|Matrices and their Applications
||Adjoint, inverse - properties, computation of inverses, a solution of a 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 and rank method. A solution of linear programming problems (LPP) in two variables.
|Trigonometry and Complex Numbers
||Definition, range, domain, principal value branch, graphs of inverse trigonometric functions and their elementary properties. Complex number system - conjugate, properties, ordered pair representation. Modulus - properties, geometrical representation, 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.
|Analytical Geometry of two dimensions
||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. Equations of conic sections (parabola, ellipse and hyperbola) in standard forms and general forms- Directrix, Focus and Latus-rectum - a parametric form of conics and chords. - Tangents and normals - Cartesian form and parametric form - the equation of chord of contact of tangents from a point (x1, y1) to all the above-said curves. Asymptotes, Rectangular hyperbola - the Standard equation of a rectangular hyperbola.
||Scalar Product - angle between two vectors, properties of the scalar product, and applications of the dot product. Vector product, right handed and left handed systems, properties of vector product, applications of the cross product. The product of three vectors - Scalar triple product, properties of the scalar triple product, vector triple product, vector product of four vectors, scalar product of four vectors
|Analytical Geometry of Three Dimensions
||Direction cosines - direction ratios - the equation of a straight line passing through a given point and parallel to a given line, passing through two given points, the angle between two lines. Planes - equation of a plane, passing through a given point and perpendicular to a line, given the distance from the origin and unit normal, passing through a given point and parallel to two given lines, passing through two given points and parallel to a given line, passing through three given noncollinear 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 (co-planar lines), angle between a line and a plane.
Skew lines - the shortest distance between two lines, condition for two lines to intersect, the point of intersection, collinearity of three points.
Sphere - the equation of the sphere whose centre and radius are given, the equation of a sphere when the extremities of the diameter are given.
||Limits, continuity and differentiability of functions - Derivative as a rate of change, velocity, acceleration, related rates, derivative as a measure of the slope, tangent, normal and the angle between curves. 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 and points of inflexion. Errors and approximations - absolute, relative, percentage errors - curve tracing, partial derivatives, Euler’s theorem
|Integral Calculus and its Applications
||Simple definite integrals - fundamental theorems of calculus, properties of definite integrals.
Reduction formulae - reduction formulae for
and, Bernoulli’s formula. Area of bounded regions, length of the curve.
||Differential equations - formation of differential equations, order and degree, solving differential equations (1st order), variables separable, homogeneous and linear equations. Second order linear differential equations - second order linear differential equations with constant coefficients, finding the particular integral if f(x) = emx, sin mx, cos mx, x, x2.
||Probability - Axioms - Addition law - Conditional probability - Multiplicative law - Baye’s Theorem - Random variable - probability density function, distribution function, mathematical expectation, variance. Theoretical distributions - discrete distributions, Binomial, Poisson distributions- Continuous distributions, Normal distribution.
||Functions - Relations -Basics of counting. Mathematical logic - logical statements, connectives, truth tables, logical equivalence, tautology, a contradiction. Groups-binary operations, semigroups, monoids, groups, the order of a group, order of an element, properties of groups.