HSA Mathematics



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.

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
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

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
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
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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