HIMACHAL PRADESH BOARD OF SCHOOL EDUCATION
. Syllabus of plus two (12th) examination
3. MATHEMATICS
The syllabus in the subject of Mathematics has undergone changes from time to time in accordance with growth of the subject and emerging needs of society. Senior Secondary stage is a launching stage from where the students go either for higher academic education in Mathematics or for professional courses like engineering, physical and Bioscience, commerce or computer applications. The present revised syllabus has been designed in accordance with National Curriculum Frame Work 2005 and as per guidelines given in Focus Group on Teaching of Mathematics 2005 which is to meet the emerging needs of all categories of students. Motivating the topics from real life situations and other subject areas, greater emphasis has been laid on application of various concepts. OBJECTIVES The broad objectives of teaching Mathematics at senior school stage intend to help the pupil : c to acquire knowledge and critical understanding particularly by way of motivation of visualization of basic facts, concepts, terms, principles and symbols and mastery of underlying processes and skills. c to feel the flow of reasons while proving a result or solving a problem. c to apply the knowledge and skills acquired to solve problems and wherever possible, by more than one method. c to develop positive attitude to think, analyze and articulate logically. c to develop interest in the subject by participating in related competitions. c to acquaint students with different aspects of mathematics used in daily life. c to develop an interest in students to study mathematics as a discipline. c to develop awareness of the need for national integration, protection of environment observance of small family norms, removal of social barriers, elimination of sex biases. c to develop reverence and respect towards great Mathematicians for their contribution to the field of Mathematics. 3 9 ONE PAPER THREE HOURS M.M. 85 Units Marks I. RELATIONS AND FUNCTIONS 8 II. ALGEBRA 11 III. CALCULUS 38 IV. VECTORS AND THREEDIMENSION GEOMETRY 15 V. LINEAR PROGRAMMING 5 VI. PROBABILITY 8 TOTAL 85 UNIT I. RELATIONS AND FUNCTIONS 1. Relations and Functions : (10) Periods Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations. 2. Inverse Trigonometric Functions : (12) Periods Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary propoerties of inverse trigonometric functions. UNIT II. ALGEBRA 1. Matrices : (18) Periods Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multipliocation of matrices, simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of nonzero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries). 2. Determinants : (20) Periods Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, cofactors 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 unique solution) using inverse of a matrix. 4 0 UNIT III. CALCULUS 1. Continuity and Differentiability : (18) Periods Continuity and differentiability, derivative of composite functions, chain rule. derivatives of inverse trigonometric functions, derivative of implicit function. Concept of exponential and logarithmic functions and their derivative. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations. 2. Applications of Derivatives : (10) Periods Applications of derivatives : rate of change, increasing/decreasing functions, tangents & normals, 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 reallife situations). 3. Integrals : (20) Periods) Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type 2 2 2 2 2 2 2 2 2 dx , dx , dx , dx , dx x a x a a – x ax bx c ax bx c to be evaluated. Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals. 4. Applications of the Integrals : (10) Periods Applications in finding the area under simple curves, especially lines, areas of circles/parabolas/ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable). 5. Differential Equations : (10) Periods 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, homogenous differential equations of first 4 1 order and first degree. Solutions of linear differential equation of the type : dx dx + p(x) y = q(x), where p(x) and q(x) are functions of x, UNIT IV. VECTORS AND THREEDIMENSIONAL GEOMETRY 1. Vectors : (12) Periods Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors. 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. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors. 2. Three  Dimensional Geometry : (12) Periods Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes. (iii) a line and a plane. Distance of a point from a plane. UNIT V. LINEAR PROGRAMMING 1. Linear Programming : (12) Periods Introduction, definition of related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optional feasible solutions (up to three nontrivial constraints). UNIT VI. PROBABILITY 1. Probability : (18) Periods Multiplication theorem on probability. Conditional probability, independent events, total probability, Baye’s theorem, Random variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli) trials and Binomial distribution. PRESCRIBED BOOKS : 1. Mathematics PartI Published by H.P. Board of School Education Dharamshala. 2. Mathematics PartII Published by H.P. Board of School Education Dharamshala.
