### Syllabus of Admission Test B. Ed. (Mathematics)

- General Intelligence;
- General English : The Syllabus for B. Ed. General English course shall comprise questions on language, grammar meanings of words, use of idioms and phrases;
- One of the SCHOOL SUBJECTS.

**School Subject – Mathematics**

**VECTOR ANALYSIS AND GEOMETRY**

**General Equation of Second Degree**

Intersection of a straight line and a conic, Equation of a tangent to a conic. Condition of tangency, Pair of tangents from a point, Chord of contact of a pair of tangents, Pole and Polar, Conjugate points and lines, Chord with a given middle point, Centre of a conic and diameter, Conjugate diameters.

**Tracing of Conic and Polar Equations**

Nature of conic, Tracing of parabola, ellipse and hyperbola, Asymptotes of the hyperbola. The length and the position of axes of the conic, Polar equation of a conic when focus is at the pole, Directrices, Tracing of the conic 1/r = 1 +e cos , Asymptotes,Equation of the chord when the vectorial angles of the extremitities are given. Equation of the tangent and the normal when the vectorial angle of the point of contact is given.

**Cylinder and Cone**

Equation of a cylinder, Equation of a right circular cylinder, Equation of a cone, Equation of a cone when the vertex is at the origin. Condition for general equation of second degree to represent to a cone, Tangent plane to a cone, Conditioin of tangency, Reciprocal cone, Cone with three mutually perpendicular generators.

**Central Concoids**

The standard equation, The tangent plane, Condition of tangency of plane, Section with a given centre, Locus of the mid-points of a system of parallel chords. The polar plane, Polar lines, Enveloping cone, Classification of central coicoids (Ellipsoid, Paraboloids, Elliptic paraboloid, hyperbolic paraboloid), Conjugate diametral plane and diameters of Ellipsoid, Normals on ellipsoid, Conjugate diameters of ellipsoid.

**Vector Analysis**

Scalar and vector product of three vectors, Product of four vectors, Reciprocal vectors differentiation. Partial and directional derivative, Orientation Gradient, Divergence and curl.

**CALCULUS**

**Differential Calculus**

Unit of a function, Intermediate forms, Successive differentiation, Leibnitz theorem, Maclaurin and Taylor series expansions, Angle between radius and tangent Perpendicular from the pole on the tangent, Pedal equation, Curvature. Partial differentiation, Total differentials, Euler’s theorem on homogeneous function, Asymptotes, Concavity and Convexity, Points of inflexion, Multiple points, Tracing of curves in Cartesian and polar coordinates.

**Integral Calculus**

Gamma function and its properties, Cartesian parameteric and polar forms for rectification, Intrinsic equation for Cartesian and polar curves, Volume and surface of solids of revolution, Cartesian and polar forms.

**NUMERICAL ANALYSIS**

Solution of algebraic and transcendental equations, The bisection and Regula Falsi method, Iteration methods namely, Newton-Raphson method, Generazlied Newton’s method, Solution of system of linear equations using direct methods such as matrix inversion method, Gauss elimination method, LU decomposition including some iteration methods namely, Jacobi and Gauss Siedel methods, The algebraic eigenvalue problems using power and Householder methods.

Symbols of ∆, Ñ, E, E^{-1}, D, *m* and *d* and their relations, Newton-Gregory Forward and Backward Difference Formulae, Gauss’s, Stirling’s and Bessel’s formulae, Lagrange’s Formula, Divided Differences and their properties, Newton’s General Interpolation Formula, Inverse Interpolation Formula, Interpolation with cubic splines.

Numerical Differentiation and Integration, Numerical Differentiation of tabular and non-tabular functions including error estimations, Numerical Integration using Gauss Quadrature Formulae, Trapezoidal, Simpson’s 1/3- and 3/8-Rule, Weddle’s Rule and Newton-Cotes Formla.

Ordinary Differential Equations, Euler’s and Modified Euler’s Methods, Picard’s Method, Taylor Series Method, Runge-Kutta Methods of 2^{nd} and 4^{th} order, Multi-step Methods, Milne-Simpson Method, Adam Bashforth-Moulton Method, Boundary value Problems using Finite Difference Method.

Approximation and Difference Equations, Least Square Curve Fitting Procedures, Different types of approximations, Least Square Polynomial Approximation, Chebyshev Polynomials and its applications in various approximations. Difference Equations, Solution of simple difference equations, First order Homogeneous Linear Difference Equations. Higher order Homogenous Linear Difference Equations, Non-homogenous Linear Difference Equations. Higher Order Homogeneous Linear Difference Equations with constant and variable Coefficients.

**DIFFERENTIAL EQUATIONS**

Exact different equations, Integrating factors, change of variables. Total differential equations, Differential equations of first order but not of first degree, Equations solvable for x,y,q Equations of the first degree in x and y, Clairaut’s equations. Linear differential equations of order n, Homogenous and non-homogenous linear differential equations of order n with constant coefficients, Different forms of particular integrals, Linear differential equations with non-constant coefficients, Reduction of order method. The Cauchy-Euler’s equation of order n, Legendre’s linear equation.

Methods of undertermined coefficients and variation of paprameters, Series solution of differential equations, Frobenius method, Bessel and Legendre differential equations and their series solutions.

Solution of a system of linear differential equation with constant coefficients, An equivalent triangular system, Degenerate case Laplace transforms, Linearity of Laplace transforms, First shifting property, Transforms of Degenerate case Laplace transforms, Linearity of Laplace transforms, First shifting property, Convolution and periodic function theorems, Solution of linear differential equations using Laplace transforms.

Formation and solution of a partial differential equations easily integrable, Linear (Lagrange’s) and nonlinear partial differential equation of first order, Charpit’s methods, Homogenous partial differential equations with constant coefficients.

Non-homogenous partial differential equations with constant coefficients, Classification of second order linear partial differential equations, Method of separation of variables, Fourier series, Even and odd functions and their Fourier series, Change of interval, Vibration of stretched string, One and two dimensional heat flow, Laplace equation in Cartesian form.

The variation of a functional and Euler’s equations, Externals, Functional depending on a unknown functionals, Functionals’ depending on higher derivatives, Variational problems in parametric form, Isoparametric problem, Canaonical form of Euler’s equation, Functionals depending on functions of several variables.

**ALGEBRA**

Binary operations, Definition of group with examples and elementary properties, Subgroups. Order of group, Statement of Lagrange’s theorem. Homomorphism of groups, Kernel of Homomorphism, Defintion of isomorphism, Introduction to rings, subrings and fields with examples and elementary properties. Vector spaces. Subspaces, sum of subspaces, Span of a set, Linear dependence and independence, dimension and basis, dimension theorem, Linear transformation and its properties, range and kernel of a linear transformation, rank and nullity of linear transformation, Rank-nullity theorem, inverse of linear transformation. The space L(u,v) of linear transformation and its dimension, composition of linear transformation, matrix associated with a linear transformation. Linear transformation associated with matrix, matrix as a liner transformation and its rank and nullity. Elementary row operations and row reduced echelon form, inverse of a matrix. Application of matrices to a system of linear equations, Eigen vectors and characteristics equation of matrix, Cayley-Hamilton theorem and its use in finding inverse of matrix.

**ADVANCED CALCULUS**

**Partial Differentiation**

Functions of several variables, Partial derivatives, implicit functions, Limits and continuity, derivatives, composite functions, differential functions, homogenous functions, Euler’s theorem, Higher derivatives, Simultaneous equations, Jacobians, inverse of transformation. Dependent and independent variables, Differential and directional derivatives, Taylor’s theorem, Jacobian’s of implicit functions, Inverse of transformations, Change of variable, functional dependence and equality of cross derivatives.

**Application of Partial Differentiation**

Maximum and minima for functions of two variables, sufficient conditions, Functions of three variables, Quadratic form, Relative extrema, Lagrange’s multipliers, one relation between two variables. One relation among three variables. Two relations among three variables, Envelopes and Evolutes of families of plane curves.

**Multiple Integrals**

Definition, Properties and evolution of double as well as triple integrals, related results, Iterated integrals and change in order of integration.

**Line and Surface Integrals**

Definition, Properties and evaluation of double as well as triple integrals, relted results, Iterated integrals and change in order of integration.

**Line and Surface Integrals**

Definition, Properties and evaluation of line as well as surface integrals, related problems, Green’s theorem and deductions, Stoke’s theorem.

**Differential Calculus**

Unit of a function, Intermediate forms, Successive differentiation, Leibnitz theorem, Maclaurin and Taylor series expansions, Angle between radius and tangent Perpendicular from the pole on the tangent, Pedal equation, Curvature. Partial differentiation, Total differentials, Euler’s theorem on homogeneous function, Asymptotes, Concavity and Convexity, Points of inflexion, Multiple points, Tracing of curves in Cartesian and polar coordinates.

**Integral Calculus**

Gamma function and its properties, Cartesian parameteric and polar forms for rectification, Intrinsic equation for Cartesian and polar curves, Volume and surface of solids of revolution, Cartesian and polar forms.

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