Further Reduced ISC Class 12 Maths Syllabus 2020-21

ISC Class 12 Maths Syllabus 2020-21MATHEMATICS (860)CLASS XIIThere will be two papers in the subject:Paper I : Theory (3 hours) 80 marksPaper II: Project Work 20 marksPAPER I (THEORY) – 80 MarksThe syllabus is divided into three sections A, B and C.Section A is compulsory for all candidates. Candidates will have a choice of attempting questions fromEITHER Section B OR Section C.There will be one paper of three hours duration of 80 marks.Section A (65 Marks): Candidates will be required to attempt all questions. Internal choice will be provided intwo questions of two marks, two questions of four marks and two questions of six marks each.Section B/ Section C (15 Marks): Candidates will be required to attempt all questions EITHER from Section Bor Section C. Internal choice will be provided in one question of two marks and one question of four marks.DISTRIBUTION OF MARKS FOR THE THEORY PAPERS.No.UNITSECTION A: 65 MARKSTOTAL WEIGHTAGE1.2.Relations and FunctionsAlgebra10 Marks10 Marks3.4.CalculusProbability32 Marks13 MarksSECTION B: 15 MARKS5.6.7.8.9.10.VectorsThree - Dimensional GeometryApplications of IntegralsORSECTION C: 15 MARKSApplication of CalculusLinear RegressionLinear ProgrammingTOTAL15 Marks6 Marks4 Marks5 Marks6 Marks4 Marks80 Marks

SECTION A( x 1 y y 1 x )cos x cos y cos ( xy 1 y 1 x )sin-1 x sin-1 y sin1. Relations and Functions(i) Types of relations: reflexive, symmetric,transitive and equivalence relations. One toone and onto functions. -1Relation on a set A-Identity relation, empty relation,universal relation.--1tan-1 x tan-1 y tan-1-Types of Relations: reflexive,symmetric,transitiveandequivalence relation.222x y, xy 11 xyx y, xy 11 xyFormulae for 2sin-1x, 2cos-1x, 2tan-1x,3tan-1x etc. and application of theseformulae.2. Algebra Functions:-2similarly tan-1 x tan-1 y tan-1Relations as:--1-1Matrices and Determinants(i) MatricesConcept, notation, order, equality, types ofmatrices, zero and identity matrix, transposeof a matrix, symmetric and skew symmetricmatrices. Operation on matrices: Additionand multiplication and multiplication with ascalar. Simple properties of addition,multiplication and scalar multiplication. Noncommutativity of multiplication of matrices.Invertible matrices (here all matrices willhave real entries).One to one and onto functions.(ii) Inverse Trigonometric FunctionsDefinition, domain, range, principal valuebranch.- Principal values.- sin-1x, cos-1x, tan-1x etc. and their graphs.x.- sin-1x cos 1 1 x 2 tan 11 x21π- sin-1x cosec 1 ; sin-1x cos-1x and2xsimilar relations for cot-1x, tan-1x, etc.(ii) DeterminantsDeterminant of a square matrix (up to 3 x 3matrices), properties of determinants,minors, co-factors. Adjoint and inverse ofa square matrix. Solving system of linearequations in two or three variables (havingunique solution) using inverse of a matrix.2

- Types of matrices (m n; m, n 3),order; Identity matrix, Diagonal matrix.Symmetric, Skew symmetric.Operation – addition, subtraction,multiplication of a matrix with scalar,multiplicationoftwomatrices(the compatibility).1 11 2E.g. 0 2 AB( say ) but BA is2 21 1-not possible.Singular and non-singular matrices.Existence of two non-zero matriceswhose product is a zero matrix.-Inverse (2 2, 3 3) A 1 Differentiation- Derivativesoftrigonometricfunctions.- Derivatives of exponential functions.- Derivatives of logarithmic functions.- Derivatives of inverse trigonometricfunctions - differentiation by meansof substitution.- Derivatives of implicit functions andchain rule.- e for composite functions.- Derivatives of Parametric functions.- Differentiation of a function withrespect to another function e.g.differentiation of sinx3 with respectto x3.- LogarithmicDifferentiationxFinding dy/dx when y x x .- Successive differentiation up to 2ndorder.NOTE :Derivatives of composite functionsusing chain rule.AdjAAMartin’s Rule (i.e. using matrices)a1x b1y c1z d1a2x b2y c2z d2a3x b3y c3z d3a 1 b 1 c1A a 2 b2 c 2a 3 b3 c3d1B d2d3 xX yz-(ii) Applications of DerivativesAX B X A 1 B L' Hospital's asing functions, tangentsand normals, maxima and minima (firstderivative test motivated geometrically andsecond derivative test given as a provabletool). Simple problems (that illustrate basicprinciples and understanding of the subject aswell as real-life situations).Problems based on .3. Calculus(i) Differentiation, Derivative of compositefunctions, chain rule, derivatives of inversetrigonometric functions, derivative ofimplicit functions. Concept of exponentialand logarithmic functions.Derivatives of logarithmic and rivative of functions expressed inparametric forms. Second order derivatives.3 Equation of Tangent and Normal Increasing and decreasing functions. Maxima and minima.-Stationary/turning points.-Absolute maxima/minima-local maxima/minima-First derivatives test and secondderivatives test

-Application problemsmaxima and minima.basedonb f ( x)dx f ( x)dxa(iii) IntegralsaIntegration asdifferentiation.theinverse-Anti-derivatives of polynomials andfunctions (ax b)n , sinx, cosx, sec2x,cosec2x etc .bIntegrals of the type sin2x, sin3x,sin4x, cos2x, cos3x, cos4x.-Integration of 1/x, ex.-Integration by substitution.00a2 f ( x)dx,if f f ( x)dx a ais an even function00,if f is an odd function(iv) Differential EquationsDefinition, order and degree, general andparticular solutions of a differentialequation. Formation of differential equationwhose general solution is given. Solutionof differential equations by method ofseparation of variables solutions ofhomogeneous differential equations of firstorder and first degree. Solutions of lineardy py q,differential equation of the type:dxwhere p and q are functions of x ordx px q, where p and q areconstants.dyfunctions of y or constants.When degree of f (x) degree of g(x),e.g.x2 13x 1 1 22x 3x 2x 3x 2- Differential equations, order and degree.- Formation of differential equation byeliminating arbitrary constant(s).- Solution of differential equations.- Variable separable.- Homogeneous equations.Definite Integral- Fundamental theorem of calculus(without proof)- Properties of definite integrals.- Problems based on the followingproperties of definite integrals are tobe covered.aaa2a2 f ( x)dx, if f (2a x) f ( x) f ( x)dx 000,f (2a x) f ( x)- Integration by parts. aa f ( x)dx f (a x)dxIntegration of tanx, cotx, secx,cosecx.bcba- Integrals of the type f ' (x)[f (x)]n,f ′( x).f ( x) a f ( x)dx f (a b x)dxof-bwhere a c bIndefinite integral-c f ( x)dx f ( x)dx f ( x)dxFundamental Theorem of Calculus (withoutproof). Basic properties of definite integralsand evaluation of definite tiation. Integration of a variety offunctions by substitution, by partial fractionsand by parts, Evaluation of simple integralsof the following types and problems basedon them. a- Linear formbdy Py Q where P and Qdxare functions of x only. Similarly, fordx/dy.NOTE 1: The second order differentialequations are excluded.f ( x)dx f (t )dta4

4. ProbabilityConditional probability, multiplication theoremon probability, independent events, totalprobability, Bayes’ theorem.- Independentanddependenteventsconditional events.- Laws of Probability, addition ty.- Theorem of Total Probability.- Baye’s theorem.-Equation of x-axis, y-axis, z axis and linesparallel to them.-Equation of xy - plane, yz – plane,zx – plane.-Direction cosines, direction ratios.Angle between two lines in terms of directioncosines /direction ratios.Condition for lines to be perpendicular/parallel. SECTION B5. VectorsVectors and scalars, magnitude and directionof a vector. Direction cosines and directionratios of a vector. Types of vectors (equal, unit,zero, parallel and collinear vectors), positionvector of a point, negative of a vector,components of a vector, addition of vectors,multiplication of a vector by a scalar. Definition,Geometrical Interpretation, properties andapplication of scalar (dot) product of vectors,vector (cross) product of vectors.- As directed line segments.- Magnitude and direction of a vector.- Types: equal vectors, unit vectors, zerovector.- Position vector.- Components of a vector.- Vectors in two and three dimensions.- iˆ, ˆj , kˆ as unit vectors along the x, y andthe z axes; expressing a vector in terms of theunit vectors.- Operations: Sum and Difference of vectors;scalar multiplication of a vector.- Scalar (dot) product of vectors and itsgeometrical significance.- Cross product - its properties - area of atriangle, area of parallelogram, collinearvectors. Lines- Cartesian and vector equations of a linethrough one and two points.- Coplanar and skew lines.- Conditions for intersection of two lines.- Distance of a point from a line.Planes- Cartesian and vector equation of aplane.- Direction ratios of the normal to theplane.- One point form.- Normal form.- Intercept form.- Distance of a point from a plane.- Intersection of the line and plane.7. Application of IntegralsApplication in finding the area bounded b ysimple curves and coordinate axes. Areaenclosed between two curves.-Application of definite integrals - areabounded by curves, lines and coordinate axesis required to be covered.-Simple curves: lines,polynomial functions.parabolasandSECTION C8. Application of CalculusApplication of Calculus in Commerce andEconomics in the following:- Cost function,- average cost,- marginal cost and its interpretation- demand function,- revenue function,NOTE: Proofs of geometrical theorems by usingVector algebra are excluded.6. Three - dimensional GeometryDirection cosines and direction ratios of a linejoining two points. Cartesian equation and vectorequation of a line, coplanar and skew lines.Cartesian and vector equation of a plane.Distance of a point from a plane.5