KEAM Syllabus for Math
Relation, Sets and Functions
Sets and their representations: Complement of a set, universal set, subsets, empty set, operations on sets (intersection and difference of set, union), Venn diagrams, power set, equal sets, infinite and finite sets. Applications of sets: Cartesian product of two sets, ordered pairs. Relations: Range, domain and co-domain. Functions: on to, into, binary operations, invertible functions, identity function, one-one on to functions, composition of functions, one-one in to, constant function.
Representation of complex number as a point in the plane, algebra of complex numbers, polar representation of complex number cube roots of unity, modulus and argument of a complex number, triangle inequality, square root of a complex number, complex number in the form, complex conjugate, imaginary and real parts of a complex number, argand diagram.
Symmetric functions of roots, nature of roots, solution of a quadratic equation in the complex number system by 1st factorization and 2nd using formula, equations reducible to quadratic forms, formation of quadratic equations with given roots, relation between roots and coefficients,
Sequences and Series
Examples and sequence of infinite and finite sequences. Arithmetic progression (A.P.): Common difference, first term, sum of n terms and nth term of A.P. Arithmetic mean (A.M.), insertion of arithmetic means between any two given numbers. Geometric progression (G.P.): Common ratio and nth term, first term, sum to n terms and sum of infinite numbers as geometric series, geometric mean, insertion of geometric means between any two given numbers, harmonic mean, harmonic progression, relationship G.M. H.M. and A.M. Arithmetico geometric series: Sum of infinite number of terms of an arithmetico geometric series and sum to n term, series exponential and logarithmic and logarithms series.
Laws of logarithms including change of base, characteristic and mantissa, longrithmic tables, growth and decay, logarithmic function log e x and its graph, meaning of logarithm of a number to a given base, common, logarithms, antilogarithms, simple applications of logarithms to problems of compound interest, concept of ‘e’ as the sum of an infinite series.
Binomial Theorem and Mathematical Induction, Permutations, Combinations
Proof of binomial theorem for positive integral exponent using principle of mathematical induction and by combinatorial method too. Properties of binomial coefficients, application of binomial theorem, simple application, general and middle terms in binomial expansions, binomial theorem for any index (without proof), the principle of mathematical induction, statement of binomial theorem, meaning of C (n.r.), meaning of P (n.r.), the factorial notation, application of permutations and combinations, combination, permutation as an arrangement, fundamental principle of counting.
Matrices and Determinants
Determinant of a square matrix, singular and non-singular matrices, transpose, consistency and inconsistency of a system of linear equations, minors and cofactors, applications of determinants in:
- Finding the area of a triangle
- Solving a system of linear equations (Cramer’s rule)
Solving, adjoint and inverse of a matrix, statement of important results on operations of matrices and their verifications by numerical problem only, concept of a matrix, equality of matrices, scalar multiplication and multiplications of matrices, types of matrices, system of linear equations in two or three variables using inverse of matrix, operations of addition.
Graphical solutions of linear inequations in two variables, solution of linear inequation in one variable and its graphical representation, solutions of system of system of linear ineqations in two variables, solution of system of linear inequations in one variable.
Mathematical Logic and Boolean Algebra
Boolean algebra as an algebraic structure, Boolean function, valid arguments, application of Boolean algebra to switching circuits, principle of duality, conditional and biconditional statements, statements, negation operation, basic logical connectives and compound statements including their negations, tautology, algebra of statements, use of Venn diagram in logic, truth tables, application of logic in solving simple problems.
Trigonometric Function and Inverse Trigonometric Functions
Concept of periodicity of trigonometric functions, degree measures and Radian measure of positive and negative angles, value of trigonometric functions of x for x, periodic functions, definition of trigonometric functions with the help of a unit circle, relation between degree measure and radian measure.
Trigonometric functions of sum and difference of numbers. Trigonometric functions of multiple and submultiples of numbers. Solution of trigonometric equations of the type sin x = sin a, conditional identities for the angles of a triangle, Tan x = Tan a and equations reducible to these forms, Cos x = Cos a.
Inverse Trigonometric Functions: y = Cos x, y = Sin x, y = a Cos bx, y = Tan x, y = a Sin bx, y = a Cos x, y = a Sin x.
Problems on heights and distances.
Cartesian System of Rectangular Coordinates
Intercepts of a line on the coordinate axes, slope of line, area of a triangle, distance formula, locus and its equation, parallel and perpendicular lines, condition for the collinearity of these points in a plane, Cartesian system of coordinates in a plane, centroid and incentre.
Lines and Family of Lines
Distance of a point from a line, angles between two lines, intersection of lines, normal form, the slope point form, various forms of equations of a line parallel to axes, equations of family lines through the intersection of too lines, condition for concurrency of three lines, equation of bisectors of angle between two lines, general for, intercept form, slope-intercept form.
Circles and Family of Circles
Condition for a line to be a tangent to the given circle, equation of circle when the end points of a diameter are given, general form of the equation of a circle, standard form of the equation of a circle and its center and radius, equation of a tangent to a circle and length of the tangent, points of intersection of a line circle with centric at origin, equation of the circle in the parametric form.
Equation of conic sections (Hyperbola, Parabola and Ellipse ) in standard form, sections of a cone.
Application of vectors in geometry, projection of a vector on a line, vectors and scalars, scalar triple product and its applications, coplanarity of three vectors or four points using scalar triple product, scalar product of two vectors, vector product of two vectors application of dot and cross product in:
- finding work done by force
- finding area of a triangle and a parallelogram
- problem of plane geometry and trigonometry
- vector moment of a vector about a point
moment of a vector about a line, vector triple product, magnitude and direction of a vector, position vector of a point, collinear and parallel vectors, compounds of a vector, multiplication of a vector by scalar, types of vectors (unit, equal and zero vector), free and localized vector, negative of a vector, addition of vectors, position vector of point dividing a line segment in a given ratio.
Three Dimensional Geometry
- two lines
- two planes
- a line and a plane,
co-ordinate axes and co-ordinate planes in three dimensional space, length of perpendicular of a point form a plane by both Cartesian and vector methods, co-ordinate of a point in space, Cartesian and vector equation of a sphere and its centre and radius diameter form the equation of a sphere, distance between two points, direction ratios and direction cosines of a line joining two points, angle between two lines whose direction ratios are given, collinearrity of three points, shortest distance between two lines.
Carterian and vector equation of plane:
- When normal vector and the distance of the plane from the origin is given
- Passing though a point and perpendicular to a given vector
- Passing through a point and parallel to two given lines through the intersection of two other planes
- Containing two lines
- Passing through three points,
Section formula, projection of the join of two points on a given line.
Cartesian and vector equation of a line through:
- A point and parallel to given vector
- Through two points,
condition for the intersection of two lines, skew and coplanar lines, collinearity of hree points.
Statistics and Probability
Random experiments and sample space, occurrence of an event, multiplication rule, exhaustive events, meaning of equality likely outcomes, probability of an event, addition rule, mean deviation for ungrouped data, standard deviation, events and subset of a sample space, sure and impossible events, algebra of events, mutually exclusive events, theorems on probability, variance for grouped and ungrouped data. Independent experiments and events, probability distribution of a random variable, finding P (A or B) P (A and B).
Limits, Functions and Continuity
Composite functions, greatest integer function, inverse of a function, concept of a real function and its range and domain, signum functions, inverse trigonometric functions and trigonometric functions and their graphs.
Left and right hand limits, limit of a function, fundamental theorems on limits without proof, meaning and related notations. Logarithmic and inverse trigonometric functions, trigonometric, continuity of special functions-polynomial, exponential, product and quotient of continuous functions, continuity of a function at a point, over closed/an open interval, limits at infinity and infinity limits.
Derivatives of sum, product and quotient of functions, difference, derivatives of polynormial, exponential, trigonometric, logarithmic, logarithmic differentiation, inverse trigonometric and implicit functions, derivatives of functions expressed in parametric form, derivatives of second order, chain rule and differentiation by substitution, relationship between continuity and differentiability, derivatives of a function and its physical and geometrical significance, derivatives of polynomial, exponential , logarithmic and inverse trigonometric functions from first principles, basic trigonometric.
Application of Derivatives
Approximation by differentials, rate of quantities, increasing and decreasing functions and sign of the derivatives, curve sketching of simple curves, Roll’s theorem and mean value theorem, Tangents and Normals, greatest and least values.
Integration by parts, exponential and logarithmic, integrals involving algebraic, integration as inverse of differentiation, integration by substitution, trigonometric, properties of integrals. Partial fractions and their use in integration, integration of rational functions.
Fundamental theorems of integral calculus without proof, definite integral as limit of a sum, evaluation of definite integrals by substitution. Line and circle, application of definite integrals in finding areas bounded by a curve, line and parabola: ellipse and line, circle, parabola and ellipse in standard form between two ordinates and x-axis.
Homogeneous differential equations of first order and their solutions, degree and order, definition, solution of differential equations by method of separation of variables, formation of differential equations whose general solution is given, particular and general solutions of a differential equation. Solution of second order differential equations and solution of linear differential equations of the type.