Civil Service Mathematics Main Exam Paper Two Syllabus
PAPER-I
SECTION-A
1. Linear Algebra: Vector spaces over R and C, linear dependence and independence, subspaces bases,dimension;Linear transformation ,rank and nullity, matrix of a linear transformation
Algebra of Matrices; Row and Colum reduction, Echelon form, congruence’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigen values and eigen vectors, characteristic polynomial, Cayley-Hamilton theorem,Symmetric,Skew-symmetric Hermitian,Skew-hermitian,orthogal 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 intergrals;Infinite and improper integrals Double and triple integrals (evaluation techniques only); Areas, surface and volumes.
3. Analytic Geometry: Cartesian and polar coordinates in three dimensions, second degree
equation in three variables, reduction to canonical forms,straihnt lines, shortest distance between two skew lines; Plane, sphere,cone,cylinder,paraboloid,ellipsoild,hyperboloid of one and two sheets and their properties.
4 Ordinary Differential Equations: Formation 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 linear equations with constant coefficients, complementary function particular integral and general solution.
Second order linear equations with coefficients, Euler-Cauchy equation; Determination 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 function. Application to initial value problems for 2nd order linear equations with constant coefficients.
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;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 equations
Application to geometry: Curves in space, Curvature and torsion;Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities
PAPER-I
SECTION-A
1. Linear Algebra: Vector spaces over R and C, linear dependence and independence, subspaces bases,dimension;Linear transformation ,rank and nullity, matrix of a linear transformation
Algebra of Matrices; Row and Colum reduction, Echelon form, congruence’s and similarity; Rank of a matrix; Inverse of a matrix; Solution of system of linear equations; Eigen values and eigen vectors, characteristic polynomial, Cayley-Hamilton theorem,Symmetric,Skew-symmetric Hermitian,Skew-hermitian,orthogal 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 intergrals;Infinite and improper integrals Double and triple integrals (evaluation techniques only); Areas, surface and volumes.
3. Analytic Geometry: Cartesian and polar coordinates in three dimensions, second degree
equation in three variables, reduction to canonical forms,straihnt lines, shortest distance between two skew lines; Plane, sphere,cone,cylinder,paraboloid,ellipsoild,hyperboloid of one and two sheets and their properties.
4 Ordinary Differential Equations: Formation 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 linear equations with constant coefficients, complementary function particular integral and general solution.
Second order linear equations with coefficients, Euler-Cauchy equation; Determination 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 function. Application to initial value problems for 2nd order linear equations with constant coefficients.
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;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 equations
Application to geometry: Curves in space, Curvature and torsion;Serret-Frenet’s formulae. Gauss and Stokes’ theorems, Green’s identities
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