuallye identifyimpossibleeventsandsureeventsIdentifysi xclusiveeventsAx mpleandcompoundevents iomatic(settheore tic)probability identifymutuallyexclusiveeventsid entifyexhaustiveevents

14.4 Probabilityofanevent getfamiliarwithindependentevents,equallylikelyevents,andcom plementaryevents* findtheprobabilityofoccurrenceofanevent Oddsofan event 14.5 Oddsinfavourofanevent Oddsagainstanevent. 14.6 Probabilityofoc FindtheprobabilityofcomplementofaneventusingtherelationP currenceofaco (E)=1 P(E) mplementaryev ents

14.7 Resultson )=0,P(S)=1 probability

IfE E thenp(E 1 2 1 )P(E -E )=P(E )- 14.8 Addition 2 1 2 1 theorem P(E E ) 1 2 AdditiontheoremfortwoeventsP(A B)=P(A)+P(B) P(A B) Additiontheoremfor threeevents P(A B C)=P(A)+P(B)+P(C) P(A B)-P(BC)-P(C A)+P(A B C) Additiontheoremformutuallyexclusiveevents. Mathematics CLASSXII

RelationsandFunctions

Unit1:Relationsa Typesofrelations:reflexive,symmetric,transitiveandequivalencerelation ndFunctions s.One tooneand on tofunction,compositefunctions,inverseofafunction.Binary operations.Conceptofexponentialand logarithmicfunction tothebase e, logarithmicfunction asinverseofexponentialfunctionandgraphs. Contents

LearningOutcomesStude ntswillbeableto: 1.1 Typesof identifyreflexiverelation,illustratereflexiverelation. relations:reflexi identifysymmetricrelationidentify ve,symmetric,tr transitiverelationidentifyantisym metricrelation ansitiveandequi understandtheconditionsforanequivalencerelationdetermi valencerelations nea relationisequivalence 1.2 Onetooneandontofunc tions identifyone- 1.3 Composite onefunctionsidentifyontofuncti functions onsidentifybijectivefunctions definecompositionoftwofunctions 1.4 Inverseofafunction understandthatgiventwofunctionsfand g,fogmaynot beequaltogof andbinaryoperatio definetheinverseofagivenfunction,ifexists.understandthedef ns initionofbinaryoperationonaset determineifagivenoperationisabinaryoperationo nagivensetdetermin ethetotalnumberofbinaryoperationsonagivenfiniteset determineifagivenbinaryoperationiscommutativedeter mineifa givenbinaryoperationisassociativedetermineifa givenbinaryoperationisdistributivedetermine 1.5 Conceptofexpon theidentityelementforabinaryoperationdeterminethein entialandlogarit verseelementofagivenelementunderstandtheproperties hmicfunctiontot oflookingatitsgraph. hebasee, understandthepropertiesof logarithmicfunctionlookingatitsgraph definelogarithmicfunctionasinverseofexponentialfunctionsketchthegra phofexponentialfunction logarithmicf unctionasinv erseofexpone ntialfunction andtheirgra phs Definitionofinversetrigonometricfunctioninunitcircles,range,domai Unit2:Inverset nprincipalvaluebranches.Graphsofinversetrigonometricfunctions,El rigonometricf ementarypropertiesofinversetrigonometricfunctions. unctions Contents LearningOutcomes

Studentswillbeableto: 2.1 Definitionofin defineallinversetrigonometricfunctionusingaunitcircl versetrigonom e etricfunction inaunitcircle 2.2 Range,do main,princ ipalvalueb statethedomainandrangeofinversetrigonometricfunctio ns ranches statetheprincipalvaluebranchofinversetrigonomet 2.3 Graphsofinver ricfunctionsandneighbouringbranches. setrigonometri sketchthegraphsofsixinversetrigonometricfunctions. cfunctions 2.4 Elemen tarypro pertieso provethepropertiesofinversetrigonometricfunctions finverse trigono metricf unction s usethetrigonometricpropertiestosolvetrigonometric 2.5 Problemsbasedo equationsandtoprovetrigonometricidentities. nproperties Matrices andDeterminants

Unit3:MatricesConcept,notation,order,equality,typesofmatrices:zeromatrix,transposeofmatrix, symmetricandskewsymmetricmatrices,Addition,multiplicationandscalar multiplicationofmatrices,simplepropertiesofaddition,multiplicationandsc alarmultiplication,Non- commutativityofmultiplicationofmatrixandexistenceof non- zeromatriceswhoseproductisthezeromatrix(restricttosquarematricesofor ders).Conceptofelementaryrow andcolumnoperations,invertiblematricesandproofoftheuniquenessofinve rse,ifitexists.(Hereallmatriceswillhaverealentries) Contents LearningOutcomes

Studentswillbeableto: 3.1 Matrices definematrices usematrixnotation determinetheorderof amatrixidentifytypesofmatrices: - Rowmatrix - Columnmatrix - Squarematrix - Diagonalmatrix - Scalarmatrix - Identity orunit - Null(zero)matrix - Uppertriangularmatrix - Lowertriangularmatrix 3.2 Equalityof understandtheconditionfortheequalityof matrices matrices

3.3 performadditionandsubtractionofmatricesunderstandt Operationonmatrices hepropertiesofaddition ofmatrices 3.4 Mul performmultiplicationofamatrixbyascalaridentifythepr opertiesofscalarmultiplication tiplica- understandtheconditionsorderofmatrices tionofmatr tomultiplythemperformmultiplicationofmatrices,whereverpos ices sibleidentifyproperties of matrixmultiplication canillustratenonzeromatriceswhose productisthezeromatrixsolveproblemsbasedonapplicationofmatrices 3.5 Transposeof writethetransposeofamatrix matrix verifythepropertiesoftranspose

3.6 Symmetric identifysymmetricmatricesidentifys and skew- kew-symmetricmatrices understandthat forasymmetricmatrixa =a symmetric ij ji matrices understandthatinaskewsymmetricmatrix,diagonalelementsarezero. constructasymmetricand skew symmetric matrix. provethateverysquarematrixcanbeexpresseduniquelyassumofsymmetric andskewsymmetricmatrix. writeagivensquarematrixassumofsymmetricandskewsymmetricmatrix. applyelementaryrow andcolumntransformationsonamatrixoforder2 2 and3 3 3.7 Conce invertiblematrices(AB=BA=I). ptof findtheinverseofamatrixusing columnorrowtransformations elementaryro wandcolumnt ransformatio ns Determinantofasquarematrix(upto3matrices)propertiesofdeterminants, minors,cofactorsandapplicationsofdeterminantsinfindingtheareaofatrian Unit4:Determ gle,collinearityofpoints.Consistency,inconsistencyandnumberofsolutions inants(10Peri ofsystemoflinearequationsby examples.Solving system oflinearequationsintwoorthree ods)

applicationsonwordproblems. Contents LearningOutcomes

4.1 Determinantofasqu findthevalueofadeterminantoforder arematrix(upto3×3 determinetheminorofanelementofasquarematrixdeterminethe matrices) cofactorofanelementofasquarematrixdeterminethedeterminan tofasquarematrixoforder3x3 4.2 Applicationofdeterm inant use determinantstofind theareaofatriangle 4.3 Propertiesofdetermi usedeterminantstodeterminethecollinearityofthreepointsverifythe nants propertiesofdeterminants applythepropertiesofdeterminantstosolveproblems 4.4 s Unit5:Adjointan Adjointofasquarematrixoforder 2 2and3 dInverseofamatr 3.Propertiesofadjointofamatrix.Inverseofasquarematrix.Consistency,in ix consistencandnumberofsolutionsofasystemoflinearequations byexamples.Solvingthesystemoflinearequationsintwoandthreevariables bymatrixmethodanditsapplicationin wordproblem. Contents

Le ar ni ng O ut co m es St ud en ts wi llb ea bl et o:

5.1 Adjointofamat 5.3 Invertible rix matrices

5.4 Solvinga 5.2 Singular systemoflinearequationsbyMatrixmethod andNons ingularm atrix f e d oramatrixAofordern. adj(AB)=(adjB) i r j (adjA) n i ( d f A a t y ) d h t j e h . A a e A ) d p T j r = = o o a i p A d n e I j t r n A o t f T f i a e d a s j s o A q f u a = a d A r j e o n m i - a n 1 t t understandthedefinitionofsingularandnonsingularmatri r : ces.identifysingular andnonsingularmatrix. i A x ( u understandtheconditionforamatrixinordertobeinve p a rtibleprovethateveryinvertiblematrixpossessesaun t ique d o inversefindtheinverseofamatrixusingdefinition o j findtheinverseofamatrixwhenitsatisfiessomematrixequ r ation.verifyresultsoninvertiblematrices A d solveasystemoflinear e ) equationintwovariables(havinguniquesolution)usinginver r seofamatrix 3 = understandtheconditionsfor consistencyandinconsistencyofsystemoflinearequation ( s 3 solveasystemoflinearequationsinthreevariables(havingun v a iqusolution)usinginverseofamatrix s inearequa o tionswhen l theinverse v ofcoeffici e entmatrixi a sobtained s fromsome y givenrelat s ion t solveprob e lemsonap m plicationo o f f simultane ouslineare l quations

Calculus Derivativesoflogarithmicandexponentialfunctions. Logarithmic differentiation derivative of functionsexpressedinparametricformssecondsorderderivatives.

Contents

LearningOutcomesStude ntswillbeableto: 6.1 Differentia- understandthemeaningofdifferentiabilityofafunction bility determinethedifferentiabilityofafunctionatagivenpointdetermin erelationbetweencontinuityanddifferentiability.derivativeatapoi nt geometricalsignificance ofderivativeasslopeoftangent. physicalsignificanceofderivativeasarateofchangeofywithrespecttox. derivativeofafunctionbyfirstprinciple.deri vativeofalgebraicfunctionsderivativeofsca larmultipleofafunction derivativeofsumanddifferenceoffunctionsderivativeofap olynomial. ProductRule 6.2 derivativeof productoffunction 6.3 Q derivativeofquotientoffunctiondifferentiabl uotient Rule egivenanimplicitfunction 6.4 Derivatives ofimplicitfun ctions determinethederivativesoflogarithmicandexponentialfunction 6.5 Derivativeof logarithmic andexponen tialfunctions 6.6 Deriv findthederivativeofthegivenInfiniteseries ativeofInfini teSeries

6.7 Logarithmic differentiatethefunctionsoftheusing logarithmicdifferentiation differentiation

6.8 Differen- differentiateonefunctionwithrespecttoanotherfunction tiationofonef unctionwith respect toanother. differentiatefunctionsgiveninparametricform 6.9 Deriv atives offunctionsex pressedinpar ametricform determinesecondorderderivativeofagivenfunction 6.10 Secon

dorderderiva Applications of derivativesrate ofchangeincreasing/decreasing tive Unit7:Applications tangentsandnormalapproximation,maximaandminima(firsderivativetest Local ofDerivatives Maxima/LocalMinimaandsecondderivativetesAbsoluteMaxima/Absolut eMinima).Simple problems (thaillustratebasicprinciplesandunderstandingtothesubjectaswellareal- lifesituations). Contents

Le ar nin gO utc om esS tu de nts wil lbe abl eto : Rateof solveproblemsonrateofchangeofy chang withrespecttoxwherey=f(x)isafunctionofx e determineifafunctionisstrictlyincreasinginagive Increasing/ decre nintervaldetermineifafunctionisstrictlydecreasin asingf ginagivenintervaldetermineifafunctionisincreasi uncti ng/decreasing ons identifunderstandth check the y estatementof applicabilityofRolle'stheoremforagivenfunctioninagive necessRolle'stheore ninterval arym d verifyRolle'stheorem cientunderstandth onditiegeometricali foragivenfunctioninagivenintervalapplyRolle'stheo on nterpretation remtosolveaproblem montoofRolle'stheor em nicity understandthestatementofLagrange'smeanvaluetheor ofafun ction emunderstandthegeometricalinterpretationofLagrang

find e'smeanvalue interv al in ch nction is asing r asing

prove he notoni cityof a ion a interv al

7.3 eirgeo oll metric s alinter theo pretati re onand Lagr simple ange 'sme an lue eo m( thou tpro of) nd application theorem

verifyLagrange'smeanvalue theoremforagivenfunction applyLagrange'smeanvaluetheoremtosolveproblems

7.4 Tangents determinetheslopeoftangentsandnormalstoagivencurveatagivenpoint andnormals determinepointsonagivencurveatwhichtangentisparalleltoagivenline

determinepointsonagivencurveatwhichtangentisperpendiculartoagivenlin e

determinetheequationofthetangentto agivencurveatagivenpoint

determinethe equationofthenormalto agivencurveatagivenpoint

determine the angleof intersection oftwocurvesi.e.theanglebetweenthetangentstothetwocurves

7.5 Approxim- understandthetermsabsoluteerror,relativeerror,percentageerror ations solve problemsbased on applicationofdifferentiation underapproximation.

7.6 Maximaand understandthedefinitionoflocalmaximaandminimaunderstandth minima(local edefinitionofabsolutemaximaandminima maxima/loca lminimaanda understand thealgorithmfor the first derivativetest forlocalmaximaand bsolutemaxi minima

ma/absolute determinethepointsoflocalmaximaandlocalminimaforagivenfunction minima),first derivativetes determinethepointsofinflexionforagivenfunctionunderstandthealgo t,secondderiv rithmforsecondderivativetest ativetest,sim determinethemaximumand pleproblems( minimumvaluesofafunctioninaclosedinterval that determinethepointsof

absolutemaximaandminimasolvepracticalproblemsonm

aximaandminima ,

sin, 8.3 evaluatetheintegrationbyparts: Integrati onbypart evaluate e f f1( s

9.3 Basicpropert identifythebasicpropertiesofdefiniteintegralspropertyI iesofdefinite integralsand b b evaluationof .,Integrationisindependentofthechange definiteinte a a grals ofvariable. propertyII

a b i.e.,ifthelimitsofadefiniteareinterchangedthenit svaluechangesbyminussignonly.

propertyIII

b c b . a a c

property IV

a,b, Ifisacontinuousfunctiondefinedon then b b

propertyV f x a, If 0, th a a en

propertyVI f x If is acontinuousfunctiondefinedon then

a x isan evenfunction

, if f x isan oddfunction

propertyVII

f x 0,2a, If isacontinuousfunctiondefinedon 2a f x f 2a

0 , if f 2a

9.4 Definite evaluatethedefiniteintegralsusingabovementionedproperties.understa integralsasali ndtheconceptof limitofsum. mit ofsum. evaluatedefiniteintegral asalimitofsum(linear, quadratic,cubicand exponentialfunctions)

Unit10:Application Applicationsin findingthe areaboundedbyacurveandaline. of theintegrals Areaboundedbetweenlines. Areasboundedbetweentwocurves.Areasofcircles/ ellipses (in y standardformonly). Areaunderthe curvey , (theregion shouldbeclearly identifiable)

Studentswillbeableto: 10.1 Finding theareas determinetheareaenclosedinacircledeterminethear

ofcircles/parabolas/el eaenclosed inanellipse lipses

(instandardformonly ) 10.2 Area determinetheareaboundedbyacurveandaline a boundedbya lineandtheaxes curveandalin e twolinesandanaxis

determinetheareaofatriangle

determinetheareaboundedbymodules functionandgivenlines

10.3 Area determinetheareaboundedbetweentwocurves bounded ' betweentwo determinetheareaunderthecurve'y curves f

find the areaboundedbyy=sin x&y=cosxundergivenconditions Definition orderand degree,generaland particularsolutionsofdifferential Unit11:Diffe equation,Formation of differential rentialEqua tions equationwhosgeneralsolutionisgivenSolutionisdifferentialequationsbymet hodofseparationofvariables,homogeneousdifferentialequationsoffirst orderandfirstdegreesolutionsoflineardifferentialequationofthetype: dy wherep andqarefunctionsofxand dx dx wherepandqarefunctionsofy dy Contents

11.1 Definition, identifyadifferential equation order telltheorderand degreeofadifferentialequation anddegree verifythatthegivensolutionissolutionofagivendifferentialequation formadifferentialequationgivenitsgeneralsolution 11.2 Generalandparticula rsolutionsofadifferen tialequation 11.3 Formationofdifferen tialequation solvedifferentialequationsinvariableseparableformdeterminepar ticularsolution,when initialvaluesaregiven 11.4 Homog- solvedifferentialequationsthatarereducibleto variableseparableform enousdiffer identifyhomogenousdifferentialequationoffirstdegreeandfirstorder entialequati solvethehomogenousdifferentialequationoffirstdegreeandfirstorder onoffirstor derandfirst degree 11.5 Linear solvelineardifferentialequationofthetype equationoff dy pandqarefunctionsofx irstorder py qdx solvelineardifferentialequationofthetype dx px pandqarefunctionsofy q,dy 11.6 Applications solveproblemsofapplicationongrowthanddecay ofdifferent solveproblemsonvelocity,acceleration,distanceand ialequation timesolvepopulationbasedproblemsonapplicationofdifferentialequ ations solveproblemsofapplicationonco-ordinategeometry VectorsandThreeDimensionalGeometry

Unit12:VectorsVectorsandScalars,Magnitude anddirectionofa vector.Representationofvectors,typesofvectors,positionvectorofapoint,com ponentsofavector,Additionofvectors(propertiesofaddition,lawsofaddition), Multiplicationofavectorbyascalar,positionvectorofapointdividingalineseg mentinagivenratio.Scalar(dot)productofvectors,projectionofavectoronaline .Vector(cross)productofvectors,scalartripleproduct. Contents

LearningOutcomesStude ntswillbeableto: 12.1 Vectorsand differentiateScalarandVectorsquantities. Scalars representa vector.

findmagnitudeofvector.

representnegativeofa vector.

12.2 Magnitudea defineandillustratevarioustypeofvectors,e.g.parallelvector,coinitialvectors nddirection ,coterminouscollinearvectors,likeandunlikevectors,equalvectors ofavector 12.3 Position writethepositionvectorofapoint vectorofapoi nt 12.4 Compon- identifycomponentsofavectorin entsofavecto r twodimensionidentifycomponentsofavectorinthreedimens ion

identifycomponentsofavectorintermsofcoordinatesofitsendpoints

12.5 Additionof understandsand canusetrianglelawofvectoraddition. vectors understandsandcanuseparallelogramlawofadditionofvectorsprovecom 12.6 Properties mutativepropertyunderaddition ofaddition proveassociativepropertyunderadditionfindad ofvectors ditiveidentity

findadditiveinverseofagivenvector.

solveproblemsbasedonvectoraddition 12.7 Multiplica- multiplythevectorbyascalar tionofavect appreciatethefollowingpropertiesofmultiplicationofvectorsa, b orbyascala byascalarm,n r n (i)

(ii) m na (iii) m (iv) m a (v)

provethesectionformulaforinternaldivisionan dexternaldivisionofvectors

usetheappropriatesectionformulatofindtheposit ionvectorofapoint dividingthegivenlinesegmentingivenratio.

12.8 Positi find the positionvector of a point on dividingtheline vecto segment(internallyandexternally) rofap ointdi vidin galine segme ntina given finddirectionratiosofavector ratio 12.9 Direct ion determinethedirectioncosinesofavector cosine findtheunitvectorinthedirectionof sandd givenvectordefinescalarproduct irectio nratio understandthegeometricalinterpretation sofvec tor ofscalarproductfindthescalarproductoftwogiv 12.10 Scala r(dot envectors produ ct)ofv ectors

applythescalarproduct

(10.i) tochecktheperpendicularityoftwovectors

(10.ii) todeterminetheanglebetweentwovectors (10.iii) tofindth

(10.iv) tofind eworkd one given rce ection displace ment understandandapplythefollowingpropertiesof scalarproduct

(i) commutativity (ii) distributivityofscalarproductovervectoraddition

(iii) a.b aisperpendicularto bwhere aand b arenon zerovectors (iv) Foranyvectora, a.a

(v) ma .b

Wherea,barevectorsandmisscalar (vi) ma.nb mna. b mna .b a.mnb

Wherea,barevectorsandm,narescalars (vii) a.

(viii)

b. a b a 2 2 a 12.11 Projection ofavectorona numberline 12.12 Vector definethevectorproductofgivenvectors product understandthegeometricalinterpretationofthevectorproduct

findthevectorproductofgivenvectors

understandandapplythepropertiesofvectorproduct a b Let and bevectorsand m,nbescalars

(i) ma (ii) ma (iii) a (iv) b a (v) (vi) a aandbarenonzerovectors use thevectorproductto

checkthecollinearityof twovectors

findunitvectorperpendiculartovectorsa&bboth findmom entofaforceindirectionofdisplacement

findareaofparallelogramformedbyadjacentvectorsa&b

findtheareaoftrianglewith adjacentsidesa&b

find the areaofquadrilateralwith diagonalsd &d 1 2

12.13 Scalartriple definescalartripleproduct. product understandgeometrical

interpretationofscalartripleproductfindthescalartripleproduct.

find thevolumeoftheparallelepipedhavingadjacentedgesusingscalartripleproduct

findthevolumeoftheparallelepipedwiththegivenvertices

usescalartripleproducttoshow thatthreevectorsarecoplanar.

Unit13:Three- Directioncosinesanddirectionratiosofalinejoiningtwopoints.Cartesianandvect dimensionalGe orequationofaline.Coplanarand ometry skewlines.Shortestdistancebetweentwolines.Cartesianandvectorequationofapl ane.Anglebetweentwolines.Anglebetweentwo planes.Anglebetweenaline andaplane.Distanceofapointfromaplane Contents LearningOutcomes Studentswillbeableto:

13.1 Briefrecall recallthedirectionratiosofalinepassingthroughtwopointsrecallthedirecti ofdirectionc osinesanddi oncosinesofalinepassingthroughtwopoints rectionratio findtheanglebetweentwovectorsintermsoftheirdirectionscosinesfindtheangl sofaline ebetweentwovectorsintermsoftheirdirection ratios

13.7 Equationofa find equation of plane passing through a given point planepassin andperpendiculartoagivenvector(bothvectorandCartesianform) gthroughagi findequationofplanepassingthroughtwopointsandparallelto venpoint agivenline(bothvectorandCartesianform)

findequationofaplane throughagivenpointand paralleltotwogivenlines

findequationof a plane

containingtwolines(both vectorandCartesianform)

findequationofaplanepassingthroughthreepoin ts(bothvectorandCartesian form)

13.8 Equat 13.12 Distanceofapointfromaplane ionof 13.13 Imageofa planei pointinplane ninter 13.14 Coplanarlines ceptfo rm 13.9 E quatio nof planei ngene ralfor m 13.10 Equati onofpl aneth rough theint ersect ionoft wopla nes 13.11 Angle betwe entwo plane s f saregive ndapl findthedistanceofapointfromaplanein(Cartesianform i n ane andvectorform) n d e gen findtheimageofthepointinagivenplane q eral u equ a atio t n conditionforthe coplanarityoftwolines and i ofth equationofthe planecontaining them. o epla n nea o ndit f sred a ucti p ont ono l rma a lfor n m e w h o s e findequa i tionofapl n anepassi t ngthroug e hintersec r tionoftw c ogivenpl e anes(bot p hvectora t ndCartes s ianform) o n c o find o angl ebet r wee d ntw i opla n nes a t find e angl a ebet x wee e nali nea Probability

Unit14:Pro Multiplicationtheoremonprobability. Conditional probability, bability

variableanditsprobabilitydistribution,meanandvarianceofrandomvariable .Repeatedindependent(Bernoulli)trialsandBinomialdistribution. Contents

LearningOutcomesStude ntswillbeableto: 14.1 Conditional understandthemeaningofconditionalprobability Probability derivetheformulaP A ofconditionalprobabilityusingmultiplicationtheorem B P A B P AB assumethatP P BA P B P A

assumethatP(A)

understandand use the

propertiesofconditionalprobability.solvetheproblemsbasedoncon

ditionalprobability.

14.2 M understandthatifAandBaretwoeventsassociatedwitharandomexperiment, then ultipli- cation theoremon P A B P A PB A P B P A B),giventhat probability . ( ). ( / P(A) PB

understandtheextensionofmultiplicationtheoremthatif A ,A ,A ,.....A areneventsassociatedwitharandomexperiment,then 1 2 3 n P A P A A P A A A ...... P A A A.....A 1 n

A A A ....A .. n 1 n

14.3 Independent Identifytheindependenceordependenceofevents Events usetheformulaP A B

forindependenteventsunderstandthatP(A/B)=P(A),P(B) 0 for

independent

&P(B/A)=P(B),P(A)0 eventsA&B findtheprobabilityofsimultaneousoccurrenceforindependentevent s findprobabilityofoccurrenceofatleastoneeventforindependent events

14.4 Total findtheprobabilityofaneventwhencertainconditionalprobabilitiesofthateve Probability ntaregiven.

14.5 usetheconditionalprobabilitytomakepredictionsinreverse theorem 14.6 Rando understandthemeaningofrandomvariable m variableandi writeprobabilitydistributionofrandomvariables tsprobabilit ydistributio n findmeanofadiscreterandomvariablefindm 14.7 Meanand varianceofra eanofcontinuousrandomvariablefindvaria ndomvariab le nceofdiscreterandomvariable 14.8 Repeated independent(Bernoulli)trialsandBinomialdistribution know thedefinition ofBernoullitrial. findtheprobabilitiesforBernoullitrialsusingbinomialprobabilityformula