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HIGHER SECONDARY MATHEMATICS XII STANDARD SCIENCE STREAM

1. SYSTEMS OF EQUATIONS

S.No. Content Expected Outcome Transactional Strategy No. of

Periods

1.1 Systems oflinear equations Presentation in Matrix form;computing the rank of matrix

and determining cases ofi. a unique solutionii. a set of solution

iii. no solution

Sets of simultaneousequations of at most three

variable only to beprescribedGraphical interpretation

wherever possible .Discriminating between

inconsistent anddependent equations.

1.2 Methods of

solution

Computing the unique

solution of a system of

equations when it exist byi)Cramer s Rule and ii)

Inverse matrix method

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2. APPLICATIONS OF MATRICES

S.No. Content Expected Outcome Transactional Strategy No. ofPeriods

2.1 Matrices for

transformations:

Matrices forTranslation,Reflection,Rotation, Glide

reflection,Shear and

Stretch

Recognising matrices as tool

to study specific geometrical

notions.Applying transformationmatrices to derive geometricresults

Correlation with Pure

and Analytical geometry

and results inTrigonometry

2.2 Isometry andsimilarity

matrices for thesame.

Identifying points, lines etcremaining invariant under a

transformation

Correlation withgeometrical notions

studied in earlier classes.

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3. VECTOR ALGEBRA


Periods

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3.1 Vectors andScalars

Representationsof vectors andoperations of

addition and

subtractions

Definition vector addition,multiplication by scalars,

linear relation amongvectors, orthogonaldecomposition; 3

dimensional Cartesian

coordinates: directioncosines.

Concept to be supportedby Geometrical

interpretation.Relation to velocity,acceleration, resultants

etc to be introduced

3.2 Scalar andvector productsTriple products

and products 4vectors

Ability to do simplemanipulative problems.Ability to use appropriate

product in a given situation

Geometrical meaning tobe explained.Use of suitable 3-D

diagrams.

3.3 Applications to

mechanics

Applying formulae for Work

don by force and Moment ofa force using vectors

Relating the results to

actual problems inmechanics in the relevant

areas.

3.4 Applications toGeometryParallel &

Perpendicularvectors. Angle

between lines,Equations oflines and planes

Derivations of the equationsand applying them in simpleproblems.

Ability to apply the ideas toderive standard trigonometric

results too.

Translation intoanalytical results of twodimensions wherever

possible.Collincarity &

coplanarity to bediscussed appropriately

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4. COMPLEX NUMBERS


4.1 ComplexAlgebra

Fundamentaloperations oncomplex

numbers

Ability to separate real andimaginary parts; compute

absolute value; multiplicativeinverse of a complexnumber, conjugation:

Triangle inequality

Emphasis to be given onComplex numbers as a

vector.Interpretation throughArgand diagram

4.2 Applications De Moivre s theorem: Roots

of a complex number; Euler,

formula, Statement andmeaning of Fundamental

Thm. Of Agebra.

Complex solutions to be

illustrated by simple

examples and diagrams

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5. ANALYTICAL GEOMETRY


Periods

5.1 Definition of aconic

Focus-directrix definitionGiven the equation to find

Tracing Parabola, Ellipseand Hyperbola using the

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Derivation ofthe standard

equation ofParabola,Ellipse,

Hyperbola and

RectangleHyperbola.

the foci, directrices.Eccentiricity, latus recta etc

of the conic.

standard equations andexplaining the special

features

5.2 Chords,Tangents &

Normals

Chord joining two point onthe conic. Tangent and

normal at a point on theconic. Condition for a line to

be tangent to a conic: chordof contact of tangent. Chordwith a given mid point (Not

by r method)

Use of equations toillustrate simple

geometrical results

5.3 Parametricrepresentation Representing point on theconic in terms of parametric

co-ordinates.

Results on chords andtangents to be explained

in terms of parametriccoordinates.

5.4 Asymptotes Derive the equations of

asymptotes of hyperbola andidentify their properties

Explaining the role of

asymptotes in tracing theconic.

6. APPLICATIONS OF DIFFERNTIATION


Periods6.1 Derivative as a

rate measurer

Rate of change of quantities;

interpretation of velocity andacceleration using distance-

time formulae and solvingproblems involving them.

Majority of examples to

be chosen from scienceand Engineering areas.

6.2 Derivative as ameasure of

slope

Solving problems connectedwith slope of a curve at a

point: Equations of tangentand normal, angle between

curves.

Comparing results ofAnalytical geometry with

the ones derived.

6.3 Maxima andminima Solving problems related to:Increasing and decreasingfunctions: Stationary values:

local and global maxima andminima; point of inflexion

Graphical approachwherever possible withstress on applications to

Science and Engineering.

6.4 Mean Value

Theorems

Statements of Rolle s

Theorem, Lagrange s MeanValue Theorem. Taylor s and

Maclaurin s Theorems,Taylor s and Maclaurin sseries Statement and

No formal proofs to be

given. Geometricalinterpretation of Rolle s

and Mean ValueTheorems.

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application of L Hospital ruleto ideterminate forms.

6.5 Errors and

Approximations

Comprehending Absolute,

Relative and Percentageerrors. Computing smallchanges

Illustrations from

Geometry, Trigonometryand Science to beprovided

6.6 Curve tracing Obtaining an idea of theapproximate shape of a curvewithout actually plotting

points.

Use of symmetry, meetson axes, pass-ing throughorigin, real and

imaginary values,extension to infinity

turning points etc.

6.7 Partialderivatives

Handling functions of 2 or 3variables; Chain rule: UsingEuler s Theorem on

homogeneous functions(without proof)

Illustrative problemsfrom Science andEngineering.

7. APPLICATIONS OF INTEGRATION


Periods

7.1 DefiniteIntegral

Identifying Definite Integralas the limit of a sum;Deriving and using

properties of definite integral

Geometricalinterpretation

?/2

?/2

Evaluation of sinmx dxcosmx dx

O

O

7.2 Application ofdefinite integral

Applying to solve problemson I) Area under a curve. ii)

Length of are of a curve, andiii) Surface and volume ofrevolution

Use of ideas of curvetracing in identifying

parts of the curve to beused in the problem

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8. DIIFFERENTIAL EQUATIONS


Periods

8.1 Formation ofDifferential

Equations

Formation of DifferentialEquations; identifying order

& degree; discriminatingbetween general andparticular solutions.

Using Graphs of familiesof curves

8.2 I order :

Variablesseparable.

Applying Method of

separation of variables;Reducing to variables

separable type.

Geometrical

interpretation of results

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8.3 I Order:Homogeneous

Equation

Reducing to the type ofVariables separable by

proper substitution

Emphasis on how thistype can be identified

8.4 I Order: ExactEquations

Ability to identify and solveexact equation by inspection.

Introducing the idea of anIntegrating factor tomake an equation exact.

8.5 I Order: LinearEquations Solving equations of theform y2 + Py=Q where P, Qare functions of x

Explanation for use ofIntegrating factor.

8.6. II Order: Linear

equations withconstant

coefficient

Solving equations of type ay

+ by + cy = 0

(with a, b, c,???? R, a ? o) anday + by + cy = f(x)

f(x) to be restricted to

the formx, x2? emx or sin mx or cos

mx (m??R)

8.7 Applications Geometrical applicationsinvolving slope, tangent

normal etc.: Simple

applications involvingmovement of a particle,

Radioactive decay, Heatconduction, Electric circuits.

Interpretation of simplesolutions such as that of

simple harmonic

equation to the form x = -n2x

9. PROBABILITY DISTRIBUTIONS


9.1 Random

variable

Definition and illustrations.

Discriminating between andworking with discrete andcontinuous random variables

Projecting Random

variable as a real valuedfunction throughexamples.

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9.2 Probability

functions

Definitions and illustrations

for (i) probability massfunction (ii) probability

density function (iii)distribution functions

Verification of properties

through a variety ofexamples.

9.3 MathematicalExpectations

Definition and justificationon properties for discrete and

continuous cases.

Straightforwardapplication of E (X), E

(X2) and Var (X)

9.4 Discretedistributions

Definitions and applicationof Binomial and poisson

distribution

Special attention to theparameters Mean,

Variance and S.D. of thedistribution

9.5 Continuousdistribution

Definition and applicationof Normal distribution.

Properties to be correctedwith the form of Normal

curve and itscharacteristics.

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10. ALGEBRAIC STRUCTURES


Periods

10.1 Group structure Illustrations from Numbersystems, matrices, functions,

transformations. Etc.Justifying main properties of

a group and applying them insimple problems. Identifyingorder of a group and order of

a group element.Definition and examples of a

cyclic group

Varied examples to bechosen

Subgroup not to betreated

Non-examples also to begiven

10.2 Rings. IntegralDomains and

Fields

Illustrating structure throughexamples from Number

system only. (No theorems

are to be proved)

Use of different numbersystems to bring out the

differences among

various structures

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COMMERCE STREAM

1.SYSTEMS OF EQUATIONS


1.1 Systems of

linear equations

Presentation in Matrix form,

Computing the rank ofmatrix and determining casesof

(i) a unique solution(ii) a set of solutions(iii) no solution

Discriminating betweenInconsistent and

dependent equations

Sets of simultaneous

equations of at most threevariables only to bepresented.

Graphical interpretationwherever possible.

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1.2 Methods ofsolution

Computing the uniquesolution of a system of

equations, when it exists, by(i) Cramer s Rule and (ii)Inverse matrix method

-Do-

2. APPLICATIONS OF MATRICES


Periods

2.1 StoringInformation

Using matrices to storeinformation.Applying matrix algebra to

Variety in examples to beadoptedRelation matrices, Route

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manipulate such matrices. matrices, and Probabilitymatrices are also to be

used for illustration

2.2 Input-OutputAnalysis

Comprehension of themeaning and basicassumptions; framing and

studying Transaction table;verification of viability of an

input-output system.

Hawkins-Simonsviability conditions to bestated (without proof)

and used.

2.3 Transitionmatrices for

market share

Interpreting ProbabilityTransition matrices and

using them.

Multiplication ofProbability Transition

Matrices used forforecasting.

3. ANALYTICAL GEOMETRY


Periods

3.1 Definition ofa conic

Derivation ofthe standardequations

Focus-directrix definitionUsing it to derive the equation

of a conic in general;Equation of Parabola, Ellipse,Hyperbola and Rectangle

hyperbola

Using the illustration of a double cone to explain

the idea of conic

3. 2 StandardEquations Derivation of the standardequation of Parabola, Ellipse,

Hyperbola and RectangleHyperbola

Training in the skill offinding the foic,

directrices, eccentricity,latus-recta etc. when the

standard equation isgiven

3.3 Tracing theconics

Introduction to tracing ofcurves

Tracing of Parabola, Ellipseand Hyperbola in their

standard form.

Appropirate graphicalillustrations to be given

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4. SEQENCES AND SERIES


4.1 Progressions

and Numbersums

Recall of AP, GP and HP and

formulae for ? n,?? n2,?? n3

Geometrical illustrations

to be given whereverpossible

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4.2 Application in Commerce

Working with concepts of(i) Discounting

(ii) Annuities &Sinking funds,

(iii) Interest paid

continuously

(iv) Present Value andInvestmentAnalysis

Use of information fromstandard financial

institutions to be used forillustration.

5. APPLICATIONS OF DIFFERENTIATION


Periods

5.1 Function inEconomicsand

Commerce

Identifying and manipulatingsupply, Demand, Cost,Revenue, Production and

Elasticity functions.Interpreting Market

Equilibrium

Detailed exposition ofdependent andindependent variables in

the case of each function

5.2 Derivative asa ratemeasurer

Rate of change of quantities,interpretation solving problemsprogrammes involving them.

Majority of examples tobe chosen fromCommerce and

Economics.

5.3 Derivative as

a measure ofslope

Solving problems connected

with: Slope of a curve at apoint. Equations of tangentand normal

Comparing results of

Analytical geometry withthe once derived

5.4 Maxima and

Minima

Solving problems related to:

Intereasing and decreasingfunctions. Stationary values;Local and global maxima and

minima; points of inflexion

Graphical approach

wherever possible withstress on applications toCommerce and

Evonomics

5.5 Application ofMaxima and

Minima

Solving problems on ProfitMaximisationd. Inventory

Control and Economics Order

Quantity

Attention to be drawn tothe constraints in each

such problems

5.6 Partialderivatives

Handling functions of 2 or 3variables. Using Euler s

Theorem (without proof)

Illustrative problemsfrom Commerce and

Economics

5.7 Application ofpartial

Derivatives

Production function of twovariables , Marginal

productivities of Labour andCapital, Partial Elasticities of

Demand.

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5.8 Errors andApplications

Comprehending Absolute,Relative and Percentage errors.

Computing small changes,

dy

Use of concept: ? ?y = dx

? ?x

6. APPLICATIONS OF INTEGRATION


6.1 Definite

Integral

Identifying Definite Integral as

the limit of a sum; Derivingand using properties of definite

integrals

Geometrical

interpretation; statementof Fundamental theorem

of Integral Calculus

6.2 Area Under aCurve

Applying Definite integral tosolve problems on Area under

a curve.

Use of ideas of curvetracing in identifying

parts of the curve to be

used in the problem6.3 Applications

of Definite

IntegralComputing

Consumer sSurplus andProducer s

surplus

scanning Total Inventory carryingCost =

He?ToI(x)dx

Where I (x) is

inventory on hand and

He is unit holding cost.

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7. DIFFERNTIAL EQUATIONS


Periods

7.1 Formula of

DifferentialEquations

Formation identifying order &

degree discriminating generaland particular solution.

Using Graphs of families

of curves

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7.2 I Order:Variables

separable.

Appling Method of separationof variables; Reducing to

Variables separable type.

Geometricalinterpretation of results

7.3 I Order:Homogeneous

Equations.

Reducing to the types ofVariables separable by proper

substitution

Emphasis on how thistype can be identified.

7.4 I Order: ExactEquations

Ability to identify and solveexact equations by inspection

Introducing the idea of anIntegrating factor to

make an equation exact.

7.5 I Order:LinearEquations

Solving equations of the formy + Py = Q where P,Q arefunctions of x

Explanation for use ofIntegral factor to make anequation exact

7.6 II Order:

Linearequations

with constant

Solving equations of type ay

+ by +cy = 0 (with a, b, c ? R,

a?? o) ay + by +cy =f(x)

f(x) to be restricted to the

form x or x2 orexponential form.

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coefficients

7.7 Applications Solving Models involving

Investment, Price adjustment,Spread of disease etc.,

Usual relationships

involving cost,Production etc. to besolved as illustrations

8. PROBABILITY DISTRIBUTIONS


Periods

8.1 Randomvariable

Definition and illustrations.Discriminating between and

working with discrete andcontinuous random variables

Projecting Randomvariable as a real valued

function throughexamples.(Most illustrations in this

topic to be fromCommerce and

Economics)

8.2 Probabiltyfunctions

Definitions and illustrations for(i) probability mass function(ii) probability density function

(iii) distribution function

Verification of propertiesthrough a variety ofexamples.

8.3 Mathematical

Expectation

Definition and justification of

properties for discrete ndcontinuous cases

Straight forward

application of E (X),E (X2) and Vary (X)

8.4 Discretedistribution s

Definitions and application ofBinomial and Poisson

distributions.

Special attention to theparameters mean,

Variance and S.D of thedistributions.

8.5 Continuousdistributions

Definition and application ofNormal distribution

Properties to becorrelated with the form

of Normal curve and itscharacteristics.

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9. SAMPLING TECHNIQUES


9.1 Concept ofSampling

Definition andtypes

Classifying as RandomStratified, systematic. Multi-

stage Also as Non-random:purposive, Quota, Cluster &

Sequential

Simulation applyingMonte-Carlo method and

Random Numbers. 25

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9.2 Errors Discriminating sampling andnon-sampling errors

9.3 Sampling

distributions

Illustrating with Distributions

of sample Mean and SampleProportions.Computing Standard Error in

simple cases

Central limit theorem to

be stand and explainedwithout proof.

9.4 Estimation Meaning of Statisticalestimation

Computing confidenceintervals

Both point and intervalestimation to be

illustrated

9.5 Hypothesis

testing

Identifying levels significance

Determining critical region

Statistical inference to be

illustrated in very simplecases.

9.6 Qualitycontrol charts

Classifying causes forvariation in the quality of

product into those (i) of chance

and (ii) assignable DefiningProcess control & Productcontrol

Presentation of techniquefor drawing a control

chart explaining its

underline principles.

10. FORCASING TECHNIES & DECISION THEORY


Periods

10.1 LinearProgramming

Dealing with objectivefunction with not more than 3

constraints and 2 variables

To be illustrated throughgraphical approach only

10.2 Correlation &

Regression

Applying method of least

squared to perform curvefitting

Explaining estimates

through the concept ofapproach curve of best fit

103. Time Seriesand

determinationof trend

Identifying differentcomponents of Time series

Applying (i) Free and (ii)Semi-average (iii) Movingaverage & (iv) Least squares

methods.

Graphical illustration tobe provided for

explanation.

10.4 IndexNumbers

Use of formulae of (i)Laspeyre, (ii) Paasche, and (iii)

Fisher. Testing Reversal tests

to be satisfied by an indexnumber

(i) Aggregate expendituremethod & (ii) Family

budget method to

compute Cost of livingindex

10.5 Decision

Theory

Identifying basic criteria for

making decision: EMV, Pay-offs, EOL, Using Maximin,

and Minimax and Baye sPrinciples

Role of Decision trees to

be highlighted anddecision diagram to be

illustrated

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