Masters in Engineering with three published papers in International journals.
Associate Member of Institution of Engineers (India)
Programming Languages- C, C++
GIS Software- MULTISPEC, eCognition,
Concepts-< strong> Operatin g Systems, MS Office
Embedded Programming- Assembly Programming for 8051 microcontrollers, 8085 & 8086 microprocessor, Multisim, Express PCB.
Experience: 2 years
Sets and their representations.Empty set.Finite and Infinite sets.Equal sets.Subsets.Subsets of the set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and intersection of sets.Difference of sets. Complement of a set, Properties of Complement sets.Types of relations: Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function.Binary operations.
2. Relations and Functions
Ordered pairs, Cartesian product of sets. Number of elements in the Cartesian product of two
finite sets. Cartesian product of the reals with itself (uptoR × R × R).
Definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as
a special kind of relation from one set to another. Pictorial representation of a function, domain,
co-domain and range of a function. Real valued function of the real variable, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum and greatest integer
functions with their graphs. Sum, difference, product and quotients of functions.
3. Trigonometric Functions
Positive and negative angles. Measuring angles in radians and in degrees and conversion from
one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the Trigonometric Identitites, for all x. Signs of trigonometric functions and sketch of their
graphs.Definition, range, domain, principal value branches.Graphs of inverse trigonometric functions.Elementary properties of inverse trigonometric functions.
1. Principle of Mathematical Induction
Process of the proof by induction, motivating the application of the method by looking at natural
numbers as the least inductive subset of real numbers. The principle of mathematical induction
and simple applications.
2. Complex Numbers and Quadratic Equations
Need for complex numbers, especially , to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers.Argand plane and polar representation of complex numbers.Statement of Fundamental Theorem of Algebra, solution of quadratic equation in the complex number system, Square-root of a Complex number.
3. Linear Inequalities
Linear inequalities, Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Solution of system of linear inequalities in two variables - graphically.
4. Permutations and Combinations
Fundamental principle of counting. Factorial n. Permutations and combinations derivation of
formulae and their connections, simple applications.
5. Binomial Theorem
History, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle,general and middle term in binomial expansion, simple applications.
6. Sequence and Series
Sequence and Series. Arithmetic Progression (A.P.), Arithmetic Mean (A.M.), Geometric
Progression (G.P.), general term of a G.P., sum of n terms of a G.P. Arithmetic and geometric
series, infinite G.P. and its sum, geometric mean (G.M.). Relation between A.M. and G.M. Sum
ton terms of the special series .
Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non-commutativityof multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).
Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.
9: Business math : Compound Interest ( for infinite and finite period), Simple Interest , EMI calculation.
1: Angles and Triangles :
Types of angles theirs properties , Congruency of triangles and properties of triangles, Circle: radius, arc lengths , angles inside circle, sectors.
2. Straight Lines
Brief recall of 2-D from earlier classes, shifting of origin. Slope of a line and angle between two
lines. Various forms of equations of a line: parallel to axes, point-slope form, slope-intercept
form, two-point form, intercepts form and normal form. General equation of a line. Equation of
family of lines passing through the point of intersection of two lines. Distance of a point from a
3. Conic Sections
Sections of a cone: Circles, ellipse, parabola, hyperbola, a point, a straight line and pair of
intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola.Standard equation of a circle.
4. Introduction to Three-dimensional Geometry
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance
between two points and section formula.
1.Limits and Derivatives
Derivative introduced as rate of change both as that of distance function and geometrically,
intuitive idea of limit. . Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives
of polynomial and trigonometric functions.
2.Continuity and Differentiability
Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretations.
3. Applications of Derivatives
Applications of derivatives: Rate of change, increasing/decreasing functions, tangents and normal, approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).
Integration as inverse process of differentiation.Integration of a variety of functions by substitution, by partial fractions and by parts, to be evaluated.
Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic
properties of definite integrals and evaluation of definite integrals.
5.Applications of the Integrals
Applications in finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses (in standard form only), area between the two above said curves (the region should be clearly identifiable).
Definition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given.Solution of differential equations by method of separation of variables, homogeneous differential equations of first order and first degree.
VECTORS AND THREE-DIMENSIONAL GEOMETRY
Vectors and scalars, magnitude and direction of a vector.Direction cosines/ratios of vectors. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors, scalar triple product.
2. Three-dimensional Geometry
Direction cosines/ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane.Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.
Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constrains).
Mathematically acceptable statements. Connecting words/phrases - consolidating the understanding of “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “and”, “or”, “there exists” and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words - difference between contradiction, converse and contrapositive.