BA IN MATHEMATICSUniversity of Delhi
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- New delhi
BA IN MATHEMATICS
PAPER I : Algebra and Calculus
SECTION - I
Definition and examples of a vector space, Subspace and its properties,
Linear independence and dependence of vectors, basis and dimension of a
vector space. Types of matrices. Rank of a matrix. Invariance of rank under
elementary transformations. Reduction to normal form- Solutions .of linear
homogeneous and non-homogeneous equations with number of equations and
unknowns upto four. Cayley-Hamilton theorem, Characteristic roots and
SECTION - II
De Moivres theorem (both integral and rational index). Solutions of equations
using trigonometry, Expansion for Cos nq. Sin nq in terms of powers of Sin q,
Cosq, and Cosq, Sinq in terms of Cosine and Sine of multiples of q, Summation
of series, Relation between roots and coefficients ofnth degree equation.
Solutions of cubic and biquadratic equations, when some conditions on roots
of the equation are given, Symmetric functions of the roots for cubic and
biquadratic equations. Transformation of equations.
SECTION - III
Limit and.Continuity, Types of discontinuities. Differentiability of functions.
Successive differentiation, Leibnitz s theorem, Partial differentiation, Euler s
theorem on homogeneous functions.
SECTION - IV
Tangents and normals, Curvature, Asymptotes, Singular points, Tracing of curves.
SECTION - V
Rolle s theorem, Mean Value Theorems, Taylor s Theorem with Lagrange s &
Cauchy s forms of remainder. Taylor s series, Maclaurin s series of sin x, cos x,
ex, log(l+x), (l+x)m, Applications of Mean Value theorems to Monotonic functions
and inequalities. Maxima & Minima. Indeterminate forms.
SECTION - VI
Reduction formulae, Integration of irrational and trigonometric functions.
Properties of definite integrals. Quadrature, Rectification of curves, Volumes
and areas of surfaces of revolution.
Geometre, Differential Equations and Algebra
UNIT-I : Geometry
Techniques for sketching parabola, ellipse and hyperbola. Reflection
properties of parabola, ellipse and hyperbola and their applications to signals,
classification of quadratic equation representing lines, parabola, ellipse and
UNIT-II : 3-Dimensional Geometry and Vectors
Rectangular coordinates in 3-space; spheres, cylindrical surfaces cones.
Vectors viewed geometrically, vectors in coordinate system, vectors determine by
length and angle, dot product, cross product and their geometrical properties.
Parametric equations of lines in plane, planes in 3-space.
UNIT-III : Ordinary differential equations
First order exact differential equations including rules for finding
integrating factors, first order higher degree equations solvable for x, y, p, Wronskian
and its properties, Linear homogeneous equations with constant coefficients, Linear
non-homogeneous equations. The method of variation of parameters. Euler s
equations. Simultaneous differential equations. Total differential equations.
Applications of ordinary differential equations to Mixture Problems, Growth
and Decay, Population Dynamics and Orthogonal trajectories.
UNIT-IV: Partial differential equations
Order and degree of partial differential equations, Concept of linear and
non-linear partial differential equations, formation of first order partial differential
equations. Linear partial differential equations of first order, Lagrange s method,
Charpit s method, classification of second order partial differential equations into
elliptic, parabolic and hyperbolic through illustrations only.
Applications to Traffic Flow.
Integers modulo n, Permutations, Groups, subgroups, Lagrange's Theorem,
Euler's Theorem, Symmetry Groups of a segment of a line, and regular n-gons for
n=3, 4, 5 and 6. Rings and subrings in the context of C[0,1] and Zn.
UNIT-1 Analysis (38 marks)
Order completeness of Real numbers, open and closed sets, limit point of
sets, Bolzano Weierstrass Theorem, properties of continuous functions,
Sequences, convergent and Cauchy sequences, sub-sequences, limit
superior and limit inferior of a sequence, monotonically increasing and
decreasing sequences, infinite series and their convergences, positive term
series, comparison tests, Cauchy s nth root test, D Alembert s ratio test,
Raabe s test, alternating series, Leibnitz s test, absolute and conditional
Riemann integral, integrability of continuous and monotonic functions,
improper integrals and their convergences, comparison tests, Beta and
Gama functions and their properties, Pointwise and uniform convergence of
sequences and series of functions, Weierstrass M-test, Uniform convergence
and continuity, Statement of the results about uniform convergence and
integrability or differentiability of functions, Power series and radius of
convergence, Fourier series.
UNIT-2 Computer Programming
Programming: Preliminaries, constants, variables, type declaration,
expressions, assignment statements, input-output statements, Control
statements, functions, Arrays, simple programs using these concepts.
Control statements, functions, arrays, Format specification.