Subject-wise Syllabus & Books for UPSC Optional Mathematics

UPSC Optional Mathematics PAPER – I

(1) Linear Algebra: 

• Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; linear transformations, rank and nullity, matrix of a linear transformation. Algebra of Matrices; row and column reduction, echelon form, congruence, and similarity; the rank of a matrix; the inverse of a matrix; solution of a system of linear equations; eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem, symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices, and their eigenvalues.
• Books:
  • SCHAUM SERIES – Seymour Lipschutz 
  • LINEAR ALGEBRA – Hoffman and Kunze 

(2) Calculus: 

• Real numbers, functions of a real variable, limits, continuity, differentiability, mean value theorem, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes; curve tracing; functions of two or three variables: limits, continuity, partial derivatives, maxima, and minima, Lagrange’s method of multipliers, Jacobian. Riemann’s definition of definite integrals; indefinite integrals; infinite and improper integrals; double and triple integrals (evaluation techniques only); areas, surface, and volumes.
• Books:
  • MATHEMATICAL ANALYSIS – S C Malik and Savita Arora
  • ELEMENTS OF REAL ANALYSIS – Shanti Narayan and M D Raisinghania

(3) Analytic Geometry: 

• Cartesian and polar coordinates in three dimensions, second-degree equations in three variables, reduction to canonical forms, straight lines, the shortest distance between two skew lines; plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets, and their properties.
• Books:
  • ANALYTICAL SOLID GEOMETRY – Shanti Narayan and P K Mittal
  • SOLID GEOMETRY – P N Chatterjee

(4) Ordinary Differential Equations: 

• Formulation of differential equations; equations of the first order and first degree, integrating factor; orthogonal trajectory; equations of first order but not of the first degree, Clairaut’s equation, singular solution. Second and higher-order linear equations with constant coefficients, complementary functions, particular integrals, and general solutions. Second-order linear equations with variable coefficients, Euler-Cauchy equation; determination of complete solution when one solution is known using the method of variation of parameters. Laplace and inverse Laplace transforms and their properties; Laplace transforms of elementary functions. Application to initial value problems for 2nd order linear equations with constant coefficients.
• Books:
  • ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS – M D Raisinghania

(5) Dynamics & Statics: 

• Rectilinear motion, simple harmonic motion, motion in a plane, projectiles; constrained motion; work and energy, conservation of energy; Kepler’s laws, orbits under central forces. Equilibrium of a system of particles; work and potential energy, friction; common catenary; the principle of virtual work; stability of equilibrium, equilibrium of forces in three dimensions.
• Books:
  • KRISHNA SERIES 

(6) Vector Analysis: 

• Scalar and vector fields, differentiation of vector field of a scalar variable; gradient, divergence and curl in cartesian and cylindrical coordinates; higher order derivatives; vector identities and vector equations. Application to geometry: curves in space, curvature, and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities.
• Books:
  • SCHAUM SERIES – Murray R. Spiegel

UPSC Optional Mathematics PAPER-II

(1) Algebra: 

• Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, homomorphism of groups, basic isomorphism theorems, permutation groups, Cayley’s theorem. Rings, subrings and ideals, homomorphisms of rings; integral domains, principal ideal domains, Euclidean domains, and unique factorization domains; Fields, quotient fields.
• Books:
  • CONTEMPORARY ABSTRACT ALGEBRA – Joseph Gallian

(2) Real Analysis: 

• Real number system as an ordered field with the least upper bound property; sequences, the limit of a sequence, Cauchy sequence, completeness of real line; Series and its convergence, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Continuity and uniform continuity of functions, properties of continuous functions on compact sets. Riemann integral, improper integrals; fundamental theorems of integral calculus. Uniform convergence, continuity, differentiability, and integrability for sequences and series of functions; partial derivatives of functions of several (two or three) variables, maxima, and minima.
• Books:
  • MATHEMATICAL ANALYSIS – S C Malik and Savita Arora
  • ELEMENTS OF REAL ANALYSIS – Shanti Narayan and M D Raisinghania

(3) Complex Analysis: 

• Analytic functions, Cauchy-Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, power series representation of an analytic function, Taylor’s series; singularities; Laurent’s series; Cauchy’s residue theorem; contour integration.
• Books:
  • SCHAUM SERIES – Speigel, Lipschitz, Schiller, Spellman

(4) Linear Programming: 

• Linear programming problems, basic solution, basic feasible solution, and optimal solution; graphical method and simplex method of solutions; duality. Transportation and assignment problems.
• Books:
  • LINEAR PROGRAMMING AND GAME THEORY – Lakshmishree Bandopadhyay

(5) Partial differential equations: 

• Family of surfaces in three dimensions and formulation of partial differential equations; solution of quasilinear partial differential equations of the first order, Cauchy’s method of characteristics; Linear partial differential equations of the second order with constant coefficients, canonical form; equation of a vibrating string, heat equation, Laplace equation, and their solutions.
• Books:
  • ADVANCED DIFFERENTIAL EQUATIONS – M D Raisinghania 

(6) Numerical Analysis and Computer programming: 

• Numerical methods:
  • solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi, and Newton-Raphson methods; solution of a system of linear equations by Gaussian elimination and Gauss-Jordan (direct), Gauss-Seidel(iterative) methods. Newton’s (forward and backward) interpolation, and Lagrange’s interpolation. Numerical integration: Trapezoidal rule, Simpson’s rules, Gaussian quadrature formula. Numerical solution of ordinary differential equations: Euler and Runge Kutta-methods.
• Computer Programming:
  • Binary system; Arithmetic and logical operations on numbers; Octal and Hexadecimal systems; Conversion to and from decimal systems; Algebra of binary numbers. Elements of computer systems and concept of memory; Basic logic gates and truth tables, Boolean algebra, normal forms. Representation of unsigned integers, signed integers and reals, double precision reals, and long integers. Algorithms and flow charts for solving numerical analysis problems.
• Books:
  • COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES – M.Goyal
  • NUMERICAL METHODS – Jain, Iyengar, and Jain 
  • DIGITAL LOGIC AND COMPUTER DESIGN – M. Morris Mano

(7) Mechanics and Fluid Dynamics: 

• Generalized coordinates; D’ Alembert’s principle and Lagrange’s equations; Hamilton equations; Moment of inertia; Motion of rigid bodies in two dimensions. Equation of continuity; Euler’s equation of motion for inviscid flow; Stream-lines, the path of a particle; Potential flow; Two-dimensional and axisymmetric motion; Sources and sinks, vortex motion; Navier-Stokes equation for a viscous fluid.
• Books:
  • KRISHNA SERIES

Free Courses for UPSC Optional Mathematics

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• Free Linear Algebra Course: Link to the Video Lectures
• Free Real Analysis Course: Link to the Video Lectures

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Subject-wise Book List for HTET Exam 

For all levels ie PRT (I Level), TGT (II Level), and PGT (III Level) the following books shall be necessary and sufficient (ie. very helpful)

• Child Development and Pedagogy:
  • RBD Publication
• English:
  • Wren & Martin
  • Class IX or X Grammar Book
  • SP Bakshi (Arihant Publication)
• Hindi:
  • Hindi Vyakran Book ( Class VIII or Class IX )
  • Diamond Publication
• General Intelligence and Reasoning Ability:
  •  Dr. R.S. Agarwal ( S Chand Publication )
• Quantitative Aptitude:
  • Dr. R.S. Agarwal ( S Chand Publication )
• Haryana G.K.:
  • Know Your State Haryana
• Mathematics:
  • R D Sharma Objective Mathematics
  • Books for DSSSB Mathematics
  • Class X-XII NCERTs

Exam Pattern

Exam Pattern Varies slightly as per the three different levels of the HTET examination. It has been detailed level-wise as follows:

PRT ( I Level )

TGT ( II Level )

PGT ( III Level )

Syllabus

1. Child Development and Pedagogy 
   • In Child Development and Pedagogy, Child psychology focuses on the mind and behavior of children from parental development through adolescence. Child psychology deals with the growth of children physically, mentally, and emotionally, and it also helps in social development.
     • Child Development
       • Growth & development and its relationship with learning
       • Heredity & Environment
       • Socialization processes, Piagel
     • Theories
       • Jean Piaget’s Moral Development Theory
       • Bruner’s Theory of Cognitive Development Theory
       • Kohlberg’s Moral Development Theory
       • Havighurst’s Moral Development Theory
       • Vygotsky’s Socio-Cultural Development Theory
       • Erickson’s Theory of Psychosocial Development
       • Bronfenbrenner’s Ecological Theory
       • David Elkind’s Theory of Adolescence Egocentrism
     • Kohlberg, and Vygotsky
       • Constructs and critical perspectives
       • Concepts of child-centered and progressive education
       • Multi-Dimensional Intelligence, Language & Thought
       • Gender roles, gender-bias and educational practice
       • Individual differences among learners
       • Assessment & Evaluation
       • Formulating appropriate questions
       • Critical thinking in the classroom, and for assessing learner achievement.
     • Inclusive education
       • Understanding children with special needs
       • Diverse backgrounds including disadvantaged and deprived & Talented
       • Creative, Specially abled Learners
     • Learning and Motivation
       • Basic processes of teaching and learning
       • Strategies of learning
       • Understanding children’s ‘errors’
       • Cognition & Emotions, Motivation, and learning
         
2. Language-Hindi
   • हिंदी भाषा परिचय
   • संज्ञा
   • सर्वनाम
   • विशेषण
   • क्रिया
   • वचन
   • लिंग
   • वाच्य
   • अव्यय
   • पदबंध
   • संधि
   • उपसर्ग एवं प्रत्यय
   • वाक्य निर्माण (सरल, संयुक्त एवं मिश्रित वाक्य)
   • पर्यायवाची
   • काल
   • विपरीपार्थक
   • अनेकार्थक
   • समानार्थी शब्द
   • मुहावरे एवं लोकोक्तियाँ
   • अलंकार
   • तत्सम, तद्भव
   • देशज एवं विदेशी शब्द
   • समास
     
3. Language-English
   • Articles
   • Adverbs (Types)
   • Modals
   • Syntax (Subject Verb Agreement)
   • Narration
   • Pronouns
   • Adjective (Degree)
   • Verb
   • Prepositions
   • Tenses
   • Punctuation
   • Voice (Active & Passive)
   • Vocabulary
     • One Word
     • Idioms
     • Antonym & Synonyms
   • Phrases
   • Figures of Speech
   • Sentences (Conditional Sentences)
   • Clauses
     
4. Quantitative Aptitude
   • Number System, Fractions
   • Algebra, Ratio, and Proportion
   • Understanding Elementary Shapes (2-D and 3-D)
   • Mensuration, Profit & loss
   • Average, problem on age
   • Speed distance &Time
   • Percentage, SI & CI
   • Time & Work
     
5. Reasoning
   • Analogies, similarities
   • Syllogism, space visualization
   • Problem-solving, Blood Relation
   • Arithmetical reasoning
   • Figure classification
   • Logical Reasoning
     
6. Haryana GK
   • Haryana districts
   • Stadium, culture
   • Folk dance, food
   • All historical places, etc.
     
7. Mathematics 
   • Complex Numbers
   • Series
     • Arithmetic Progression (AP)
     • Geometric Progression (GP)
     • Harmonic Progression (HP)
   • Functions
   • Binomial Theorem
   • Coordinate Geometry
   • Trigonometry
     • Trigonometric Ratios & Identities
     • Properties of Triangles & Circles Connected with Trigonometry
     • Heights & Distances
     • Trigonometric Equations
   • Probability
   • Calculus
     • Limits
     • Continuity
     • Differentiability
     • Indefinite & Definite Integrals
     • Areas of Bounded Regions
     • Differential Equations
   • Vectors
     • Algebra of Vectors
     • Scalar Product of 2 vectors & 3 vectors
     • Vector Product of 2 vectors & 3 vectors
   • Three Dimensional Geometry
     • 3-D Coordinate System
     • Planes & Straight Lines in Space
       
8. Environment Studies 
   • Food: Sources, Components of food
   • Materials, The World of the Living
   • Moving Things People and Ideas
   • How things work, Electric current and circuits
   • Magnets, Natural Phenomena
   • Natural Resources

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