Topics to be covered from July to September

Topics to be covered from October to December

CC-1:

Calculus, Geometry & Vector Analysis

Unit-1: Calculus

SGD

Hyperbolic functions, higher order derivatives, Leibnitz rule and its applications to problems of type curvature, concavity and points of inflection, envelopes, rectilinear asymptotes (Cartesian & parametric form only), curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves, L’Hospital’s rule, applications in business, economics and life sciences.

Reduction formulae, derivations and illustrations of reduction formulae of the type , . Parametric equations, parametrizing

a curve, arc length of a curve, arc length of parametric curves, area under a curve, area and volume of surface of revolution.

Unit-2: Geometry

MI

Classification of conics, tangent and normal, polar equations of conics

Equation of Plane : General form, Intercept and Normal forms. The sides of a plane. Signed distance of a point from a plane. Equation of a plane passing through the intersection of two planes. Angle between two intersecting planes. Parallelism and perpendicularity of two planes.

Straight lines in 3D: Equation (Symmetric & Parametric form). Direction ratio and direction cosines. Canonical equation of the line of intersection of two intersecting planes. Angle between two lines. Distance

of a point from a line. Condition of coplanarity of two lines. Equation of skew lines. Shortest distance between two skew lines.

Spheres. Cylindrical surfaces. Central conicoids, paraboloids, plane sections of conicoids, generating lines, classification of quadrics, illustrations of graphing standard quadric surfaces like cone, ellipsoid. Tangent and normals of conicoids.

Unit-3: Vector Analysis

MDG

Triple product, vector equations, applications to geometry and mechanics.

Introduction to vector functions, limits and continuity of vector functions, differentiation and integration of vector functions of one variable

Graphical Demonstration(Teaching Aid)

SG/MDG/AM/MI/SGD

• Plotting of graphs of function , log(ax+b), 1/(ax+b), sin(ax+b), cos(ax+b), |ax+b| and to illustrate the effect of a and b on the graph.

• Plotting the graphs of polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them.

• Sketching parametric curves (Eg. trochoid, cycloid, epicycloids, hypocycloid).

• Obtaining surface of revolution of curves.

• Tracing of conics in cartesian coordinates/ polar coordinates.

• Sketching ellipsoid, hyperboloid of one and two sheets, elliptic cone, elliptic, paraboloid, and hyperbolic

paraboloid using cartesian coordinates.

CC-2:

Algebra

Unit-1: Complex Numbers,

Theory of Equations, Inequality, Linear Difference Equations

SG

Polar representation of complex numbers, n-th roots of unity, De Moivre’s theorem for rational indices and its applications. Exponential, logarithmic, trigonometric and hyperbolic functions of complex variable.

Theory of equations: Relation between roots and coefficients, transformation of equation, Descartes rule of signs, Sturm’s theorem, cubic equation (solution by Cardan’s method) and biquadratic equation (solution by Ferrari’s method).

Inequality : The inequality involving AM ³ GM ³ HM, Cauchy-Schwartz inequality.

Linear difference equations with constant coefficients (up to 2nd order).

Unit-2:

Relations, Mappings, Integers

AM

Relation : equivalence relation, equivalence classes & partition, partial order relation, poset, linear order relation.

Mapping : injective, surjective, one to one correspondence, invertible mapping, composition of mappings,

relation between composition of mappings and various set theoretic operations. Meaning and properties of , for any mapping f :X®Y and BÍY.

Well-ordering property of positive integers, Principles of Mathematical induction, division algorithm, divisibility

and Euclidean algorithm. Prime numbers and their properties, Euclid’s theorem. Congruence

relation between integers. Fundamental Theorem of Arithmetic. Chinese remainder theorem. Arithmetic

functions, some arithmetic functions such as f, t, s and their properties.

Unit-3:

Rank of a Matrix, System of Linear Equations

MDG

Rank of a matrix, inverse of a matrix, characterizations of invertible matrices.

Systems of linear equations, row reduction and echelon forms, vector equations, the matrix equation AX =

B, solution sets of linear systems, applications of linear systems.

GE-1

Unit-1: Algebra-I

SG

Complex Numbers : De Moivre’s Theorem and its applications. Exponential, Sine, Cosine and Logarithm of a complex number. Definition of (a¹ 0). Inverse circular and Hyperbolic functions.

Polynomials : Fundamental Theorem of Algebra (Statement only). Polynomials with real coefficients, the n-th degree polynomial equation has exactly n roots. Nature of roots of an equation (surd or complex rootsoccur in pairs). Statement of Descarte’s rule of signs and its applications.

Statements of : (i) If a polynomial f(x) has opposite signs for two real values a and b of x, the equation f(x) = 0 has odd number of real roots between a and b. If f(a) and f(b) are of same sign, either no real root or an even number of roots lies between a and b. (ii) Rolle’s Theorem and its direct applications. Relation between roots and coefficients, symmetric functions of roots, transformations of equations. Cardan’s method of solution of a cubic equation.

Rank of a matrix: Determination of rank either by considering minors or by sweep-out process. Consistency

and solution of a system of linear equations with not mo re than 3 variables by matrix method.

Unit-2: Differential Calculus-I

MDG

Rational numbers, Geometrical representations, Irrational number, Real number represented as point on a

Line--Linear Continuum. Acquaintance with basic properties of real number (No deduction or proof is included).

Real-valued functions defined on an interval, limit of a function (Cauchy’s definition). Algebra of limits. Continuity of a function at a point and in an interval. Acquaintance (on proof) with the important properties of continuous functions no closed intervals. Statement of existence of inverse function of a strictly monotone function and its continuity.

Derivative - its geometrical and physical interpretation. Sign of derivative-Monotonic increasing and decreasing functions. Relation between continuity and derivability. Differential - application in finding approximation.

Successive derivative - Leibnitz’s theorem and its application.

Applications of Differential Calculus: Curvature of plane curves. Rectilinear Asymptotes (Cartesian only). Envelope of family of straight lines and of curves (problems only). Definitions and examples of singular points (Viz. Node. Cusp, Isolated point).

AM

Functions of two and three variables: their geometrical representations. Limit and Continuity (definitions only) for function of two variables. Partial derivatives. Knowledge and use of chain Rule. Exact differentials (emphasis on solving problems only).

Functions of two variables--Successive partial Derivatives : Statement of Schwarz’s Theorem on Commutative property of mixed derivatives. Euler’s Theorem on homogeneous function of two and three variables.

Unit-3: Differential Equation-I

SGD

Order, degree and solution of an ordinary differential equation (ODE) in presence of arbitrary constants, Formation of ODE.

First order equations : (i) Exact equations and those reducible to such equation. (ii) Euler’s and Bernoulli’sequations (Linear). (iii) Clairaut’s Equations : General and Singular solutions.

Second order linear equations : Second order linear differential equation with constant coefficients. Euler’s Homogeneous equations.

Second order differential equation: (i) Method of variation of parameters, (ii) Method of undetermined coefficients.

Unit-4:

Coordinate

Geometry

MI

Transformations of Rectangular axes: Translation, Rotation and their combinations. Invariants.

General equation of second degree in x and y: Reduction to canonical forms. Classification of conic.

Pair of straight lines : Condition that the general equation of 2nd degree in x and y may represent two straight lines. Point of intersection of two intersecting straight lines. Angle between two lines given by . Equation of bisectors. Equation of two lines joining the origin to the points in which a line meets a conic.

Equations of pair of tangents from an external point, chord of contact, poles and polars in case of General conic : Particular cases for Parabola, Ellipse, Circle, Hyperbola.

Polar equation of straight lines and circles. Polar equation of a conic referred to a focus as pole. Equationof chord joining two points. Equations of tangent and normal.

Sphere and its tangent plane. Right circular cone.