COURSE STRUCTURE

Class XI

One Paper Three Hours Max Marks. 100

Units Marks

I. SETS AND FUNCTIONS 29

II. ALGEBRA 37

III. COORDINATE GEOMETRY 13

IV. CALCULUS 06

V. MATHEMATICAL REASONING 03

VI. STATISTICS AND PROBABILITY 12

100

61

UNIT-I: SETS AND FUNCTIONS

1. Sets : (12) Periods

Sets and their representations. Empty set. Finite & Infinite sets. Equal sets.Subsets. Subsets

of the set of real numbers especially intervals (with notations). Power set. Universal set.

Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set.

2. Relations & Functions: (14) Periods

Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of

two finite sets. Cartesian product of the reals with itself (upto R x R x R). Definition of

relation, pictorial diagrams, domain. codomain and range of a relation. Function as a

special kind of relation from one set to another. Pictorial representation of a

function, domain, co-domain & range of a function. Real valued function of the real variable,

domain and range of these functions, constant, identity, polynomial, rational, modulus,

signum and greatest integer functions with their graphs. Sum, difference, product and

quotients of functions.

3. Trigonometric Functions: (18) Periods

Positive and negative angles. Measuring angles in radians & in degrees and conversion

from one measure to another. Definition of trigonometric functions with the help of

unit circle. Truth of the identity sin2x + cos2x=1, for all x. Signs of trigonometric

functions and sketch of their graphs. Expressing sin (x+y) and cos (x+y) in terms of

sinx, siny, cosx & cosy. Deducing the identities like the following:

Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric

equations of the type sinθ = sin α, cosθ = cos α and tanθ = tan α.

UNIT-II: ALGEBRA

1. Principle of Mathematical Induction: (06) Periods

Processes of the proof by induction, motivating the application of the method by looking

at natural numbers as the least inductive subset of real numbers. The principle of

mathematical induction and simple applications.

2. Complex Numbers and Quadratic Equations: (10) Periods

62

Need for complex numbers, especially , to be motivated by inability to solve every

quadratic equation. Brief description of algebraic properties of complex numbers. Argand

plane and polar representation of complex numbers. Statement of Fundamental Theorem

of Algebra, solution of quadratic equations in the complex number system.

3. Linear Inequalities: (10) Periods

Linear inequalities. Algebraic solutions of linear inequalities in one variable and their

representation on the number line. Graphical solution of linear inequalities in two variables.

Solution of system of linear inequalities in two variables- graphically.

4. Permutations & Combinations: (12) Periods

Fundamental principle of counting. Factorial n. (n!)Permutations and combinations,

derivation of formulae and their connections, simple applications.

5. Binomial Theorem: (08) Periods

History, statement and proof of the binomial theorem for positive integral indices. Pascal's

triangle, General and middle term in binomial expansion, simple applications.

6. Sequence and Series: (10) Periods

Sequence and Series. Arithmetic progression (A. P.). arithmetic mean (A.M.) Geometric

progression (G.P.), general term of a G.P., sum of n terms of a G.P., geometric mean

(G.M.), relation between A.M. and G.M. Sum to n terms of the special series Σn, Σn2 and

Σn3.

UNIT-III: COORDINATE GEOMETRY

1. Straight Lines: (09) Periods

Brief recall of 2D from earlier classes. Slope of a line and angle between two lines. Various

forms of equations of a line: parallel to axes, point-slope form, slope-intercept form, twopoint

form, intercepts form and normal form. General equation of a line. Distance of a

point from a line.

2. Conic Sections: (12) Periods

Sections of a cone: circle, ellipse, parabola, hyperbola, a point, a straight line and pair of

intersecting lines as a degenerated case of a conic section. Standard equations and simple

properties of parabola, ellipse and hyperbola. Standard equation of a circle.

3. Introduction to Three -dimensional Geometry (08) Periods

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point.

Distance between two points and section formula.

UNIT-IV: CALCULUS

1. Limits and Derivatives: (18) Periods

63

Derivative introduced as rate of change both as that of distance function and geometrically,

intuitive idea of limit. Definition of derivative, relate it to slope of tangent of the curve,

derivative of sum, difference, product and quotient of functions. Derivatives of polynomial

and trigonometric functions.

UNIT-V: MATHEMATICAL REASONING

1. Mathematical Reasoning: (08) Periods

Mathematically acceptable statements. Connecting words/ phrases - consolidating the

understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or",

"implied by", "and", "or", "there exists" and their use through variety of examples related to

real life and Mathematics. Validating the statements involving the connecting wordsdifference

between contradiction, converse and contrapositive.

UNIT-VI: STATISTICS & PROBABILITY

1. Statistics: (10) Periods

Measure of dispersion; mean deviation, variance and standard deviation of ungrouped/grouped

data. Analysis of frequency distributions with equal means but different variances.

2. Probability: (10) Periods

Random experiments: outcomes, sample spaces (set representation). Events: occurrence

of events, 'not', 'and' and 'or' events, exhaustive events, mutually exclusive events Axiomatic

(set theoretic) probability, connections with the theories of earlier classes. Probability of

an event, probability of 'not', 'and' & 'or' events.

CLASS XII

One Paper Three Hours Marks: 100

Units Marks

I. RELATIONS AND FUNCTIONS 10

II. ALGEBRA 13

III. CALCULUS 44

IV. VECTORS AND THREE - DIMENSIONAL GEOMETRY 17

V. LINEAR PROGRAMMING 06

VI. PROBABILITY 10

Total 100

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 properties 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

multiplication of matrices, simple properties of addition, multiplication and scalar

multiplication. Non-commutativity 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.

65

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 variaty of functions by

substitution, by partial fractions and by parts, only simple integrals of the type

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,

homogeneous differential equations of first order and first degree. Solutions of

linear differential equation of the type:

+ py = q, where p and q 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, optimal 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.

## Wednesday, December 10, 2008

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