# Syllabus of plus two (12th) examination- MATHEMATICS subject

### HIMACHAL PRADESH BOARD OF SCHOOL EDUCATION

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Syllabus of plus two (12th) examination

3. MATHEMATICS

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.

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ONE PAPER THREE HOURS M.M. 85

Units Marks

I. RELATIONS AND FUNCTIONS 8

II. ALGEBRA 11

III. CALCULUS 38

IV. VECTORS AND THREE-DIMENSION 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 non-zero

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.

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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 real-life

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

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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 THREE-DIMENSIONAL 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 non-trivial 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 Part-I Published by H.P. Board of School Education

Dharamshala.

2. Mathematics Part-II Published by H.P. Board of School Education

Dharamshala.