1) ALGEBRA : a) Functions: Types of functions – Definitions – Inverse functions and Theorems – Domain, Range, Inverse of real valued functions.
b) Mathematical Induction : Principle of Mathematical Induction & Theorems – Applications of Mathematical Induction – Problems on divisibility.
c) Matrices: Types of matrices – Scalar multiple of a matrix and multiplication of matrices – Transpose of a matrix – Determinants – Adjoint and Inverse of a matrix- Consistency and inconsistency of Equations- Rank of a matrix – Solution of simultaneous linear equations.
d) Complex Numbers: Complex number as an ordered pair of real numbers- fundamental operations – Representation of complex numbers in the form a+ib – Modulus and amplitude of complex numbers –Illustrations – Geometrical and Polar Representation of complex numbers in Argand plane- Argand diagram. e) De Moivre’s Theorem: De Moivre’s theorem- Integral and Rational indices – nth roots of unity- Geometrical Interpretations – Illustrations. f) Quadratic Expressions: Quadratic expressions, equations in one variable – Sign of quadratic expressions – Change in signs – Maximum and minimum values – Quadratic inequations.
g) Theory of Equations: The relation between the roots and coefficients in an equation – Solving the equations when two or more roots of it are
connected by certain relation – Equation with real coefficients, occurrence of complex roots in conjugate pairs and its consequences – Transformation of equations – Reciprocal Equations. h) Permutations and Combinations: Fundamental Principle of counting – linear and circular permutations- Permutations of ‘n’ dissimilar things taken ‘r’ at a time – Permutations when repetitions allowed – Circular permutations – Permutations with constraint repetitions – Combinations-definitions and certain theorems. i) Binomial Theorem: Binomial theorem for positive integral index – Binomial theorem for rational Index (without proof) – Approximations using Binomial theorem. j) Partial fractions: Partial fractions of f(x)/g(x) when g(x) contains non –repeated linear factors – Partial fractions of f(x)/g(x) when g(x) contains repeated and/or non-repeated linear factors – Partial fractions of f(x)/g(x) when g(x) contains irreducible factors.
2) TRIGONOMETRY: a) Trigonometric Ratios upto Transformations : Graphs and Periodicity of Trigonometric functions – Trigonometric ratios and Compound angles – Trigonometric ratios of multiple and sub- multiple angles – Transformations – Sum and Product rules.
b) Trigonometric Equations : General Solution of Trigonometric Equations – Simple Trigonometric Equations – Solutions.
c) Inverse Trigonometric Functions: To reduce a Trigonometric Function into a bijection – Graphs of Inverse Trigonometric Functions – Properties of Inverse Trigonometric Functions.
d) Hyperbolic Functions: Definition of Hyperbolic Function – Graphs – Definition of Inverse Hyperbolic Functions – Graphs – Addition formulae of Hyperbolic Functions.
e) Properties of Triangles: Relation between sides and angles of a Triangle – Sine, Cosine, Tangent and Projection rules – Half angle formulae and areas of a triangle – Incircle and Excircle of a Triangle.
3) VECTOR ALGEBRA: a) Addition of Vectors : Vectors as a triad of real numbers – Classification of vectors – Addition of vectors – Scalar multiplication – Angle between two non zero vectors – Linear combination of vectors – Component of a vector in three dimensions – Vector equations of line and plane including their Cartesian equivalent forms.
b) Product of Vectors : Scalar Product – Geometrical Interpretations – orthogonal projections – Properties of dot product – Expression of dot product in i, j, k system – Angle between two vectors – Geometrical Vector methods – Vector equations of plane in normal form – Angle between two planes – Vector product of two vectors and properties – Vector product in i, j, k system – Vector Areas – Scalar Triple Product – Results.
4) PROBABILITY: a) Measures of Dispersion – Range – Mean deviation – Variance and standard deviation of ungrouped/grouped data – Coefficient of variation and analysis of frequency distribution with equal means but different variances.
b) Probability : Random experiments and events – Classical definition of probability, Axiomatic approach and addition theorem of probability – Independent and dependent events – conditional probability- multiplication theorem and Bayee’s theorem.
c) Random Variables and Probability Distributions: Random Variables – Theoretical discrete distributions – Binomial and Poisson Distributions.
5) COORDINATE GEOMETRY: a) Locus : Definition of locus – Illustrations – To find equations of locus – Problems connected to it.
b) Transformation of Axes : Transformation of axes – Rules, Derivations and Illustrations – Rotation of axes – Derivations – Illustrations.
c) The Straight Line : Revision of fundamental results – Straight line – Normal form – Illustrations – Straight line – Symmetric form – Straight line – Reduction into various forms – Intersection of two Straight Lines – Family of straight lines – Concurrent lines – Condition for Concurrent lines – Angle between two lines – Length of perpendicular from a point to a Line – Distance between two parallel lines – Concurrent lines – properties related to a triangle.
d) Pair of Straight lines: Equations of pair of lines passing through origin – angle between a pair of lines – Condition for perpendicular and coincident lines, bisectors of angles – Pair of bisectors of angles – Pair of lines – second degree general equation – Conditions for parallel lines – distance between them, Point of intersection of pair of lines – Homogenizing a second degree equation with a first degree equation in X and Y.
e) Circle : Equation of circle -standard form-centre and radius of a circle with a given line segment as diameter & equation of circle through three non collinear points – parametric equations of a circle – Position of a point in the plane of a circle – power of a point-definition of tangent-length of tangent – Position of a straight line in the plane of a circle – conditions for a line to be tangent – chord joining two points on a circle – equation of the tangent at a point on the circle- point of contact-equation of normal – Chord of contact – pole and polar-conjugate points and conjugate lines – equation of chord with given middle point – Relative position of two circles- circles touching each other externally, internally- common tangents –centers of similitude- equation of pair of tangents from an external point.
f) System of circles: Angle between two intersecting circles – Radical axis of two circles- properties- Common chord and common tangent of two circles – radical centre – Intersection of a line and a Circle.
g) Parabola: Conic sections –Parabola- equation of parabola in standard form-different forms of parabola- parametric equations – Equations of tangent and normal at a point on the parabola ( Cartesian and parametric) – conditions for straight line to be a tangent.
h) Ellipse: Equation of ellipse in standard form- Parametric equations – Equation of tangent and normal at a point on the ellipse (Cartesian and parametric)- condition for a straight line to be a tangent.
i) Hyperbola: Equation of hyperbola in standard form- Parametric equations – Equations of tangent and normal at a point on the hyperbola (Cartesian and parametric)- conditions for a straight line to be a tangent- Asymptotes.
j) Three Dimensional Coordinates : Coordinates – Section formulae – Centroid of a triangle and tetrahedron.
k) Direction Cosines and Direction Ratios : Direction Cosines – Direction Ratios.
l) Plane : Cartesian equation of Plane – Simple Illustrations.
6) CALCULUS: a) Limits and Continuity: Intervals and neighbourhoods – Limits – Standard Limits – Continuity.
b) Differentiation: Derivative of a function – Elementary Properties – Trigonometric, Inverse Trigonometric, Hyperbolic, Inverse Hyperbolic Function – Derivatives – Methods of Differentiation – Second Order Derivatives.
c) Applications of Derivatives: Errors and approximations – Geometrical Interpretation of a derivative – Equations of tangents and normals – Lengths of tangent, normal, sub tangent and sub normal – Angles between two curves and condition for orthogonality of curves – Derivative as Rate of change – Rolle’s Theorem and Lagrange’s Mean value theorem without proofs and their geometrical interpretation – Increasing and decreasing functions – Maxima and Minima.
d) Integration : Integration as the inverse process of differentiation- Standard forms -properties of integrals – Method of substitution- integration of Algebraic, exponential, logarithmic, trigonometric and inverse trigonometric functions – Integration by parts – Integration- Partial fractions method – Reduction formulae.
e) Definite Integrals: Definite Integral as the limit of sum – Interpretation of Definite Integral as an area – Fundamental theorem of Integral Calculus – Properties – Reduction formulae – Application of Definite integral to areas.
f) Differential equations: Formation of differential equation-Degree and order of an ordinary differential equation – Solving differential equation by i) Variables separable method, ii) Homogeneous differential equation, iii) Non – Homogeneous differential equation, iv) Linear differential equations.