A.P., G.P., H.P.:Summation of first n-terms
Logarithms: Definition; General properties; Change of base.
Arithmetic/Geometric series, A.M., G.M. and their relation; Infinite G.P. series and its sum.
Solution of quadratic equation in complex number system.
Complex Numbers: Definition and properties of complex numbers; Complex conjugate; Triangle inequality;
Square root of complex numbers; Cube roots of unity; De Moivre’s theorem (statement only) and its elementary applications.
Quadratic Equations: Quadratic equations with real coefficients; Relations between roots and coefficients;
Nature of roots; Formation of a quadratic equation, sign and magnitude of the quadratic expression ax2 +bx+c (where a, b, c are rational numbers and a â‰ 0).
Matrices: Concepts of m x n (m â‰¤ 3, n â‰¤ 3) real matrices, operations of addition, scalar multiplication and multiplication of matrices. Transpose of a matrix. Determinant of a square matrix. Properties of determinants
(statement only). Minor, cofactor and adjoint of a matrix. Nonsingular matrix. Inverse of a matrix. Finding area of a triangle. Solutions of system of linear equations. (Not more than 3 variables).
Permutation and combination: Permutation of n different things taken r at a time (r â‰¤ n). Permutation of n things not all different. Permutation with repetitions (circular permutation excluded). Combinations of n different things taken r at a time (r â‰¤ n). Combination of n things not all different. Basic properties. Problems involving both permutations and combinations.
Principle of mathematical induction
Relation and its properties. Equivalence relation â€” definition and elementary examples, mappings, range and domain, injective, surjective and bijective mappings, composition of mappings, inverse of a mapping.
Binomial theorem (positive integral index): Statement of the theorem, general term, middle term, equidistant terms, properties of binomial coefficients.
Sets, Relations and Mappings: Idea of sets, subsets, power set, complement, union, intersection and difference of sets, Venn diagram, De Morgan’s Laws, Inclusion / Exclusion formula for two or three finite sets, Cartesian product of sets.
Statistics and Probability:Measure of dispersion, mean, variance and standard deviation, frequency distribution. Addition and multiplication rules of probability, conditional probability and Bayesâ€™ Theorem, independence of events, repeated independent trails and Binomial distribution.
Trigonometry:Trigonometric functions, addition and subtraction formula, formula involving multiple and sub multiple angles, general solution of trigonometric equations. Properties of triangles, inverse trigonometric functions and their properties.
Two dimensions Coordinate geometry: Distance formula, section formula, area of a triangle, condition of colinearity of three points in a plane.
Polar coordinates, transformation from Cartesian to polar coordinates and vice versa. Parallel transformation of axes, concept of locus, elementary locus problems. Slope of a line. Equation of lines in different forms, angle between two lines. Condition of perpendicularity and parallelism of two lines. Distance of a point from a line.
Distance between two parallel lines. Lines through the point of intersection of two lines.
Equation of a circle with a given center and radius. Condition that a general equation of second degree in x,y may represent a circle. Equation of a circle in terms of endpoints of a diameter . Equation of tangent, normal and chord. Parametric equation of a circle. Intersection of a line with a circle. Equation of common chord of two intersecting circles.
Definition of conic section, Directrix, Focus and Eccentricity, classification based on eccentricity. Equation of Parabola, Ellipse and Hyperbola in standard form, their foci, directrices, eccentricities and parametric equations.
Three dimensions Co-ordinate geometry: Direction cosines and direction ratios, distance between two points and section formula, equation of a straight line, equation of a plane, distance of a point from a plane.
Differential calculus: Functions, composition of two functions and inverse of a function, limit, continuity, derivative, chain rule, derivative of implicit functions and functions defined parametrically.
Rolle’s Theorem and Lagrange’s Mean Value theorem (statement only). Their geometric interpretation and elementary application. L’Hospital’s rule (statement only) and applications. Second order derivative.
Integral calculus: Integration as a reverse process of differentiation, indefinite integral of standard functions. Integration by parts. Integration by substitution and partial fraction. Definite integral as a limit of a sum with equal subdivisions. Fundamental theorem of integral calculus and its applications. Properties of definite integrals.
Differential Equations: Formation of ordinary differential equations, solution of homogeneous differential equations, separation of variables method, linear first order differential equations.
Application of Calculus: Tangents and normals, conditions of tangency. Determination of monotonicity, maxima and minima. Differential coefficient as a measure of rate. Motion in a straight line with constant acceleration. Geometric interpretation of definite integral as area, calculation of area bounded by elementary curves and Straight lines. Area of the region included between two elementary curves.
Vectors: Addition of vectors, scalar multiplication, dot and cross products, scalar triple product.