SubsetsofR open interval. Venn diagrams Unionand Differenceof sets Lawsof Practical representsetsusingvenndiagrams. findtheintersection of setsandunion ofsets. showtheintersectionandunionofsetsusing Venndiagrams.identifydisjointsetsanditsrepresentationusing venn diagram. findthedifferenceofsetsandtheirrepresentationusing venndiagram. applythefollowinglawsofalgebra onsets: lawsofintersectionofsets applypropertiesofcomplementsets. solvepracticalproblemsonunionandintersectionofsets. applyresultsandsolveproblemsonnumberofelementsof setsusingpropertieslike 1) 2) Functions onesetto another.Pictorialrepresentationofafunction,domain,co-domain&rangeof functions,Evenandoddfunction s
Contents LearningOutcomes OrderedPairs Studentswillbeableto: identifyanorderedpair. identifytheequalityoftwoorderedpairs. Cartesian ProductofSets nonemptysets.identifythetwosetsgiventheir R R). identifythenumberofelementsinthecartesianproductoftwofinitesets. identifycartesianproductofsetofallrealnumberswithitself. understandrelationoftwosetsasasubsetoftheircartesianproduct. ArrowDiagram pictorialrepresentationofarelationbetween twosets. Pictorial identifyfunctionasaspecialkindofrelationfromonesettoanother. determinewhen arelationis afunction. describeandwritefunctionalrelationshipsforgivenproblemsituations. understandthatf R tounderstandthat everygraphdoes notrepresentafunction. odd functions. findspecificfunctionvalues g(x)=g(x) f(x) (fg)(x)=f(x).g(x)
UNIT 7: Permutation andCombination Solutionoflinearinequalityintwovariablesandthe graphofitssolutionset Solutionofsystemof linearinequalitiesintwovariablesandthegraphofits solutionset inequalitiesinvolvingmodulusfunction. understandwavycurve methodfor2nddegree and higher degreepolynomialsexpressedintheform (x+a)(x+b) ......(thenumber of suchterms correspondingtothedegree ofthepolynomial) Fundamental principle of counting. Factorial n. (n!)Permutationsandcombinations. Propertiesof combination,derivation offormulae and theiconnections,simpleapplications. Contents LearningOutcomes Students willbeableto: knowthefundamentaladditionprinciplesofcountingandapply ittofindout numberofwaysparticular eventcanoccur know thefundamentalmultiplicationprinciplesofcountingandapply ittofindoutnumberofwaysparticular eventcanoccur Factorialn(n!) knowthe meaningoffactorial anditssymbol knowhowtocompute factorial Permutation familiaritywith themeaningofpermutation derive theformulaeofalinearpermutationnP use the formula of permutation to find other results Combinations familiarwiththemeaningofcombination distinctionbetweencombinationandpermutation derivetheformulasofcombinationnC of familiarityanduseofproperties ofcombination n combination For 0 ,wehave Cr
sectionofacone sectionsofadoublenappedconebyaplane. radiusr. equationofacirclehavingcentreatoriginandradius r. equationofacirclewhentheendpointsofadiameteraregiven. Circle(general form) generalequationofacirclewithcentreat(-g,-f)andradius find the equationofthecircleusinggivenconditions. find theconditionforalinetobeatangenttoacircle Parabola (standardform) findtheaxis,vertex,focus,directrixandthelatusrectumofthestandard parabola Parabola (generalform) aparabola reductionofgeneralformofparabolatothestandardform. findtheequationofparabolaundergivencondition. Ellipse identifytheverticalandhorizontalellipse. findtheequationofanellipseundergivenconditions. Hyperbole (standardform) standardform(alsoconjugatehyperbola)
rectum. Hyperbole (generalform) identifythegeneralformofhyperbole. reductionofgeneralformofhyperbolatostandardform. findtheequationofhyperboleundergivencondition Applicationof conicsection Co-ordinate axes andco-ordinateplanesinthreedimensions.Co-ordinatesofapointinspace. Distance betweentwopoints and section Contents LearningOutcomes Studentswillbeableto: Distance inspace anglebetweentwolines.
Contents Random Events: & 'not'events Exhaustive listthesamplespacesofarandomexperiment findtheprobabilityofoccurrenceofanevent Oddsofan event Oddsinfavourofanevent Oddsagainstanevent. P(E) Resultson probability Addition theorem E2thenp(E1 E2) 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. P
toanother. Derivatives Unit7:Applications ofDerivatives determinesecondorderderivativeofagivenfunction Applications of derivativesrate ofchangeincreasing/decreasing problems Contents Rateof change Increasing/ withrespecttoxwherey=f(x)isafunctionofx identifythe necessaryandsufficientconditionformontonicityofafunction findaninterval inwhichafunctionisincreasingordecreasing provethemonotonicityofafunctiononagiven interval Rolle's check the applicabilityofRolle'stheoremforagivenfunctioninagiveninterval verifyRolle'stheorem foragivenfunctioninagivenintervalapplyRolle'stheoremtosolveaproblem
application theorem verifyLagrange'smeanvalue Tangents andnormals determinetheslopeoftangentsandnormalstoagivencurveatagivenpoint determinepointsonagivencurveatwhichtangentisparalleltoagivenline determinepointsonagivencurveatwhichtangentisperpendiculartoagivenline determinetheequationofthetangentto agivencurveatagivenpoint determinethe equationofthenormalto agivencurveatagivenpoint determine the angleof intersection oftwocurvesi.e.theanglebetweenthetangentstothetwocurves understandthetermsabsoluteerror,relativeerror,percentageerror solve problemsbased on applicationofdifferentiation underapproximation. understand thealgorithmfor the first derivativetest forlocalmaximaand minima determinethepointsoflocalmaximaandlocalminimaforagivenfunction determinethemaximumand minimumvaluesofafunctioninaclosedinterval determinethepointsof
2a f x f 2a 0 , if f 2a Definite ofsum. limitofsum. evaluatedefiniteintegral asalimitofsum(linear, quadratic,cubicand exponentialfunctions) theintegrals Applicationsin findingthe areaboundedbyacurveandaline. Areaboundedbetweenlines. Areasboundedbetweentwocurves.Areasofcircles/ ellipses (in standardformonly). Areaunderthe curvey , y (theregion shouldbeclearly identifiable) Contents LearningOutcomes Finding theareas Area inanellipse determinetheareaboundedbyacurveandalinea lineandtheaxes twolinesandanaxis determinetheareaofatriangle determinetheareaboundedbymodules functionandgivenlines Area bounded determinetheareaboundedbetweentwocurves determinetheareaunderthecurve'y ' f find the areaboundedbyy=sin x&y=cosxundergivenconditions Definition orderand degree,generaland particularsolutionsofdifferential
Studentswillbeableto: VectorsandThreeDimensionalGeometry Unit12:VectorsVectorsandScalars,Magnitude anddirectionofa Contents Vectorsand Scalars differentiateScalarandVectorsquantities. representa vector. findmagnitudeofvector. representnegativeofa vector. Position Compon- writethepositionvectorofapoint identifycomponentsofavectorin twodimensionidentifycomponentsofavectorinthreedimension identifycomponentsofavectorintermsofcoordinatesofitsendpoints Properties understandsand canusetrianglelawofvectoraddition. findadditiveinverseofagivenvector. solveproblemsbasedonvectoraddition
Multiplica- multiplythevectorbyascalar appreciatethefollowingpropertiesofmultiplicationofvectorsa, b byascalarm,n n (i) (ii) (iii) (iv) (v) m na m m a provethesectionformulaforinternaldivisionandexternaldivisionofvectors usetheappropriatesectionformulatofindthepositionvectorofapoint dividingthegivenlinesegmentingivenratio. Position Direction Scalar(dot find the positionvector of a point dividingtheline segment(internallyandexternally) finddirectionratiosofavector determinethedirectioncosinesofavector understandthegeometricalinterpretation ofscalarproductfindthescalarproductoftwogivenvectors applythescalarproduct tochecktheperpendicularityoftwovectors todeterminetheanglebetweentwovectors tofindtheprojectionof avectoronaline
understandandapplythefollowingpropertiesof scalarproduct commutativity distributivityofscalarproductovervectoraddition a.b aisperpendicularto bwhere aand b arenon zerovectors Foranyvectora, a.a ma .b Wherea,barevectorsandmisscalar ma.nb mna. b mna .b a.mnb Wherea,barevectorsandm,narescalars a. (viii) Projection ofavectorona numberline 2 2 a b. a b a Vector product definethevectorproductofgivenvectors understandthegeometricalinterpretationofthevectorproduct findthevectorproductofgivenvectors understandandapplythepropertiesofvectorproduct Let aand bbevectorsand m,nbescalars (i) ma (ii) ma (iii) a (iv) b
Probability Conditional probability, Contents Conditional Probability understandthemeaningofconditionalprobability derivetheformulaofconditionalprobabilityusingmultiplicationtheorem A B P A B P AB assumethatP P BA P B P A assumethatP(A) understandand use the Multipli- cation understandthatifAandBaretwoeventsassociatedwitharandomexperiment, then theoremon probability P A B P A P(A) PB .PB A P(B).P(A/B),giventhat understandtheextensionofmultiplicationtheoremthatif A1,A2,A3,.....Anareneventsassociatedwitharandomexperiment,then P A1 A A.....An P A P A A P A A A ...... AnA1 A ....An .. A B forindependenteventsunderstandthatP(A/B)=P(A),P(B) 0 for independent eventsA&B findprobabilityofoccurrenceofatleastoneeventforindependent events