Unit-I: Sets and Functions
1.Sets
Sets and their representations, Empty set, Finite and Infinite sets, Equal sets, Subsets, Subsets of a set of real numbers especially intervals (with notations). Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement.
2.Relations & Functions
Ordered pairs. Cartesian product of sets. Number of elements in the Cartesian product of two finite sets. Cartesian product of the set of reals with itself (upto R x R x R).Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.
3.Trigonometric Functions
Positive and negative angles. Measuring angles in radians and 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. Domain and range of trigonometric functions and their graphs. Expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications. Deducing identities like the following:
tan(x ± y) = tan x ± tan y 1 ∓ tan x tan y
1 , cot(x ± y) =1 cot x cot y ∓ 1
cot y ± cot x sinα ± sinβ = 2sin
2 (α ± β)cos 2 (α ∓ β) 1 cosα + cosβ = 2cos
2 1 (α + β)cos 2 1 (α − β) 1 𝑐𝑜𝑠𝛼 − 𝑐𝑜𝑠𝛽 = −2𝑠𝑖𝑛
(𝛼 + 𝛽)𝑠𝑖𝑛 2 (𝛼 − 𝛽) 2
Identities related to sin2x, cos2x, tan2 x, sin3x, cos3x and tan3x.
Unit-II: Algebra
1. Complex Numbers and Quadratic Equations
Need for complex numbers, especially√−1, to be motivated by inability to solve some of the quadratic equations. Algebraic properties of complex numbers. Argand plane
2.Linear Inequalities
Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line.
3.Permutations and Combinations
Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for nPr and nCr and their connections, simple applications.
4. Binomial Theorem
Historical perspective, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle, simple applications.
5.Sequence and Series
Sequence and Series. Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between
A.M. and G.M.
Unit-II: Algebra
Complex Numbers and Quadratic Equations
Need for complex numbers, especially√−1, to be motivated by inability to solve some of the quadratic equations. Algebraic properties of complex numbers. Argand plane
2.Linear Inequalities
Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line.
3.Permutations and Combinations
Fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for nPr and nCr and their connections, simple applications.
4.Binomial Theorem
Historical perspective, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle, simple applications.
5.Sequence and Series
Sequence and Series. Arithmetic Mean (A.M.) Geometric Progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between
A.M. and G.M.
Unit-III: Coordinate Geometry
1. Straight Lines
Brief recall of two dimensional geometry from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point -slope form, slope-intercept form, two-point form, intercept form, Distance of a point from a line.
2.Conic Sections
Sections of a cone: circles, ellipse, parabola, hyperbola, a point, a straight line and a 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
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points.
Unit-IV: Calculus
1. Limits and Derivatives
Derivative introduced as rate of change both as that of distance function and geometrically. Intuitive idea of limit. Limits of polynomials and rational functions trigonometric, exponential and logarithmic functions. Definition of derivative relate it to scope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.
Unit-V Statistics and Probability
1. Statistics
Measures of Dispersion: Range, Mean deviation, variance and standard deviation of ungrouped/grouped data.
2.Probability
Events; occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events.
Unit-I: Relations and Functions
- Relations and Functions
Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.
- Inverse Trigonometric Functions 15 Periods Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Unit-II: Algebra
1.Matrices
Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations 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-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
2.Determinants
Determinant of a square matrix (up to 3 x 3 matrices), 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 unique solution) using inverse of a matrix.
Unit-III: Calculus
1. Continuity and Differentiable
Continuity and differentiability, chain rule, derivative of inverse trigonometric functions,
𝑙𝑖𝑘𝑒 sin−1 𝑥 , cos−1 𝑥 and tan−1 𝑥, derivative of implicit functions. Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.
2. Applications of Derivatives
- Applications of derivatives: rate of change of quantities, increasing/decreasing functions, 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
Integration as inverse process of differentiation. 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.
dx dx dx dx dx
∫ x2 ± a2, ∫ √x2 ± a2 , ∫ √a2 − x2 , ∫ ax2 + bx + c , ∫ √ax2 + 𝑏𝑥 + 𝑐
px + q
∫ ax2 + bx + c px + q
dx, ∫ dx, ∫ √a2 ± x2 dx, ∫ √x2 − a2 dx
√ax2+bx + c
∫ √𝑎𝑥2 + 𝑏𝑥 + 𝑐 𝑑𝑥,
Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.
1.Applications of the Integrals
Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)
2.Differential Equations
Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:
dy dx d𝑥 d𝑦
+ py = q, where p and q are functions of x or constants.
+ px = q, where p and q are functions of y or constants.
Unit-IV: Vectors and Three-Dimensional Geometry
- Vectors 15 Periods
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.
- Three – dimensional Geometry 15 Periods Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.
Unit-V: Linear Programming
1. Linear Programming
Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).
Unit-VI: Probability
1. Probability
Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.