Union, intersection, and complement of sets and their algebraic properties.
Relation, Types of relations, equivalence relations.
Functions; one-one, into and onto functions, the composition of functions.
Unit 2: Complex Numbers and Quadratic Equations
Complex numbers as ordered pairs of reals.
Representation 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.
Quadratic equations in real and complex number system and their solutions.
The relation between roots and coefficients, nature of roots, the formation of quadratic equations with given roots.
Unit 3: Matrices and Determinants
Matrices: Algebra of matrices, types of matrices, and matrices of order two and three.
Determinants: Properties of determinants, evaluation of determinants, the area of triangles using determinants.
Adjoint 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 variables using determinants and matrices.
Unit 4: Permutations and Combinations
The fundamental principle of counting.
Permutation as an arrangement and combination as selection.
The meaning of P (n,r) and C (n,r). Simple applications.
Unit 5: Mathematical Induction
The principle of Mathematical Induction and its simple applications.
Unit 6: Binomial Theorem
Binomial theorem for a positive integral index.
General term and middle term.
Properties of Binomial coefficients and simple applications.
Unit 7: Sequence and Series
Arithmetic and Geometric progressions, insertion of arithmetic.
Geometric means between two given numbers.
The relation between A.M. and G.M.
Sum up to n terms of special series: Sn, Sn2, Sn3.
Arithmetic Geometric progression.
Unit 8: Limit, Continuity and Differentiability
Real-valued 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 up to two.
Rolle’s and Lagrange’s Mean Value Theorems.
Applications 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 antiderivative.
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.
Evaluation of simple integrals:
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.
Formation of differential equations.
The solution of differential equations by the method of separation of variables.
The solution of homogeneous and linear differential equations of the type:
Unit 11: Coordinate Geometry
Cartesian system of rectangular coordinates in a plane, distance formula, section formula, locus and its equation, translation of axes, the slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes.
Straight lines: Various forms of equations 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 the centroid, orthocentre, and circumcentre of a triangle, equation of the family of lines passing through the point of intersection of two lines.
Circles, 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 endpoints of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent.
Sections of cones, equations of conic sections (parabola, ellipse, and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency.
Unit 12: 3D Geometry
Coordinates of a point in space, the distance between two points.
Section formula, direction ratios and direction cosines, the 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, the intersection of a line and a plane, coplanar lines.
Unit 13: Vector Algebra
Scalars and Vectors. Addition, subtraction, multiplication and division of vectors.
Vector’s Components in 2D and 3D space.
Scalar products and vector products, triple product.
Unit 14: Statistics and Probability
Measures of Dispersion: Calculation of mean, mode, median, variance, standard deviation, and mean deviation of ungrouped and grouped data.
Probability: Probability of events, multiplication theorems, addition theorems, Baye’s theorem, Bernoulli trials, Binomial distribution and probability distribution.
Unit 15: Trigonometry
Identities of Trigonometry and Trigonometric equations.
Functions of Trigonometry.
Properties of Inverse trigonometric functions.
Problems on Heights and Distances.
Unit 16v Mathematical Reasoning
Statements and logical operations: or, and, implied by, implies, only if and if.
Understanding of contradiction, tautology, contrapositive and converse.
JEE Advanced Maths Syllabus
Unit 1 Algebra
Algebra of complex numbers, addition, multiplication, conjugation.
Polar representation, properties of modulus and principal argument.
Triangle inequality, cube roots of unity.
Quadratic equations with real coefficients.
Relations between roots and coefficients.
Formation of quadratic equations with given roots.
Symmetric functions of roots.
Sequence and Series
Arithmetic, geometric, and harmonic progressions.
Arithmetic, geometric, and harmonic means.
Sums of finite arithmetic and geometric progressions, infinite geometric series.
Sums of squares and cubes of the first n natural numbers.
Logarithms and their properties.
Permutation and Combination
Problems on permutations and combinations.
Binomial theorem for a positive integral index.
Properties of binomial coefficients.
Matrices and Determinants
Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix.
Determinant of a square matrix of order up to three, the inverse of a square matrix of order up to three.
Properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties.
Solutions of simultaneous linear equations in two or three variables.
Addition and multiplication rules of probability, conditional probability.
Bayes Theorem, independence of events.
Computation of probability of events using permutations and combinations.
Unit 2 Trigonometry
Trigonometric functions, their periodicity, and graphs, addition and subtraction formulae.
Formulae involving multiple and submultiple angles.
The general solution of trigonometric equations.
Inverse Trigonometric Functions
Relations between sides and angles of a triangle, sine rule, cosine rule.
Half-angle formula and the area of a triangle.
Inverse trigonometric functions (principal value only).
Unit 3 Vectors
Properties of Vectors
The addition of vectors, scalar multiplication.
Dot and cross products.
Scalar triple products and their geometrical interpretations.
Unit 4 Differential Calculus
Real-valued functions of a real variable, into, onto and one-to-one functions.
Sum, difference, product, and quotient of two functions.
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