Mathematics syllabus for KEE
Matrices And Determinants
Types of matrices, addition and multiplication of matrices-Properties, computation of inverses, solution of system of linear equations by matrix inversion method. Rank of a Matrix – Elementary transformation on a matrix, consistency of a system of linear equations, Cramer's rule, Non-homogeneous equations, homogeneous linear system, rank method.
Theory of Equations, Sequence and Series
Quadratic equations – Relation between roots and coefficients – Nature of roots – Symmetric functions of roots – Diminishing and Increasing of roots – Reciprocal equations. Arithmetic, Geometric and Harmonic Progressions-Relation between A.M., G. M ., and H.M. Special series: Binomial, Exponential and Logarithmic series – Summation of Series.
Vector Algebra
Scalar Product – Angle between two vectors, properties of scalar product, applications of dot products. Vector Product – Right handed and left handed systems, properties of vector product, applications of cross product. Product of three vectors – Scalar triple product, properties of scalar triple product, vector triple product, vector product of four vectors, scalar product of four vectors. Lines – Equation of a straight line passing through a given point and parallel to a given vector, passing through two given points, angle between two lines. Skew lines – Shortest distance between two lines, condition for two lines to intersect, point of intersection, collinearity of three points. Planes – Equation of a plane, passing through a given point and perpendicular to a vector, given the distance from the origin and unit normal, passing through a given point and parallel to two given vectors, passing through two given points and parallel to a given vector, passing through three given non-collinear points, passing through the line of intersection of two given planes, the distance between a point and a plane, the plane which contains two given lines, angle between two given planes, angle between a line and a plane. Sphere – Equation of the sphere whose centre and radius are given, equation of a sphere when the extremities of the diameter are given.
Complex Numbers & Trigonometry
Complex number system, conjugate – properties, ordered pair representation. Modulus – properties, geometrical representation meaning, polar form principal value, conjugate, sum, difference, product quotient, vector interpretation, solutions of polynomial equations, De Moivre's theorem and its applications. Roots of a complex number – nth roots, cube roots, fourth roots. Angle measures-
Circular function-Trigonometrical ratios of related angles – Addition formula and their applications – Trigonometric equations – Inverse trigonometric functions-Properties and solutions of triangle.
Analytical Geometry
Definition of a Conic – General equation of a conic, classification with respect to the general equation of a conic, classification of conics with respect to eccentricity. Parabola – Standard equation of a parabola tracing of the parabola, other standard parabolas, the process of shifting the origin, general form of the standard equation, some practical problems. Ellipse – Standard equation of the ellipse, tracing of the ellipse (x^2/a^2 )+(y^2/a^2 ) = 1 (a> b). Other standard form of the ellipse, general forms, some practical problems Hyperbola – standard equation, tracing of the hyperbola (x^2/a^2 )-(y^2/a^2 ) = 1
, other form of the hyperbola, parametric forms of a conics, chords, tangents and normals – Cartesian
form and parametric form, equation of chord of contact of tangents from a point (x1 ,y1 ) Asymptotes, Rectangular Hyperbola –standard equation of a rectangular hyperbola.
Differential Calculus
Derivative as a rate measure – rate of change – velocity-acceleration – related rates – Derivative as a measure of slopetangent, normal and angle between curves. Maxima and Minima. Mean value theorem- Rolle's Theorem – Lagrange Mean Value Theorem – Taylor's and Maclaurin's series, L' Hospital's Rule, Stationary Points – Increasing, decreasing, maxima, minima, concavity convexity points of inflexion. Errors and approximations – absolute, relative, percentage errors, curve tracing, partial derivatives – Euler's theorem.
Integral Calculus and Its Applications Methods of Integration Standard Types
Properties of definite integrals, reduction formulae for sin^n (x) and cos^n (x) , Area, length, volume and surface area.
Differential Equations
Formation of differential equations, order and degree, solving differential equations (1st order) – variable separable homogeneous, linear equations. Second order linear equations with constant co-efficient f (x)=e^m(x), sin mx, cos mx,x, x^2.
Discrete Mathematics
Mathematical Logic – Logical statements, connectives, truth tables, tautologies, sets, algebraic properties, relations, functions, permutation, combination, Induction. Binary Operations – Semi groups – monoids, groups (Problems and simple properties only), order of a group, order of an element.
Probability Distributions
Probability, axioms, theorems on probability, conditional probability, Random Variable, Probability density function, distribution function, mathematical expectation, variance, discrete distributions-Binomial , Poisson, continuous distribution – Normal.
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