WBPSC/WBPCS/WBCS Maths Optional Syllabus Paper-1

**PAPER-I **

**(1) Linear
Algebra:** Vector spaces over R and C, linear
dependence and independence, subspaces, bases, dimensions, 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 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.

**(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 integral; Double and triple integrals (evaluation techniques only);
Areas, surface and volumes.

**(3) Analytic
Geometry:** Cartesian and polar coordinates in
three dimensions, second degree equations in three variables, reduction to
Canonical forms; straight lines, shortest distance between two skew lines,
Plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and
two sheets and their properties.

**(4) Ordinary
Differential Equations:** Formulation
of differential equations; Equations of first order and first degree, integrating
factor; Orthogonal trajectory; Equations of first order but not of first
degree, Clairaut’s equation, singular solution. Second and higher order liner
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 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.

**(5) Dynamics and
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;
Principle of virtual work; Stability of equilibrium, equilibrium of forces in
three dimensions.

**(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 equation. Application to geometry: Curves in space,
curvature and torsion; Serret-Frenet’s formulae. Gauss and Stokes’ theorems,
Green's identities.