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Vector spaces, Linear independence bases
Dimensions of a finitely generated space.
Lineartransformation. Matrices and their
operations. Row & column reduction Echelon
form, Rank and nullity of a linear transfoamation.
System of linear equations.
2. Differential Calculus - Convergence
of sequences and series; Limit and continuity of
functions. Differentiability, Mean Value theorem.
Taylor's theorem. Expansion in Taylor's and
Maclauries Series, Maxima and Minima,
successive differentiation. Leibnitz's theorem.
Functions. of several variables Partial derivatives,
transformation. Eular's theorem.jacobian.
3. Integral Calculus - Integration of
rational , irrational and transcendental functions.
Definite integrals; Double and tripe integrals, Beta
and Gamma functions.
4. Differential Equations - Ordinary
differential equations of first and higher degree
Homogeneous equations of first degree Integrating
factors; Linear differential equations with constant
co-efficients. Complimentary functions and
perticular integrais, Linear differential equations
with variable co-efficients.
5. Vector Analysis - Algebra of vectors ,
Double and triple products (Scalar and Vectors )
, Differentiation of vectors, Directional direvatives
Gradient divergence and curl and their elementary
properties. Integratioin of vector functions along
curves.
6. Statistics - Statistical population and
random sample Collection, and presentation of
data measures of locaton dispersion Moments
and shepard's corrections, cummulants. Measures
of skewness and kurtosis.
PAPER - II
1. Analytical Geometry- Direction
cosines, plane and straight line, sphere cone and
cylinder. Tangent planes and normal lines.
2. Advanced Calculus -Tangents and
Normals. Asymptotes Curvature and Tracing of
conics. Determination of areas, lengths surfaces
and volumes (Cartesian and polar curves)
3. Real Analysis - Least upper bound and
greatest lower bounds of sets; limit point of a set
. Bolzanoweirstrass theorem. open and closed
sets and their porperties, Metric spaces. Cauchy
sequences and complete metric spares . Compact
sets and Heine-Borel theorem.
4. Boolean Algebra - Difinition and
properties. Demorgan's Laws. Switching circuits.
Boolean fuctions conjunctive and disjunctive
forms.
5. Mechanics - Equilibrium of three forces,
friction. Newton's second law of motion, impulsive
force. kinetic energy of particles angular velocity,
tragential and normal velocity and accleration
projectiles on horizontal and inclined planes
Colliston of elastic bodies, Direct and bolique
impacts.
(20)
6. Partial Differential Equations - Partial
differential equations of first order, Standards
forms, Linear partial differential equations of higher
degree with constant co-efficients. Laplace,
diffusion and vibration equation. Simple cases of
solutions of these equations as boundary value
problems.
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14.MECHANICAL
ENGINEERING
PAPER - I
1. Statics and Dynamics :
Analysis of framed structures, method of
sections. Simple harmonic motions, simple and
compound pendulum, work, energy , power,
Gyroscopic motion.
2. Theory of Machines :
Higher and lower pairs, inversions, steering
mechanisms, Hooks joint, velocity and
acceleration of links inertia-forces, Cams,
Conjugate action of gearing and interference , gear
trains , epicyclic gears, Belt drives, brakes ,
dynamometers, Fly wheels, Governors Balancing
of rotating and reciprocating masses. Free forced
and damped vibratiions for a single degree of
freedom. critical speed and whirling of shafts.
3. Mechanics of solids:
Stess and strain in two dimensions, Mohr's
circle. Theories of failure . Deflection of beams,
Buckling of columns. Combined bending and
torsion. Castigliano's theorem, Thin cylinders,
closed coil helical springs, leaf springs.
4. Manufacturing Science : Taylor's
equation. Machinability Unconventional machining
methods including EDM, ECM and ultrasonic
machining . Use of laser and plasmas. Surface
roughness, gauging comparators, jigs and Fixtures.
5. Production Management :
Work simplification, work sampling. Value
engineering . Line balancing , ABC analysis.
Economic order quantity including finite porduction
rate Graphical and simplex methods for linear
programming , transporation model, elementary
queuing theory Quality control and its uses
inproduct designs Use of X, R,σ, p and C charts.
PAPER - II
1. Theromodyamics
Applications of the first and second laws of
thermodynamics. Detailed analysis of thermo
dynamic cycles & their limitations.
2. Fluid Mechanics
Continuity momentum and energy
equations, Velocity distribution in laminar and
turbulent flow. Dimensional analysis. Boundary
layer on a flat plate; Adiabatic and insentropic flow,
Mach number.
3. Heat transfer :
Critical thickness of insulation . Conduction
in the presence of heat source. Heat transfer from
fins. Momentum and energy equations for
boundary layer on a flat plate. Dimentionless
numbers. Free and forced convection . Boiling
and condensation. Nature of radiant heat, Stafan-
Boltzmann law, Configuration. factor, Logarithmic
mean temperature difference Heat exchanger
effectiveness.
4. Energy Conversion
Combustion phenomenon in C.I. and S.I.
engines, Carburation and atomisation.
Classification of of hydraulic turbines, specific
speed, Impulses and Reaction turbines, velocity
tringles, diagram efficiency, open and closed cycle
, Gas turbine plants, Nuclear power and MHD
systems, Solar' thermal systems.
5. Environmental Control
Vapour, Compression, absorption and air
refrigeration systems. Properties and
characteristics of important refrigerants . Use of
Psychromatric chart and comfort chart. Estimation
of cooling and heating loads. Calculation of supply
air, state and rate. Summer & winter, Air-
Conditioning plants . layout.
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