Exam Scheme

Note :-

1. Negative marking: For every wrong answer, 1/3 of the marks prescribed for that particular question shall be deducted.
2. Paper includes questions from: Senior Secondary Level, Graduation Level, Post Graduation Level, Educational Psychology, Pedagogy, Teaching Learning Material, and Use of Computers and Information Technology in Teaching Learning.

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Syllabus : Mathematics -II

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1. Part- I Senior Secondary Level

1. Algebra
   1. Quadratic equation with real coefficients, relation between roots and coefficients, formation of quadratic equation with given roots.
   2. Symmetric functions of roots, linear and quadratic inequations.
   3. Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequalities, cube roots of unity, geometric interpretations.
   4. Arithmetic and geometric progressions, arithmetic and geometric means, infinite geometric series, Arithmetico-Geometric Progression.
   5. Sum of the first 'n' natural numbers, sums of squares and cubes of the first 'n' natural numbers.
   6. Fundamental principle of counting.
   7. Factorial n. Permutations and Combinations and simple applications.
   8. Exponential and logarithmic series.
   9. Binomial theorem (for positive integral index and for any index), general term and middle term, properties of binomial coefficients.
2. Matrices and Determinants
   1. Matrices, algebra of matrices, type of matrices.
   2. Determinants of order two and three, properties of determinants.
   3. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations.
   4. Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices.
3. Sets, Relations and Functions
   1. Sets and their representations. Different kinds of sets. Venn diagrams. Operation on Sets.
   2. De-Morgan's laws and practical problems based on them.
   3. Ordered pair, relations, domain and co-domain of relations, equivalence relation.
   4. Function as a special case of relation, domain, co-domain, range of functions, invertible functions, even and odd functions, into, onto and one-to-one functions.
   5. Special functions (polynomial, trigonometric, exponential, logarithmic, power, absolute value, greatest integer etc.), sum, difference, product and composition of functions.
   6. Inverse trigonometric functions (principal value only) and their elementary properties.
4. Analytical Geometry
   1. Two Dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.
   2. Equation of a straight line in various forms, angle between two lines, distance of a point from a line; lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines; Centroid, orthocenter, incenter and circumcenter of a triangle.
   3. General equation of second degree. Nature of conic. Equation of a circle in various forms, equation of tangent, normal and chord of a circle. Parametric equations of a circle, intersection of a circle with a straight line/ circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.
   4. Equation of a parabola, ellipse and hyperbola, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal. Problems based on locus. Polar equation of a conic, polar equation of tangent, normal, asymptotes, chord of contact, auxiliary circle, director circle of a conic and related problems.
   5. Three Dimensions: Distance between two points, direction cosines and direction ratios, equation of a straight line in space, skew lines, shortest distance between two lines, equation of a plane, distance of a point from a plane and a line, Cartesian and vector equation of a plane and a line. Angle between (i) two lines, (ii) two planes (iii) a line and a plane. Coplanar lines.
5. Calculus
   1. Limits, continuity and differentiability. Differentiation of the sum, difference, product and quotient of two functions. Differentiation of trigonometric, inverse trigonometric, logarithmic, exponential, composite and implicit functions; Second and third order derivatives.
   2. Rolle's and Lagrange's Mean value Theorems.
   3. Applications of derivatives: Rate of change of quantities, monotonic Increasing and decreasing functions, Maxima and minima of functions of one variable, tangent and normal.
   4. Integral as an anti-derivative, Integration of a variety of functions by substitution, by partial fractions and by Integration using trigonometric identities.
   5. Definite integral and their properties, application of definite integrals in finding the area under simple curves, especially lines, arcs of circles/parabolas/ellipses etc., area between the said curves (the region should be clearly identifiable).
6. Vector Algebra
   1. Vectors and scalars, magnitude and direction of a vector. Direction cosines/ratios of vectors.
   2. Types of vectors (equal, unit, zero, parallel and collinear vectors etc.), 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.
   3. Scalar (dot) product of vectors, projection of a vector on a line. Vector (cross) product of vectors. Scalar and Vector triple product and problems related to them.
7. Statistics and Probability
   1. Standard deviation, variance and mean deviation for grouped and ungrouped data. Dispersion and its various measures.
   2. Probability: Probability of an event, addition and multiplication theorems of probability, conditional probability, Bayes’ theorem, probability distribution of a random variate, Bernoulli trials and binomial distribution.

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2. Part-II Graduation Level

1. Abstract Algebra
   1. Definition and example of groups. General properties of groups, Order of an element of a group.
   2. Permutations: Even and Odd permutations. Groups of permutations. Cyclic group, Cayley’s theorem.
   3. Subgroups, Cosets, Lagrange’s theorem, Product Theorem of subgroups, Conjugate elements, conjugate complexes, Centre of a group, Simple group, center of group, Normaliser of an element and of a complex.
   4. Normal subgroups, quotient Groups, Group homomorphism and isomorphism with elementary basic properties, fundamental theorem of homomorphism in groups. Isomorphism theorems of groups.
   5. Ring Theory: Introduction to Rings, Zero divisors, Division ring, Ideals of a ring, Quotient rings, Integral Domain and Fields, their examples and properties.
2. Complex Analysis
   1. Functions, Limits, Continuity and Differentiability of complex functions.
   2. The extended plane and its spherical representation, Concept of an analytic function.
   3. Cartesian and Polar form of Cauchy-Riemann equations. Harmonic functions.
   4. Construction of an analytic function, Conformal mapping, Bilinear transformation and its properties, Fixed points, Cross ratio, Inverse point.
3. Calculus
   1. Polar Co-ordinates. Angle between radius vector and the tangent. Angle between curves in polar form. Length of polar sub-tangent and polar subnormal, Pedal equation of a curve, Derivatives of an arc, curvature, various formulae, Centre of curvature and chord of curvature and related problems.
   2. Partial differentiation, Euler’s theorem on homogeneous functions, Chain rule of partial differentiation, Maxima and Minima of functions of two independent variables and of three variables connected by a relation, Lagrange’s Method of undetermined multipliers.
   3. Asymptotes, double points, curve tracing, Envelopes and evolutes. Theory of Beta and Gamma functions. Quadrature and Rectification. Volume and Surfaces of solids of revolution.
   4. Differentiation and integration under the sign of integration. Evaluation of double and triple integrals and their applications in finding areas and volumes. Dirichlet’s integral. Change of order of integration and changing into polar co-ordinates.
4. Differential Equations
   1. Ordinary differential equations of first order and first degree, differential equations of first order but not of first degree, Clairaut’s equations, general and singular solutions.
   2. Linear differential equations with constant coefficients, homogeneous differential equation, second order linear differential equations, simultaneous linear differential equations of first order.
   3. Partial differential equations of the first order, Solution by Lagrange’s method.
5. Vector Calculus
   1. Curl, Gradient and Divergence & Identities involving these operators and related problems. Problems based on Stoke, Green and Gauss theorems.
6. Analytical Geometry of Three Dimensions
   1. Sphere: Equation of a sphere in various forms, Tangent Plane, Pole and Polar, Intersection of two spheres, Orthogonal spheres.
   2. Cone: Enveloping cone, Tangent plane, Reciprocal cone, Three mutually Perpendicular generators, Right circular cone.
   3. Cylinder: Right circular cylinder, Enveloping cylinder.
7. Statics and Dynamics
   1. Composition and resolution of co-planer forces, component of a force in two given directions, equilibrium of concurrent forces, parallel forces and moment, Friction, Virtual work and common catenary.
   2. Velocity and acceleration, Velocities and accelerations along radial and transverse directions, and along tangential and normal directions, simple linear motion under constant acceleration, Laws of motion, projectile, Simple Harmonic Motion, Rectilinear motion under variable laws.

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3. Part-III Post Graduation Level

1. Linear Algebra and Metric Spaces
   1. Definition and examples of a vector space, subspace of a vector space, Linear combination, Linear dependence and independence of vectors.
   2. Direct product of vector spaces and internal direct sums of subspaces. Bases and dimension of a finitely generated spaces, Quotient space, Isomorphism, Linear transformation (Homomorphism), Rank and nullity of linear transformation.
   3. The characteristic equation of a matrix, Eigen values and Eigen vectors, Cayley-Hamilton theorem.
   4. Definition and example of a metric space, Diameter of a set, Bounded set, Open sphere, Interior point and Interior of a set, Derived and Closure of set, Closed set, Closed Sphere, Properties of Open and Closed sets, Boundary point of set.
   5. Convergent and Cauchy sequences, complete metric space, Cantor’s Intersection theorem. Bolzano-Weierstrass theorem, Heine-Borel theorem, Compactness, connectedness, Cantor’s ternary set.
2. Integral Transforms and Special Function
   1. Laplace transform: Definition and its properties. Rules of manipulations, Laplace theorems of derivatives and integrals, Properties of Inverse Laplace transforms, Convolution theorem, Complex inversion formulas, applications to the solutions of ordinary differential equations with constant and variable coefficients.
   2. Fourier Transform: Definition and properties of Fourier sine and cosine and complex transforms, Convolution theorem.
   3. Legendre’s polynomial/ Functions: Legendre’s differential equation and associated Legendre’s differential equations, Simple properties of Legendre’s functions of first and second kind and the associated Legendre’s function of integral order.
   4. Bessel functions: Generating function, Integral formulae, Recurrence relations; Addition formula and other properties of Bessel functions.
3. Differential Geometry and Tensors
   1. Differential Geometry: Curves in Space- Definition of unit tangent vector, tangent line, Normal line and Normal plane. Contact of a curve and a surface.
   2. Tensors: Transformation of coordinates, Contravariant and covariant vectors, second order tensors, Higher order tensors. Addition, subtraction and multiplication of tensors, Contraction, Quotient Law, symmetric and skew symmetric tensors: Conjugate symmetric tensors of the second order, Fundamental tensor, Associated tensors, Christoffel symbols, Transformation law of Christoffel symbols, Covariant differentiation of vectors and tensors.
4. Numerical Analysis
   1. Difference operators and factorial notation, Differences of polynomial, Newton’s formulae for forward and backward interpolations. Divided differences, relation between divided differences and Simple difference.
   2. Newton’s general interpolation formulae, Lagrange interpolation formula. Central differences, Gauss, Stirling and Bessel interpolation formulae.
   3. Numerical Differentiation. Numerical integration, Newton-Cotes quadrature formula, Gauss’s, quadrature formulae, convergence, Estimation of errors.
   4. Transcendental and polynomial equations, bisection method, method of iteration, Trapezoidal, Simpson’s and Weddle’s rules.
5. Optimization Technique
   1. Convex sets and their properties. Simplex Method. Concepts of duality in linear programming. Framing of dual programming.
   2. Assignment problems, Transportation problems.
   3. Theory of Games: Competitive strategies, minimax and maximin criteria, two-person zero-sum games with and without saddle point.

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4. Part – IV (Educational Psychology, Pedagogy, Teaching Learning Material, Use of Computers and Information Technology in Teaching Learning)

1. Educational Psychology

   1. Concept, scope and functions of educational psychology.
   2. Physical, cognitive, social, emotional and moral developmental characteristics of adolescent learner and its implication for teaching-learning.
   3. Behavioural, cognitive and constructivist principles of learning and its implication for senior secondary students.
   4. Concept of mental health & adjustment and adjustment mechanism.
   5. Emotional intelligence and its implication in teaching learning.

2. Pedagogy and Teaching Learning Material (Instructional Strategies for Adolescent Learner)

   1. Communication skills and its use.
   2. Teaching models- advance organizer, concept attainment, information processing, inquiry training.
   3. Preparation and use of teaching-learning material during teaching.
   4. Cooperative learning.

3. Use of Computers and Information Technology in Teaching Learning

   1. Concept of ICT, hardware and software.
   2. System approach.
   3. Computer assisted learning, computer aided instruction.