Syl Lab Us Entrance MSC
Syl Lab Us Entrance MSC
Syl Lab Us Entrance MSC
Unit-I
- definition of the limit of a function. Basic properties of limits. Infinitesimals;
Definition with examples. Theorems on infinitesimals. Comparing infinitesimals,
Definition with examples and related theorems. Principal part of an infinitesimal and
related theorems. Continuity and basic properties of continuous functions on closed
intervals. If a function is continuous in a closed interval, then it is bounded therein. If a
function is continuous in a closed interval [a, b], then it attains its bounds at least once in
[a,b]. Differentiation, Rolles theorem with proof and its applications. Lagranges Mean
value theorem and Cauchys Mean value theorem with their applications.
Unit-II
Taylors and Maclaurins theorem with their applications. Intermediate forms.
Successive differentiation with Leibnitz theorem.
Tangents and normals (polar co-ordinates only). Pedal equations, length of arcs. Partial
differentiation of functions of two and three variables. Eulers theorem on homogeneous
functions. Curvature, radius of curvature for Cartesian and polar coordinates, double
points, Asymptotes, Cartesian and polar coordinates, envelopes, involutes and evolutes,
tracing of curves( Cartesian coordinates only).
Unit-IIII
Review of complex number system, triangle inequality and its generalization. Equation of
circle (Apollonius circle), Geometrical representation of complex numbers. De Moiveres
theorem for rational index and its application. Expansion of Sin n, Cos n etc. in terms
of powers of Sin , Cos and expansion of Sinn and Cosn in terms of multiple
angles of Sin and Cos
Functions of complex variable. Exponential, circular, Hyperbolic, Inverse hyperbolic
and Logarithmic functions of a complex variable and their properties. Summation of
trigonometric series, Difference method, C + iS method.
Unit-IV
Parabola: Equation of tangent and normal, pole and polar, pair of tangents from a point,
equation of a chord of a parabola in terms of its middle point, parametric equations of a
parabola. Ellipse; Tangents and Normals, pole and polar, parametric equations of ellipse,
Diameters, conjugate diameters and their properties.
Hyperbola: Equations of tangents and normals, equation of hyperbola referred to
asymptotes as axes, Rectangular and conjugate diameters and their properties. Tracing of
conics (Cartesian co-ordinates only).
The plane, Every first degree equation in X,Y,Z represents a plane, Equation of plane in
normal and intercepts forms , and through points. Systems of planes, Two sides of a
plane. Bisectors of angles between two planes, joint equation of two planes, Volume of a
tetrahedron in terms of the co-ordinates of its vertices. Straight line. Equation in
symmetrical and unsymmetrical form. Equation of a straight line through two points.
Transformation of the equation of a line to the symmetrical form. The condition that two
given lines may intersect.
Unit-V
Sphere; Definition and equation of a Sphere, condition for two spheres to be orthogonal.
Radical plane. Coaxial system. Simplified form of the equation of two spheres.
Definition of Cone, Vertex, guiding curve, generator, equation of cone with vertex as
origin or a given vertex and guiding curve, condition that the general equation of the
second degree should represent a cone. Angle between generators of section of a cone
and plane through vertex. Necessary and sufficient conditions for a cone to have three
mutually perpendicular generators. Definition of a cylinder, equation of the cylinder
whose generators intersect a given conic and are parallel to given line enveloping
cylinder of a sphere. Central conicoids. Tangent lines and tangent planes. Normal to
conicoid at a point on it. Normal from a point to a conicoid, polar plane. Shapes and
features of the three central coincides. Diametric planes. Generating lines of ruled
surfaces.
Unit VI
Review of the methods of integration, integration by substitution and by parts, integration
of algebraic rational functions; case of non-repeated or repeated linear factors. Case of
linear or quadratic non-repeated factors. Integration of algebraic rational functions by
substitution, integration of irrational functions, Reduction formulae.
Review of the definite integral as the limit of a sum. Summation of series with the help
of definite integrals. Quaderature. Area of a region bounded by a curve, X-axis (y-axis)
and two ordinates (abscissa), Sectorial areas bounded by a closed curve. Lengths of plane
curves. Volumes and surfaces of revolution.
Vector Analysis: Scalar and vector product of three and four vectors. Reciprocal vectors.
Vector functions of a single scalar variable, limit of a vector function, continuity. Vector
Differentiation, Gradient, Divergence and curl. Vector integration. Theorems of Gauss,
Green, Stokes and problems based on these.
Unit-VII
Degree and order of a differential equations. Equations of first order and first degree.
Equations in which the variables are separable. Homogeneous equations. Linear
equations and equations reducible to linear form. Bernoullis equations, Exact differential
equations, Symbolic operators. Linear differential equations with constant coefficients.
Differential equations of the forms f (D) y = Sin ax, eaxV, where V is any function of x.
Homogeneous linear equations.
Miscellaneous form of differential equations. First order higher degree equations solvable
for x,y,z,p. Equations from which one variable is explicitly absent, Clairuts form,
equations reducible to Clairuts form. Legendre polynomials. Recurrence relation and
differential equation satisfied by it. Bessel functions, recurrence relation and differential
equation.
Unit-VIII
Symmetric, Skew-symmetric, Hermition and skew-Hermition matrices, Diagonal, scalar
and triangular matrices, sum of matrices and properties of the addition composition.
Representation of a square matrix as a sum of a symmetric (Hermition) and a skewsymmetric (Skew-Hermition) matrix. Representation of a square matrix in the form of
P + iQ, where P and Q are both Hermition.
divided into a finite number of subintervals such that oscillation of f in each of the
subintervals can be made arbitrarily small. If a function is continuous on [a,b], and f(a)
and f(b) are of opposite in sign, then there is at least one point c in (a,b) such that f(c) = 0.
The intermediate value theorem. Darboux intermediate value theorem for derivative.
Unit-XII
Riemann- Integration: The Riemann- Integral ,Definition and existence of the Riemann
integral. Upper and lower sums. Refinement of a partition. Under a refinement , the lower
sums do not decrease and upper sums do not increase. The necessary and sufficient
condition for integrability of bounded functions. Integrability of sum, difference, product
and quotient of two functions If f is bounded and integrable on [a,b], then so is |f| and
b
f ( x )dx f ( x ) dx M (b-a).
a