Mathematics Syllabus: JEE Mains 2015:
Unit 1: Sets, Relations and Functions:
Sets and their representations, union, intersection, and complement of sets and their algebraic properties, power set, relation, types of relations, equivalence relations, functions one-one into and onto functions, composition of functions.
Unit 2: complex numbers and quadratic equations:
Complex number as ordered pairs of reals, representations of complex numbers in the form a+ib and their representation in a plane, argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a complex number, square root of a complex number, triangle inequality, quadratic equations, in real and complex number system and their solutions. Relations between roots and coefficients, nature of roots, formation of quadratic equations with given roots.
Unit 3: Matrices and Determinants:
Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants, evaluation of determinants, area of triangles using determinants. Ad-joint and evaluation of inverse of a square matrix using determinants and elementary transformations. Test of consistency and solution of simultaneous linear equations in two or three variable using determinants and matrices.
Unit 4: permutations and combinations:
Fundamental principle of counting, permutation as an arrangement and combination as selection, meaning of P(n,r) and C (n,r) simple applications.
Unit 5: Mathematical induction:
Principle of mathematical induction and its simple applications
Unit 6: Binomial Theorem and its applications:
Binomial theorem for a positive integral index, general term and middle term, properties of binomial coefficients and simple applications.
Unit 7: Sequences and Series:
Arithmetic and geometric progressions, insertion of arithmetic, geometric means between two given numbers, relation between AM and GM sum upto n terms of special series Sn Sn2 and Sn3 Arithmetico- Geometric progression
Unit 8: Limit, continuity and differentiability:
Real value functions, algebra of functions, polynomials, rational, trigonometric, logarithmic, and exponential functions, inverse functions, graphs of simple functions.
Limits, continuity and differentiability. Differentiation of the sum, difference, product and quotient of two functions. Differentiation of trigonometric, inverse trigonometric,logarithmic, exponential, composite and implicit functions, derivatives of order upto two. Rolle’s and Lagrange’s mean value theorems.
Application of derivatives: rate of change of quantities, monotonic increasing and decreasing functions, maxima and minima of functions of one variable, tangents and normals.
Unit 9: integral calculus:
Integral as an derivative. Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions. Integration by substitution by parts and by partial fractions. Integration using trigonometric identities.
Integral as limit of a sum. Fundamental theorem of calculus, properties of definite integrals, evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form.
Unit 10: Differential Equations:
Ordinary differential equations, their order and degree. Formulations of differential equations. Solutions of differential equations by the method of separation of variables, solution of homogeneous and linear differential equation of the type.
Unit 11: Coordinate Geometry:
Cartesian system of rectangular coordinates 10 in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Straight lines: various forms of equation of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocenter and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines.
Circle, conic sections: standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the end points of a diameter are given, pints of intersection of a line and a circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standar forms, condition for y=mx+b to be a tangent and pints of tangency.
Unit 12: Three dimensional geometry:
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines. Skew lines, the shortest distance between them and its equation. Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines.
Unit 13: Vector Algebra:
Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.
Unit 14: statistics and probability:
Measures of dispersion: Calculation of mean, median, mode of grouped and ungrouped data calculation of standar deviation, variance and mean deviation for grouped and ungrouped data.
Probability: probability of an event, addition and multiplication theorems of prbabilty, baye’s theorem, probability distribution of a random variate, Bernoulli trials and binomial distribution.
Unit 15: trigonometry:
Trigonometrical identities and equations. Trigonometrical functions, inverse trigonometrical functions and tier peoperties, heights and distances.
Unit 16: Mathematical Reasoning:
Statements,logical operations and or, implies, implied by if and only if, understanding of tautology, contradiction, converse and contrapositive.