Kerala PSC High School Assistant [HSA] Mathematics Exam Syllabus 2015. Detailed Exam Syllabus of Kerala PSC High School Assistant [HSA Mathematics] Examination published by Kerala public service commission.

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**HSA Mathematics Detailed Exam Syllabus**

**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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