# Mathematics Syllabus : PPSC Exam

Published on: Apr 29, 2010

**Section A****1. Linear Algebra :**Vector spaces, linear dependence and independence, subspaces, bases, dimensions. Finite dimensional vector spaces.

Matrices, Cayley-Hamilton theorem, Eigen values and Eigen vectors, matrix of linear transformation, row and column reduction, Echelon form, equivalence, congruence and similarity, reduction to Cannonical forms, rank, orthogonal, symmetrical, skew symmetrical, unitary, Hermitian, skew-Hermitian forms and their eigenvalues. Orthogonal and unitary reduction of quadratic and Hermitian forms, positive definite quardratic forms.

**2. Calculus :**Real numbers, limits, continuity, differentiability, mean-value theorems, Taylors theorem with remainders, indeterminate forms, maxima and minima, asymptotes. Functions of several variables - continuity, differentiability, partial derivatives, maxima and minima, Lagranges method of multipliers, Jacobian. Riemanns definition of definite integrals, indefinite integrals, infinite and improper intergrals, Double and triple integrals (evaluation techniques only), beta and gamma functions. Areas, surface and volumes, centre of gravity.

**1. Analytic Geometry :**Cartesian and polar coordinates in two and three dimensions, second degree equations in two and three dimensions, reduction to Cannonical forms, straight lines, shortest distance between two skew lines, plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties.

**Section B****1. Ordinary Differential Equations :**Formulation of differential equations, order and degree, equations of first order and first degree, integrating factor, equations of first order but not of first degree, Clariauts equation, singular solution. Higher order linear equations with constant coefficients. Complementary function and particular integral, general solution, Euler-Cauchy equation. Second order linear equations with variable coefficients, determination of complete solution when one solution is known, method of variation of parameters.

**2. Statics :**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.

**Degree of freedom and constraints, rectilinear motion, simple harmonic motion, motion in a plane, projectiles, constrained motion, work and energy, conservation of energy, motion under impulsive forces, Keplers laws, orbits under central forces, motion of varying mass, motion under resistance.**

3. Dynamics :

3. Dynamics :

**4. Vector Analysis :**Scalar and vector fields, triple products, differentiation of vector function of a scalar variable, gradient, divergence and curl in cartesian, cylindrical and spherical coordinates and their physical interpretations. Higher order derivatives, vector identities and vector equations.

**Section C****1. Algebra :**Groups, subgroups, normal subgroups, homomorphism of groups, quotient groups, basic isomorsphism theorems, permutation groups, Cayley theorem. Rings and ideals, principal ideal domains, unique factorization domains and Euclidean domains.

**2. Real Analysis :**Real number system, ordered sets, bounds, ordered field, real number system as an ordered field with least upper bound property, Cauchy sequence, completeness, continuity and uniform continuity of functions, properties of continuous functions. Riemann integral, improper integrals, absolute and conditional convergence of series of real and complex terms, rearrangement of series. Uniform convergence, continuity, differentiability and integrability of sequences and series of functions. Differentiation of functions of several variables, change in the order of partial derivatives, implicit function theorem, maxima and minima.

**3. Complex Analysis :**Analytic function, Cauchy-Riemann equations, Cauchys theorem, Cauchys integral formula, power series, Taylors series, Laurents Series, Singularities, Cauchys residue theorem, contour integration. Conformal mappings, bilinear transformations.

**Section D****Linear programming problems, basic solution, basic feasible solution and optimal solution, graphical method and Simplex method of solutions. Duality. Transportation and assignment problems. Traveling salesman problems.**

1. Linear Programming :

1. Linear Programming :

**Curves and surfaces in three dimensions, formulation of partial differential equations, solutions of equation of type dx/p=dy/q=dz/r ; orthogonal trajectories, partial differential equations of the first order, solution by Cauchys method of characteristics; Charpits method of solution, linear partial differential equations of the second order with constant coefficients, equations of vibrating string, heat equation, laplace equation.**

2. Partial differential equations :

2. Partial differential equations :

**3. Numerical Analysis :**

**Solution of algebraic and transcendental equations of one variable by bisection, Regula-Falsi and Newton-Raphson methods, solution of system of linear equations by Gaussian elimination and Gauss-Jordan (direct) methods, Gauss-Seidel (iterative) method. Newtons (Forward and backward) and Lagranges method of interpolation.**

Numerical Methods :

Numerical Methods :

**Numerical Integration :**Simpsons one-third rule, tranpezodial rule, Gaussian quardrature formulae.

**Euler and Runge Kutta-methods.**

Numerical solutions of ordinary differential equations :

Numerical solutions of ordinary differential equations :