# IIT Suggested Syllabus: Mathematics

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## Mathematics

• Algebra: Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.
• Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.
• 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.
• Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients.
• Matrices as a rectangular array of real numbers, equality of matrices, addition, scalar multiplication and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, 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, independence of events, computation of probability of events using permutations and combinations.
• Trignometry: Trignometric functions, their periodicity and graphs, addition and subtraction formulae, formulae consisting multiple and sub-multiple angles, general solution of trigonometric equations.
• 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) .
• Analytical geometry: Two dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.
• Equation of a straight line in various forms, angle between two lines, distance of a point from a line.
• Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines, centroid, orthocentre, incentre and circumcentre of a triangle.
• Equation of a circle in various different forms, equations of tangent, normal, and chords.
• Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and that of a circle and a straight line.
• Equation of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal.
• Locus Problems.
• Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.
• Differential calculus: Real valued functions of a real variable, into, onto and one-to-one functions, sum, difference, product and quotient of two functions, composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions.
• Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, Hospital rule of evaluation of limits of functions.
• Even and odd functions, inverse of a function, continuity of composite functions, intermediate value property of continuous functions.
• Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions.
• Derivatives of implicit functions, derivatives up to order two, geometrical interpretation of the derivative, tangents and normals, increasing and decreasing functions, maximum and minimum values of a function, applications of Rolle՚s Theorem and Lagrange՚s Mean Value Theorem.
• Integral calculus: Integration as the inverse process of differentiation, indefinite integrals of standard functions, definite integrals and their properties, application of the Fundamental Theorem of Integral Calculus.
• Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves.
• Formation of ordinary differential equations, solution of homogeneous differential equations, variables separable method, linear first order differential equations.
• Vectors: Addition of vectors, scalar multiplication, scalar products, dot and cross products, scalar triple products and their geometrical interpretations.