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HSA Mathematics

MATHEMATICS
PART A

 


Module I : Renaissance and freedom movement
Module II: General Knowledge and current affairs
Module III: Methodology of teaching the subject


♦ History/conceptual development. Need and Significance, Meaning Nature and Scope of
the Subject.
♦ Correlation with other subjects and life situations.
♦ Aims, Objectives, and Values of Teaching - Taxonomy of Educational Objectives - Old
and revised
♦ Pedagogic analysis- Need, Significance and Principles.
♦ Planning of instruction at Secondary level- Need and importance. Psychological bases of
Teaching the subject - Implications of Piaget, Bruner, Gagne, Vygotsky, Ausubel and
Gardener - Individual difference, Motivation, Maxims of teaching.
♦ Methods and Strategies of teaching the subject- Models of Teaching, Techniques of
individualising instruction.
♦ Curriculum - Definition, Principles, Modern trends and organizational approaches,
Curriculum reforms - NCF/KCF.
♦ Instructional resources- Laboratory, Library, Club, Museum- Visual and Audio-Visual
aids - Community based resources - e-resources - Text book, Work book and Hand book.
♦ Assessment; Evaluation- Concepts, Purpose, Types, Principles,
Modern techniques - CCE and Grading- Tools and techniques -
Qualities of a good test - Types of test items- Evaluation of projects,
Seminars and Assignments - Achievement test, Diagnostic test –
Construction, Characteristics, interpretation and remediation.
♦ Teacher - Qualities and Competencies - different roles - Personal
Qualities - Essential teaching skills - Microteaching - Action research.

PART B
Module I


Elementary Set Theory, Relations, Partial order, Equivalence relation, Functions, bijections,
Composition, inverse function, Quadratic equations –relation between roots and coefficients,
Mathematical induction, Permutation and combination.
Trigonometric Functions – Identities solution of triangles, heights and distances.
Geometry – Length and area of Polygons and circle.
Solids – Surface area and volume, Euler’s formula.

Module II


Theory of Numbers – divisibility, division algorithm, gcd, lcm. Relatively prime numbers (Coprimes),
Fundamental Theorem of Arithmetic, congruences, solution of linear congruences, Fermat’s
Theorem.
Matrices – Addition, Multiplication, Transpose, Determinants, singular matrices, inverse,
symmetric, skew-symmetric, hermitian, skew-hermitian, Orthogonal matrices, normal form,
echelon form, rank of a matrix. Solution of system of linear equations. Eigenvalues, eigenvectors,
Cayley Hamilton Theorem.


Module III


Calculus - Limits, Continuity, Differentiability, Derivatives, Intermediate Value Theorem, Rolle’s
Theorem, Mean value Theorem, Taylor and Maclaurin’s series, L’Hospital’s rule. Partial
differentiation, homogeneous functions, Euler’s Formula. Applications of differentiation - maxima
and minima, critical points, concavity, points of inflection, asymptotes, Tangents and normals.
Integration – methods of integration, definite integrals – properties.
Fundamental theorem of calculus.
Applications of Integration – Area between curves, volume and area of revolution.
Double and Triple Integrals
Conic sections- Standard equations – Parabola, ellipse, hyperbola, Cartesian, Parametric and polar
forms.


Module IV


Bounded sets, infinum, supremum, order completeness, neighbourhood, interior, open sets, closed
sets, limit points, Bolzano Weierstrass Theorem, closed sets, dense sets, countable sets, uncountable
sets.
Sequences – convergence and divergence of sequences, monotonic sequences, subsequences.
Series – Convergence and divergence of series, absolute convergence, Canchy’s general principle of
convergence of series. The series Σ1/np .
Tests for convergence of series – comparison test, root test, ratio test. Continuity and uniform
continuity, Riemann integrals, properties, integrability.
Complex numbers, modulus, conjugates, polar form, nth roots of complex numbers. Functions of
complex variables – Elementary functions of complex variables, Analytic functions. Taylor series,
Laurent’s Series.

Module V


Vectors – Unit vector, collinear vectors, coplanar vectors, like and unlike vectors, orthogonal triads
(i, j, k) Dot product, cross product- properties. Vector differentiation- unit tangent vector, unit
normal vector, curvature, torsion, vector fields, scalar fields, gradient divergence, curl, directional
derivatives. Vector Integration – Line Integrals, conservative fields, Green’s Theorem, Surface
Integrals, Stoke’s Theorem, Divergence Theorem.
Differential Equations – Order and degree of differential equations. First order differential
equations- solution of Linear equations, separable equations and exact equations.
Second order differential equations- Solution of homogeneous equations with constant coefficients –
various types non-homogeneous equations, solutions by undetermined coefficients.


Module VI


Data Representation: Raw Data, Classification and tabulation of data, Frequency tables, Contingency
tables; Diagrams – Bar diagrams, sub-divided bar diagrams, Pie diagrams, Graphs – Frequency
polygon, frequency curve, Ogives.
Descriptive Statistics: Percentiles, Deciles, Quartiles, Arithmetic Mean, Median, Mode, Geometric
Mean and Harmonic Mean; Range, Mean deviation, Variance, Standard deviation, Quartile
deviation; Relative measures of dispersion – Coefficient of variation; Moments, Skewness and
Kurtosis – Measures of Skewness and Kurtosis.
Probability: Random Experiment, Sample space, Events, Type of Events, Independence of events;
Definitions of probability, Addition theorem, Conditional probability, Multiplication theorem,
Baye’s theorem.


Module VII

 


Random variables and probability distributions: Random variables, Mathematical Expectation,
Definitions and properties of probability mass function, probability density function and distribution
function. Independence of random variables; Moment generating function; Standard distributions –
Uniform, Binomial, Poisson and Normal distribution.
Bivariate distribution: Joint distribution of two random variables, marginal and conditional
distributions.
Correlation and regression: Scatter Diagram, Karl Pearson’s Correlation Coefficient, Spearman’s
rank correlation coefficient. Principle of least squares – curve fitting – Simple linear regression.


Module VIII

 


Random Sampling Methods: Sampling and Census, Sampling and Non-sampling errors, Simple
random sampling, Systematic sampling, Stratified sampling.
Sampling distributions: Parameter and statistic; Standard error, sampling distributions – normal, t, F,
Chi square distributions; Central limit theorem. Estimates, Desirable properties of estimate –
Unbiasedness, consistency, sufficiency and efficiency.
Testing of hypothesis (basic concepts only) - Simple and composite hypotheses, null and alternate
hypotheses, Type I error, Type II error, Level of significance, Power of a test.


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