iirs.gov.in M.Tech / MSc Entrance Examination Syllabus : Indian Institute of Remote Sensing

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Entrance Exam : M.Tech and MSc Entrance Examination

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Syllabus for M.Tech and MSc Entrance Examination
Indian Institute of Remote Sensing, Indian Space Research Organisation
1. Common syllabus for M.Tech. /M.Sc. (Geoinformatics) / M.Sc. (NHDRM) Entrance Examination
1.1 Basic Science (Common) (25 Marks)
1.1.1 Chemistry
Matter: solid, liquid and gas; change of state – melting (absorption of heat), freezing, evaporation (Cooling by evaporation), condensation, sublimation Elements, compounds and mixtures.
Atoms and molecules, Atomic and molecular masses, Valency. Chemical formulae of common compounds.
Electrons, protons and neutrons; isotopes and isobars.
Chemical Equation, types of chemical reactions, Acids, bases and salts, concept of pH scale Metals and non metals formation and properties of ionic compounds,
Carbon compounds, Covalent bonding in carbon compounds. Periodic classification of elements.

1.1.2 Physics
Distance and displacement, velocity; uniform and non-uniform motion along a straight line; acceleration. Force and motion, Newton’s laws of motion, inertia of a body, inertia and mass, momentum, force and acceleration. Elementary idea of conservation of momentum, action and reaction forces. Gravitation; universal law of gravitation, force of gravitation of the earth (gravity), acceleration due to gravity; mass and weight; free fall. Thrust and pressure. Archimedes’ principle, buoyancy, elementary idea of relative density. Work, power, Sound, Basic principles of optics, Electromagnetic Theory Different forms of energy, conventional and non-conventional sources of energy: fossil fuels, solar energy; biogas; wind, water and tidal energy; nuclear energy. Renewable versus non-renewable sources.

1.2 Basic Mathematics – Part I (Common) (25 Marks)
1.2.1 Number Systems
Real Numbers: Euclid’s division Lemma, Fundamental Theorem of Arithmetic, proof of the resultsirrationality of , , decimal expansions of rational numbers in terms of terminating/nonterminating recurring decimals.

1.2.2 Algebra
Polynomials: Zeros of a polynomial. Relationship between zeros and coefficients of quadratic polynomials. Statement and simple problems on division algorithm for polynomials with real coefficients.

Pair of linear equations in two variables: Pair of linear equations in two variables and their graphical solution. Geometric representation of different possibilities of solutions/inconsistency. Algebraic conditions for number of solutions. Solution of pair of linear equations in two variables algebraically – by substitution, by elimination and by cross-multiplication. Simple problems on equations reducible to linear equations.

Quadratic equations: Standard form of a quadratic equation Solution of the quadratic equations (only real roots) by factorization, by completing the square and by using quadratic formula. Relationship between discriminant and nature of roots.

Arithmetic progressions: General form of arithmetic progression (AP), finite and infinite AP, common difference, derivation of standard results of finding the nth term and summation of n terms of AP.

1.2.3 TRIGONOMETRY
Introduction to trigonometry: Trigonometry ratios of an acute angle of a right-angled triangle. Proof of their existence; Values of the trigonometric ratios of 30°, 45°, 60° and 90°. Relationships between the ratios.
Trigonometric identities: Proof and applications of the trigonometric identities (e.g. =1). Trigonometric ratios of complementary angles.
Heights and distances: Simple and believable problems on heights and distances.

1.2.4 Coordinate Geometry
Lines (2-Dimensions): Concepts of coordinate geometry including graphs of linear equations. Geometrical representation of quadratic polynomials. Distance between two points and section formula (internal). Area of triangle.

1.2.5 Geometry
Triangles: Definitions, examples, counter examples of similar triangles, properties of triangles (proof and application).
Circles: Definition of tangent to a circle, properties of the tangents drawn to a circle from a point (proof and application).

1.2.6 Mensuration
Areas related to circles: The area of a circle; area of sectors and segment of a circle. Problems based on areas and perimeter/circumference of the simple figures such as triangles, simple quadrilaterals and circles.

Surface areas and volumes: Problems on finding surface areas and volumes of combinations of the following: cubes, cuboids, spheres, hemispheres and right circular cylinders/cones. Frustum of the cone. Problems involving converting one type of metallic solid into another and other mixed problems.

1.2.7 Statistics And Probability
Statistics: Mean, median and mode of a ungrouped/grouped data, SD, Correlation, Regression Cumulative frequency graph.
Probability: Simple problems of single events on classical definition of probability.

1.3 General Knowledge, Aptitude test and Reasoning (25 marks)
1.4 Optional Themes – 8 No. (choose any one of the options) ( 25 marks)

1.4.6 Satellite Imagery Analysis & Photogrammetry
1.4.6.1 Maths (Part II)
Sets And Functions
Sets: Sets and their representations, empty set, finite & infinite 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

Relations & 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 (unto R x R x 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, codomain & 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.

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto
functions, composite functions, inverse of a function. Binary operations.

Trigonometric Functions: Positive and negative angles, measuring angles in radians & in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Signs of trigonometric functions and sketch of their graphs. Expressing sin (x+y) and cos (x+y) in terms of sinx, siny, cosx & cosy. Identities related to sin 2x, cos2x, tan 2x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type sin? = sin a, cos? = cos a and tan? = tan a.

Inverse Trigonometric Functions: Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Algebra
Principle of Mathematical Induction: Processes 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.

Complex Numbers and Quadratic Equations: Need for complex numbers, especially 1 ? , 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 equations in the complex number system. 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.

Permutations & Combinations: Fundamental principle of counting. Factorial n (n!). Permutations and combinations, derivation of formulae and their connections, simple applications.

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. 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., geometric mean (G.M.), relation between A.M. and G.M. Sum to n terms of the special series.

Matrices: 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-commutativity of 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).

Determinants: Determinant of a square matrix (up to 3 x 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.

Coordinate Geometry
Straight Lines: 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, intercept form and normal form. General equation of a line. Distance of a point from a line.

Conic Sections: Sections of a cone: circle, 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.

Three-dimensional Geometry: Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula. Direction cosines and direction ratios of a line, equation of a line in space, angle between (i) two lines, (ii) two planes. (iii) a line and a plane, shortest distance between two lines, Co-planarity of two lines, distance of a point from a plane.

Vectors: 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, additions and subtractions of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio, scalar and vector products of vectors, projection of a vector on a line, scalar triple product of vectors.
Calculus

Introduction to 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.

Continuity and Differentiability: Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential 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 interpretation.

Applications of Derivatives: Applications of derivatives, rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in 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).

Integrals: Integration as inverse process of differentiation, Integration of a variety of functions by substitution, by partial fractions and by parts, simple integrals of the following type 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.

Applications of the Integrals: Applications in finding the area under simple curves, especial lines, circles/parabolas/ellipses, area between the two above said curves.

Differential Equations: 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. Solutions of linear differential equations.

Mathematical Reasoning
Mathematical Reasoning: 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 a variety of examples related to real life and Mathematics. Validating the statements involving the connecting words-difference between contradiction, converse and contrapositive.

Statistics & Probability
Statistics: Measures of dispersion; mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal means but different variances.

Probability: Random experiments: outcomes, sample spaces (set representation). Events: occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events. Axiomatic (set theoretic) probability, connections with the theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ & ‘or’ events.

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, random variables and its probability distributions, means and variance of randomvariable, repeated independent (Bernoulli) trials and Binomial distributions.

Linear Programming
Constraints, objective function, optimization, different types of Linear Programming Problems (LPPs) and its Mathematical Formulation of LPPs, 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 constraints).