WBCS Mathematics Optional Syllabus
In WBCS main exam, there are 2 papers on the optional subject - Paper I and Paper-II. Each paper in the optional subject is 200 marks, making it a total of 400 marks. As per the WBCS official notification, the syllabus of both the papers of optional mathematics is as follows.
Download WBCS Mathematics Optional Syllabus PDF
After applying for the WBCS exam, aspirants should begin their preparation for WBCS Mathematics optional immediately. To prepare for the exam, candidates must thoroughly understand the WBCS Mathematics Optional Syllabus 2022. After studying the WBCS Mathematics Optional Syllabus, candidates can look into the WBCS Exam Pattern. The WBCS Mathematics Optional Syllabus PDF is available at the link below.
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Paper -I
| Topics | |
| Linear Algebra: | Vector spaces over R and C, linear dependence and independence, subspaces, bases, dimension; the existence of a basis for finite-dimensional vector spaces; deletion and replacement theorem. Linear transformations, rank and nullity, matrix of a linear transformation. Algebra of Matrices; Row and column reduction, Echelon form, congruence’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of a system of linear equations; Eigenvalues and eigenvectors, characteristic polynomial, Cayley-Hamilton theorem.Euclidean space, Gram-Schmidt orthogonalization. Symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices and their eigenvalues. Quadratic forms, diagonalization of symmetric matrices. |
| (2) Real Analysis I: | 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, Polar to cartesian conversion, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Open sets, limit points, closed sets. Bolzano-Weierstrass theorem .Functions of a real variable, limits, continuity. Intermediate value theorem. Differentiability, Rolle’s theorem, mean-value theorem. Higher-order differentiation, Leibnitz’s formula, Taylor’s theorem with remainders. L’Hospital’s rule. Maxima and minima; asymptotes; envelopes. |
| (3) Real Analysis II: | Compact sets. Nested interval theorem. Heine Borel theorem. Uniform continuity of functions, properties of continuous functions on compact sets. Riemann Integration. Riemann’s definition of definite integrals; Darboux theorem; Indefinite integrals; Fundamental theorems of integral calculus. Improper integrals.Sequences and series of functions. Uniform convergence. Term by term differentiation and integration.Power series. Cauchy-Hadamard test. Weierstrass approximation theorem (statement only). Fourier series. |
| (4) Analytic Geometry: | Cartesian and polar coordinates in two and three dimensions. Transformation of rectangular axes.Straight lines.Conic sections: Circle, parabola, ellipse, hyperbola and pair of straight lines. Second-degree equations in two variables, reduction to canonical forms and classification of conics. Tangents and normals to conic sections.Planes in three-dimension; the shortest distance between two skew lines. Second-degree equations in three variables, reduction to canonical forms. Sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets: tangent planes and normals. Surfaces of revolution. |
| (5) 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 function, particular integral and general solution.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. Formation of partial differential equations. Solutions of 1st order PDE, Lagrange’s method and Charpit’s method. |
| (6) Statics: | Equilibrium of a system of coplanar forces, Astatic equilibrium; Stability of equilibrium, equilibrium of forces in three dimensions. Work and potential energy, friction; Principle of virtual work. |
| (7) Particle Dynamics: | Rectilinear motion, simple harmonic motion. Damped harmonic oscillation. The motion of a particle in a plane.Work and energy, conservation of energy. Orbits under central forces. Planetary motion and Kepler’s laws. Artificial satellite. |
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Paper-II :
| Topics | |
(1)Classical Algebra | Prime integers. Existence of infinitely many primes. Relatively prime integers. Congruence. Chinese remainder theorem. Fermat’s theorem. Complex numbers; de Moivre’s theorem; complex functions.Polynomial with real coefficients. Fundamental theorem of algebra.Relation between roots and coefficients. Symmetric functions of roots. Descartes’ rule of the sign. Cardan’s method of solving a cubic equation. Ferrari’s method of solving a biquadratic equation. Binomial equations and special roots. Inequalities AM ≥ GM ≥ HM and their generalizations. Cauchy Schwarz inequality. |
| (2) Abstract Algebra | Sets and relations; equivalence relations.Groups, subgroups, cyclic groups, cosets, Lagrange’s Theorem, normal subgroups, quotient groups, |
| (3) Multivariate Calculus & Vector Analysis | Vector-valued functions of one real variable. Continuity and differentiability. Velocity and acceleration.Functions of two or three variables: limits, continuity. Directional derivative, partial derivatives, Jacobian.Chain rule. Higher-order partial derivatives. Euler’s theorem. Maxima and minima, Lagrange’s method of multipliers. Double and triple integrals; Areas and volumes.Scalar and vector fields. Differentiation of vector fields. Gradient, divergence and curl. Higher-order derivatives; Vector identities and vector equations. Line integral, Surface integral. Green’s theorem and Stokes’ theorem. |
| (4) Metric Space & Complex Analysis: | Metric spaces. Open sets and closed sets. Cauchy sequence and convergence. Completeness. Total boundedness. Compactness. Continuity, uniform continuity. Connectedness. Separable metric spaces.Baire category theorem. Examples: Rn, Cn, Space of real-valued continuous functions on [a,b]. ep spaces.The extended complex plane, stereographic projection. Differentiability of complex functions; Cauchy-Riemann equations, Analytic functions, harmonic functions; the relation between analytic and harmonic functions. |
| (5) Numerical Analysis and Computer programming: | Numerical Analysis: Interpolation. Newton’s (forward and backwards) interpolation, Lagrange’s interpolation. Solution of algebraic and transcendental equations of one variable by bisection, fixed-point iteration; Regula-Falsi and Newton-Raphson methods; Progression mathematics,near equations by Gaussian elimination and Gauss-Seidel (iterative) methods. Numerical integration: Trapezoidal rule, Simpson’s 1/3rd rule, Gaussian quadrature formula. Numerical solution of ordinary differential equations: Picard, Euler and Runge- Kutta method (4-th order).Computer Programming: Positional number system, Binary, Octal, Decimal and Hexadecimal systems; Binary arithmetic, Conversion to and from decimal systems.Algorithms and flow charts: important features, Ideas about complexities of the algorithm, applications in simple problems. Boolean algebra: Huntington postulates for Boolean algebra, algebra of sets and switching algebra as examples of Boolean algebra, duality principle, disjunctive normal and conjuctive normal forms of Boolean expressions. Design of simple switching circuit.Programming using C. |
| (6) Probability & Statistics: | Probability: Classical and frequency definitions of probability. Axioms of Probability. Multiplication rule of probabilities. Conditional probability, Bayes’ theorem. Independent events. Bernouli trials and binomial law. Bivariate samples. Sample correlation coefficient. Least square regression lines and parabolas, Polar coordinates.Statistical hypothesis. Simple and composite hypothesis. The best critical region of a test. Neyman-Pearson theorem and its application to normal population. Likelihood ratio testing and its application to normal population. |
| (7) Linear Programming: | Linear programming problems, Graphical method of solutions; hyperspace, convex sets, extreme points.Basic solution, basic feasible solution and optimal solution; Fundamental theorem of LPP; Simplex method; Duality.Transportation and assignment problems. |
Best Books for WBCS Mathematics Optional Syllabus
To complete the WBCS Mathematics Optional Syllabus, candidates can refer to the books mentioned below.
Paper I : Book List
Topic | Book Name |
Linear Algebra:
| Linear Algebra by Rao Bhimasankaram Linear Algebra by Schaum Series
|
Calculus :
| Differential Calculus – Maity & Ghosh Integral Calculus – Maity & Ghosh
|
Analytical Geometry :
| Krishna Series on Analytical Geometry Krishna Series on Analytical Solid Geometry
|
Ordinary Differential Equations:
| Ordinary and Partial Differential Equations by MD Raisinghania Advanced Differential Equations by MD Raisinghania |
Dynamics and Statics
| Fluid Dynamics by M.D. RAISINGHANIA Dynamics by P N Chatterji Hydro Dynamics by Shanti Swarup Analytical Dynamics of a Particles and of Rigid Bodies by S R Gupta
|
Vector Analysis
| Krishna Series on Vector Calculus S Chand Publications – Vector Analysis
|
Paper-II Book List
Topic
| Book Name
|
Abstract Algebra
| Abstract Algebra, Ring Theory by Sen, Ghosh Mukhopadhyay Abstract Algebra by Ramji Lal
|
Complex Analysis:
| Functions of Complex Variable by S Ponnuswamy
|
Ordinary Differential Equations:
| Ordinary and Partial Differential Equations by MD Raisinghania Advanced Differential Equations by MD Raisinghania |
Partial Differential Equations
| ODE and PDE by MD Raisinghania Engineering Maths by Grewal
|
Numerical Analysis and Computer programming:
| Numerical Methods by Jain and Iyengar Numerical Analysis chapter from Grewal, Engineering Mathematics
|
How to prepare WBCS Mathematics optional syllabus?
It is important to check the optional tips on WBCS mathematics suggested by experts. Our experts themselves have qualified for the test by following these techniques. Many candidates have scored good marks after following these tips to cover the WBCS Mathematics Optional
- Thoroughly check WBCS Mathematics optional's complete syllabus and exam pattern
- Start preparing for the exam with a schedule
- Practice with the previous year's question paper and its solution
- Keep all topic revisions from the topic on a daily basis
- Check your progress with mock tests
- Study carefully and maintain your health.
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