HPSC Maths Optional Syllabus

1. Linear Algebra:

Vector space, linear dependence and independence, subspaces, bases, dimensions. Finite dimensional vector spaces. Matrices, Cayley Hamilton theorem, Eigen values and Eigenvectors. Matrix of linear transformation, row and column reduction, Echelon form, equivalence, congruence and similarity, reduction to canonical form, rank, orthogonal, symmetrical, skew symmetrical, unitary, Hermitian, skew Hermitian forms their Eigen values.  Orthogonal and unitary reduction of quadratic and Hermitian forms, positive definite quadratic form.

2. Calculus:

Real numbers, limits, continuity, differentiability, mean value theorems, Taylor’s theorem with remainders, indeterminate forms, maxima and minima, asymptotes.  Functions of several variables:  continuity, differentiability, 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).  Repeated Integrals.  Beta and Gamma functions.  Areas, surface and volumes, centre of gravity. 

3. Analytic Geometry:

Cartesian and polar coordinates in two and three dimensions, second degree equations in two and three dimensions, Reduction to canonical forms, straight lines, plane, sphere, cone, cylinder, paraboloid, ellipsoid, hyperboloid of one and two sheets and their properties. shortest distance between two skew lines.

4. 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, Clairaut’s equation, singular solution.  Higher order linear equations with constant coefficients, complementary function and particular integral, general solution. Euler-Cauchy equations.  Second order linear equations with variable coefficients, Determination of complete solution when one solution is known, method of variation of parameters. 

5. 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. Lemi’s theorem

6. Dynamics:

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, Kepler's laws, orbits under central forces, motion of varying mass, motion under resistance.

7. Hydrostatics:

Pressure of heavy fluids, equilibrium of fluids under given system of forces. centre of pressure, thrust on curved surfaces, equilibrium of floating bodies, stability of equilibrium, metacentre, pressure of gases.  Problems relating to atmosphere.