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Class 12 Revised Syllabus for 2023 Board Examination.pdf

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Class 12 Revised Syllabus for 2023 Board Examination.pdf

1. 1. SECTION A 1. Relations and Functions (i) Types of Relations Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions,compositefunctions, inverse ofa function. Binary operations. Relations as: - Relation on a set A. Identity relation, empty relation, universal relation. - Types of Relations: reflexive, symmetric, transitive and equivalence relation. (v)
2. 2. Binary operation: all axioms and properties Functions: As special relations, concept of writing "y is afunction ofx" as y =fx) ypes: 0ne to one, many to one, into, onto. - Real raluedfuncti01. - Domain and range ofafunction. Conditions of invertibility. Composite functions and invertiblefunctions (algebraicfunctions only). (ii) Inverse Trigonometric Functions Definition, domain, range, principal value branch. Graphs of inverso trigonometric functions. Elementary properties of inverse trigonometric tunctions. Principal values. sinx, coslx, tan'x etc. and their graphs. sinx =cos V1-x = tan. v1- - sin'r = cosec; sinx +cosx =and similar relations for cot x, tanl x, etc. sinxtsin'y=sin"(r/1-y2*yv1-2 cosxtcosy=cos (xy¥y1-y2 V1-x2) -1 X*,xy <1 Similarly tan' x + tany == tan xy<1 xy X-y tan' x tan y = tan Z, xy> -1 1+xy Formulae for 2 sin' x, 2 cos-x, 2 tan1x,3 tanl x etc. andapplicationof these formulae. 2. Algebra Matrices and Determinants (i) Matrices Concept, notation, order, equality, typesofmatrices, zero and identitymatrik transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices. Addition and multiplication and multiplication with a scalar Simple properties of addition, multiplication and scalar multiplication Non-commutativity of multiplication of matrices and existence of non-zt ero (vi)
3. 3. matrices whose product is the zero matrix (restrict to square matrices of order upto 3), Conceptofelementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists (here all matrices will have real entries). (i) Determinants Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having nique solution) using inverse of a matrix. - Types ofmatrices (mx n; m, n s 3), order; ldentity matrix, Diagonal matrix. Symmetric, Skew symmetric. Operation - addition, subtraction, multiplication of a matrix with scalar, multiplication of two matrices (the compatibility). AB (say) but BA is not possible. Singular and non-singular matrices. Existence of two non-zero matrices whose product is a zero matrix. Adj A Inverse (2 x2,3 x3) A-1 = 4 A Martin's Rule (i.e., using matrices) a,X +b,y + c,z = d, ax+ b,y + C,Z = a, ax+by + CZ = a b A =2 b2 a3 b C3 41 B= d2X=|y C2 AX =BX=A1 B Problems based on above. Note 1: The conditions for consistency ofequations in two and three variables, using matrices, are to be covered. Note 2: Inverse of a matrix by elementary operations to be covered. Determinants - Order Minors - Cofactors (vii)
4. 4. Expansion Applications of determinants in finding the area of triangle and collinearitu Propcrtics ofdeterminants. Problems based on properties ofdeterminants. 3. Calculus () Continuity, Differentiability and Differentiation. Continuity and difterentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic ditferentiation, derivative of functions expressed in parametric forms Second order derivatives, Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation. Continuity Continuity ofafunction at a point x =a. Continuity ofafunction in an interval. Algebra ofcontinuousfunction. Removable discontinuity. Differentiation Concept ofcontinuityanddifferentiability of |x|, [x], etc. - Derivatives of trigonometricfunctions. Derivatives ofexponentialfunctions. Derivatives oflogarithmicfunctions. Derivatives of inverse trigonometric functions-diferentiation by means of substitution. Derivatives of implicit functions and chain rulefor compositefunctions. Derivatives of Parametricfunctions. Differentiationofafunction with respect toanotherfunction e.g., differentiation ofsinx' with respect to r'. Logarithmic Differentiation-Finding dy/dx when y = r Successive differentiation up to 2nd order. Note 1: Derivatives of composite functions using chain rule. Note 2: Derivatives of determinants to be covered. L'Hospital's theorem form,form,0" from, oo formetc. 0 Rolle's Mean Value Theorem - its geometrics interpretation. .Lagrange's Mean ValueTheorem- its geometrical interpretation. (viii)
5. 5. () Application of Derivatives Application of derivatives: rate of change of bodies, increasing/ decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real- life situations). .Equation of Tangent and Normal Approximation Rate measure Increasing and decreasing functions Maxima and minima Stationary/turning points - Absolute maxima/minima Local maxima/minima - First derivatives test and second derioatives test - Point ofinflexion Application problems based on maxima and minima. (ii) Integrals Integration as inverse process of differentitation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them. Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals. Indefinite integral Integration as the inverse ofdifferentiation. Anti-derivatives of polynomials and functions (ax + b)", sinx, cosx, secx, cosecx, etc. Integrals of the type sin x, sin' x, sint x, cos* x, cos' x, cos* x Integration of1/x, e'. Integration by substitution. Integrals ofthetypef(x)Ifx), flx) Integration oftanx, cotx, secx, cosecx. Integration by parts. (iN)
6. 6. f(X) when degree g(x) -Tntegration using partial functions. Expression oftheform of fx)<degreeofg(x) x+2 A B Eg (r-3)(1+ 1) x-3 x+1 B C +2 (x-2)(r -1) x-1 (x-1) x-2 Ax+B C +3 x+1 ( +3)(x -1) - 1 When degree of flx) 2 degree of g(X), 3x+1 eg 2 +3x+2 2 +1 =1- +3x+2) Integrals of the type: arh dx, J px +9 dx axbx +c dx dx ax + bx +C and Va+rdx,[Vr2-adx, Vax +bx+c dx, J(px+q)Vax* + bx +c dx, integrations reducible to the above forms. dx acos x +bsinX dx dx dx a+bcosx 4 a+bsinx a cosx +bsinx +c r(acosx+bsinx)dx CCOSX+dsinx dx a cos x+bsin x +c 44x, dx t a nxdx, JVcotxdr etc. <
7. 7. Definite integral -Definite integralasa limit of the sum. Fundamental theorem ofcalculus (without proof) Properties ofdefinite integrals. Problems based onthefollowing propertiesofdefinite integrals are to be covered. fdxfdx odx= frdr+ fdx wherea< c<b fxdr =fa+b-xd fxdx a-xdr fCx)dr=2 frdx,iff(2a-x)=ft) 0 iff(2a-x) =-f(x) fxdr= 2fxdx,iff is an even function 0, if f is an odd function (iv) Differential Equations Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equation of first order and first degree. Solutions of linear differential equation of the type:+Py =q, wherep ax dx + px =q, wherep and q are functions dy and q are functions of x or constants. of y or constants. -Differential equations, order and degree. Formation ofdifferential equation by eliminating arbitrary constant(s). - Solution ofdifferential equations. - Variable separable. - Homogeneous equation: Linearform+Py=Q where P and Qarefunctions ofxonly. Similarlyfor dx dx/dy. - Solve problems ofapplication on growth and decay. (xi)
8. 8. Solae population based problems on application ofdifferntialequations Solae problems ofapplication on coordinate geometry. Note 1: Equations reducible to variable separable type are included. Note 2: The second order differential equations are excluded. 4. Probability Conditional probability, multiplication theorem on probability,independ. events, total probability, Bayes' theorem, Random variable and its probabilit distribution, mean and variance of random variable. Repeated independen (Bernoulli) trials and Binomiál distribution. -independent and dependent events, conditionalevents. Laus of Probability, addition theorem, nultiplication theorem, conditional probability. Theorem of Total Probability. Baye's theorem. Theoretical probability distribution, probability distribution function; mean and varianceof random variable, Repeated independent (Bernoulli trials), binominl distribution - its mean and variance. SECTIONNB 5. Vectors Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors. As directed line segments. Magnitude and direction ofa vector. Types: egual vectors, unit vectors, zero vector. - Position vector. - Components of a vector. Vectors in two and three dimensions. i, j,k as unit vectors along the x, y and the z axes; expressing a vector in terms ofthe unit vectors. Operations: Sum and Difference of vectors; scalar multiplicationof a vector. Section formula. (xii)
9. 9. -Triangle inequalities. Scalar (dot) product of vectors and its geometrical significance. Cross product-its properties-area ofatriangle, area ofparallelogram,collinear vectors. Scalar triple product-volume ofaparallelopiped,co-planarity Note: Proofs of geometricaltheorems by using Vector algebra are excluded. 6. Three-dimensional Geometry Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest ditances between two lines. Cartesian and vector equation of a plane. Angle between () two lines, (ii) two planes, (ii) a line and a plane. Distance of a point from a plane. Equation ofx-axis, y-axis, z-axis and lines parallel to them. - Equation ofxy-plane, yz-plane, zx-plane. - Direction cosines, direction ratios. Angle between two lines in terms ofdirection cosines/direction ratios. Condition for linesto be perpendicular/parallel. Lines Cartesian and vector equations ofa line through one and two points. - Coplanar and skew lines. - Conditions for intersection oftwo lines. - Distance ofa pointfrom a line. - Shortest distance between two lines. Note: Symmetric and non-symmetricforms oflinesare required to be covered. Planes - Cartesian and vector equation ofa plane. - Direction ratios ofthe normal to the plane. One pointform. Normal form. - Intercept form. -Distanceofapointfrom a plane. ntersection of the line and plane. Angle betuween two planes, a line and a plane. -Equatiom ofa plane through the intersection of two planes ie., P, +kP, = 0. (xi)
10. 10. 7. Application of lntegrals and coordinate Application in finding the area bounded by simple curves and coord:. axes. Area encloscd between two curves. - Application ofdefinite integrals-area bounded by curves, lines and coordin. axes is rcquired to be covercd. dinates Simple currves: lines, circles/parabolas/ellipses, polynomialsfunctions, functions mod.. dulus function, trigonometric function, exponentialfunctions,logarithmicfuncin SECTIONC S. Application of Calculus Application of Calculus in Commerce and Economics in the following Costfunction aterage cost, marginal cost and its interpretation demandfunction revenuefunction marginal revenuefunction and its interpretation, Profitfunction and breakeven point. Rough sketching of the following curves: AR, MR, R, C, AC, MC and their mathematical interpretation using the concept of maxima & minima and increasing-decreasing functions. Self-explanatory. Note: Application involving differentiation, integration, increasing and decreasing function and maxima and minima to be covered. 9. Linear Regression - Lines ofregression ofx and y on x. Scatter diagrams The method ofleast squares. - Lines ofbestfit. Regression coefficient ofx on y and y on x. - bb = r, 0sb, x b, s1 Identification ofregression equations Angle between regression line and properties ofregression lines. - Estimation ofthe value ofone variable using the value ofother variablefrom appropriate line of regresssion. rom Self-explanatory. (NIY)
11. 11. 10. Linear Programming Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problens, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints). Introduction, definition of related terminology such as constraints, objective function, optimization, adoantages of linear programming; limitations of linear programnming; application areas of linear programming ; different types of linear programming (L.P.) problems, mathematicalformulation ofL.Pproblems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimumfeasible solution. tobe: ansfortation (roblem is excuded X