 # IAS/IFOS Mains Mathematics (Optional) Syllabus

## Complete Details of Contents/ Synopsis

Paper – I
1. Linear Algebra
2. Calculus
3. Analytic Geometry
4. Ordinary Differential Equations
5. Dynamics & Statics
6. Vector Analysis
Paper – II
1. Algebra
2. Real Analysis
3. Complex Analysis
4. Linear Programming
5. Partial Differential Equations
6. Numerical Analysis and Computer Programming
7. Mechanics and Fluid Dynamics.

Linear Algebra:-
Vector spaces:
(i) Definition of a field
(ii) Definitions of internal and external Compositions.
(iii) Definition of a vector space over rational real and complex fields.
(iv) Properties of vector space
(v) Problems on vector space.
(vi) Miscellaneous results and Notations
(vii)  Definition of Subspaces
(viii) Theorems on subspaces
(ix) Problems on subspaces
(x) Definition of linear combination
(xi) Definition of linear space
(xii) Definition of smallest subspace containing any subset of v (f)
(xiii) Definition of linear sum of two subspaces.
(xiv) Some Theorems
(xv) Definition of linear independence (LI) and linear dependence of subsets of vector spaces
(xvii) Theorems and problems on LI and LD
Basis and Dimension:
(i) Definition of Basis
(ii) Definition of finite and infinite dimensional vector spaces.
(iii) Theorems on the above definitions.
(iv) Problems on the above definitions by using matrices tricks.
(v) Dimension of a subspace.
(vi) Definitions Row-equivalence of two matrices and column-equivalence of two matrices.
(vii) Elementary row- operations
(viii) Row space of a matrix
(ix) Problems.
Linear transformations:
(i) Definition linear transformation
(ii) Theorems and problems on the linear transformation.
(iii) Definitions of Range and Null space (Kernal) of a linear transformation.
(iv) Definitions of Rank and Nullity..
(v) Theorems and problems on Rank and nullity
(vi) Definitions of singular and Non- Singular transformations.
(vii) Theorems and problems on singular and non-singular transformations.
(viii) Definition of Matrix of linear transformation
(ix) Theorems and problems on Matrix of linear transformations.
Matrices:-
(i) Definitions of Matrix and other basic definitions with proper examples.
(ii) Definitions of equality matrices matrix addition matrix multiplication, Idempotent matrix, involuntary matrix, Nilpotent matrix, Trace of matrix, Transpose of matrix symmetric matrix and skew- symmetrix matrix with proper examples.
(iii) Some properties of symmetric and skew symmetric matrices..
(iv) Conjugate of Matrix, Tranjugate of matrix and properties
(v) Hermition matrix, skew-hermition and properties
(vi) Determiners, Minors and cofactors and properties.
(vii) Ad joint matrix, inverse of a square matrix, singular and non-singular matrices and properties.
(viii) Orthogonal and unitary matrix and properties.
(ix) Sub matrix of matrix, minors of a matrix rank of matrix, Elementary operations echelon Matrix etc. and properties
(x) Problems on the above.
Linear Equations:
(i) Definitions of non-homogeneous linear equations and homogeneous linear equations and some theorems and working rules.
(ii problems on the above.
Eigen values and Eigen vectors:
(i) Introduction
(ii) Definitions of characteristic roots and characteristic vectors.
(iii) Certain relation between characteristic roots and characteristic vectors
(iv) Linear independence of Characteristic vectors corresponding to distinct characteristic roots some.
(v) Theorems and problems
(vi) The cayley- Hamilton terms and problems
Similarity of Matrices
(i) Definition of similarity of matrix
(ii) Theorems and problems on the above
(iii) Diagonalizable matrix
(iv) Theorems and problems on the above.
(v) Algebraic and Geometric multiplicity of a characteristic roots and problems.
(vi) Inner product space and some important observations.
(vii) Inner product of two vectors norm or length of a vector unit vector orthogonal vectors orthogonal set orthogonal set, orthogonality of similarity of matrices.
(viii) Working rule for orthogonal reduction of a real symmetric matrix and problems
(ix) Unitary similar matrices and problems.
Calculus and Real Analysis
Set – I
Real number system
(i) Introduction
(ii)  Field axioms order axioms and some properties.
(iii) Intervals finite intervals infinite intervals and length of an internal
(iv) Bounded subset of real number
(v) Completeness property of IR
(VI) Complete ordered field
(vii) The Archimedian property
(viii) Absolute value (modulus of a real numbers) and properties.
(ix) Neighborhood of a point and Neighborhood of a set
(x) Theorems and problems on the above
(xi) Interior point of a set and interior of set
(xii) Theorems and problems on the above
(xiii) Open set limit point of a subset ‘S’ of IR
(xiv) Theorems and problems on the above.
(xv) Desired set, Adherent point,, closures of a set, Dense set, Dense in itself, perfect set and problems
(xvi) Bolzano- weierstress theorem closed set compact set, and theorems and problems.
Dynamics & statics
RECTILINEAR MOTION
-Basic concepts on velocity and Acceleration.
-Derivation of motion in a straight line with constant acceleration.
-Problems on motion of a train between two stations.
-Newton’s laws of motion.
-Definition of simple Harmonic motion.
-problems on simple Harmonic motion.
-Definition Hooke’s law
-Derivation of particle attached to one end of a horizontal elastic string.
-Problems on particle attached to one end of a horizontal elastic string.
-Derivation of particle suspended by an elastic string
Problems on particle suspended by an elastic string.
-Derivation of motion under inverse square law.
-Problems on motion under inverse square law.
-Problems on motion under miscellaneous laws of forces.
Projectiles
Derivation of The motion of a projectile and its Trajectory.
Latus, rectum, vertex, focus and direction of the trajectory.
Derivation of time of flight, Horizontal range and maximum height.
Derivation of velocity at any point of the trajectory.
Derivation of locus of the focus and vertax of the trajectory.
Problems on.
Derivation of projections of hit a given point.
Problems on projections to hit a given point.
Derivation of Range and time of flight on an inclined plane.
Derivation of range and time of flight down an inclined plane.
Problems on range and time of flight on an inclined plane & time of flight down an inclined plane.
Work, Energy and Impulse:-
Basic concepts of work, energy and impulse.
Definition of work done by a variable force.
Definition of power, units of work.
Definition of kinetic Energy, calculation of kinetic Energy
Derivation of the work-energy principle.
Definition & derivation of conservative and non-conservative forces.
Definition of potential energy
Definition & derivation of the principle of conservation of Energy.
Derivation of the principle of conservation of energy for the motion in a plane.
Derivation of the principle of conservation of linear momentam.
Definition of impulse
Derivation of impulse-momentum principle for a particle.
Problems on work, energy and impulse.
Problems on the principle of conservation of energy for the motion in a plane, conservation of linear momentum, impulse-momentum principle for a particle.
Constrained Motion:
Definition of constrained motion.
Derivation of motion in a vertical circle.
Problems on motion in a vertical circle.
Definition of cycloid
Derivation of motion on a cycloid.
Problems on motion on a cycloid.
Derivation of motion on the outside of a smooth cycloid with its axis vertical and vertex upwards & problems.
Definition of a simple pendulum.
Derivation of oscillations of a simple pendulum.
Problems on oscillations of a simple pendulum.
Motion in a Resisting Medium
Definition of Terminal velocity.
Derivation of a particle falling under Gravity & problems.
Derivation of a particle projected vertically upwards & problems
Derivation of a particle projected vertically down ward and problems.
Statics
Virtual work.
Basic concepts of Displacement.
-Basic concepts of virtual work & principle of virtual work.
Problems on virtual work.
Problems relating to bodies or frameworks resting on pegs or on inclined planes.
Problems involving Elastic strings.
Problems in which the nature of stress is to be found out.
Problems involving curves.
Strings in two Dimensions
Definition of category & uniform (or) common catenary.
Derivation of Intrinsic equation of the common catenary.
Derivation of Cartesian equation of the common catenary.
-Some important relations for the common catenary.
-Derivation of sag of tightly stretched wires.
Problems of sag of tightly stretched wires.
Stable and Unstable Equilibrium
-Definitions of stable and unstable equilibrium.
-Work function test for the nature of stability of equilibrium.
Potential energy test for the nature of stability.
Z-test for the nature of stability.
Derivation of a body-resting on a fixed rough surface. & problems.
Problems based upon Z-test.
Mechanics & Fluid Dynamics
Mechanics
Moments and products of Interia

• Defintions of Rigid-body, moment of Interia of a particle  moment of Inertia of a system of particles, moment of Interia of a body, radius of Gyration product of Inertia.
• Moment and product of Intertia with respect to three mutually perpendicular axes.
• Some simple propositions.
• Derivation of moments of Inertia in some simple cases
• (a) Moment of Intertia of a uniform rod of length 2a
• Moment of Intertia of a rectangular lamina.
• M.I. of a circular wire.
• M.I. of circular plate.
• M.I. of an elliptic disc
• M.I. of a uniform triangular lamina about one side
• M.I. of a rectangular parallelopiped about an axis through its centre and parallel to one of its edges.
• M.I. of a spherical shell (i.e. hollow sphere) about diameter.
• M.I. of a solid sphere about a diameter.
• M.I. of an ellipsoid.
• Problems on moment of Inertia
• Theorem of parallel axis.
• Problems on Theorem of parallel axis.
• Derivation of moment and product of Interia of a plane lamina about a line & problems.
• Derivation of moment of Intertia of a body about a line. and problem
• Derivation of principle axes & problems

D’ Alembert’s Principle

• Definitions of Impressed forces, effective force’s
• Derivation of D ‘ Alembert’s principle
•  Derivation of General Equations of motion  of a body
• Definition of linear momentum
• Derivation of motion of the centre of Inertia
• Derivation of Motion Relative to the centre of Inertia
• Problems on motion relative to the centre of Inertia.
• Definition of Impulse of a force.
• Derivation of General equations of motion under-Impulsive forces.
• Problems on General equations of motion under Impulsive forces.

Fluid Dynamics
Kinematics (Equations of Continuity)

• Definitions of basis concepts.
• Different methods on fluid motion.
• Definitions of kinds of motion.
• Definitions of some curves.
• Some definitions.
• Derivation of Boundary surface.
• Statement of equation of continuity
• Derivation of Equation of Continuity by Euler’s method.
•  Derivation of continuity by Lagrange’s method.
• Derivation of equivalence between eulerian and lagrangian forms of equations of continuity.
• Derivation of Generalised orthogonal curvilinear co-ordinates.
• Equation of continuity in generalised orthogonal curvilinear co-ordinates.
• Equation of continuity in Cartesian co-ordinates.
• Equation of continuity in spherical polor co- ordinates.
• Equation of continuity in cylindrical co-ordinates
• Certain symmetrical forms of equations of continuity.
• Solved problems related to stream lines and possible liquid motion.
• Solved problems related to boundary surface
• Solved problems related to Equation of continuity.

Equations of Motion.

• Theorem on Eulers Equation of motion.
• Theorem on pressure equation (Bernoulli’s equation for unsteady motion)
• Problems on Euler’s equation of motion and pressure equation.
• Bernoulli’s equation for steady motion.
• Lagranges equation of motion.
• Theorem on Cauchy’s Integrals.
• Theorem on Equations for Impulsive Actions
• Theorem on Kelvin’s circulation theorem.
• Permanence of irrotational motion.
• Problems on. Bernoulli’s Equation for steady motion, Lagrange’s equation of motion, Cauchy’s integrals. Impulsive action, Kelvin circulation, irrational motion.

Sources, sinks & Doublets (motion in two Dimensions)

• Motion in two dimensions.
•  Derivation of Lagrange’s stream function.
• The difference of the values of up at the two points represents the flux of fluid across any curve joining the two points.
• Theorem on irrotational motion in two dimensions.
• Problems on Lagrange’s stream function, Irrotational motion in two dimensions.
• Definition of complex potential.
• Cauchy-Riemann Equations in polar form.
• Definitions of two dimensional sources, sinks.
• Theorem on complex potential due to a source.
• Definition two dimensional doublets.
• Definition of Image.
• Significance of Image.
• Derivation of Image of a source in a circle.
• Derivation of Image of a doublet relative to a circle.
• Circle Theorem of Milne-Thomson.
• Derivation of Image of source W.R.T. a  circle of radius
• Blasius Theorem.
• Problems on Image of a source in a circle Image of a doublet relative to a circle.  Circle theorem of milne-Thomson, Image of source W.R.T. a circle of radius, Blasius Theorem.