Syllabus for written examination for
PGT(Mathematics)
Sets :
Sets and their representations. Empty set. Finite &
Infinite sets. Equal sets. Subsets. Subsets of the set of real numbers. Power
set. Universal set. Venn diagrams. Union and Intersection of sets. Difference
of sets. Complement of a set.
Relations & Functions:
Ordered pairs, Cartesian product of sets. Number of
elements in the cartesian product of two finite sets. Cartesian product of the
reals with itself (upto R x R x R). Definition of relation, pictorial diagrams,
domain. co-domain and range of a relation. Function as a special kind of
relation from one set to another. Pictorial representation a function, domain,
co-domain & range of a function. Real valued function of the real variable,
domain and range of these functions, constant, identity, polynomial, rational,
modulus, signum and greatest integer functions with their graphs. Sum,
difference, product and quotients of functions. Sets and their Representations.
Union, intersection and complements of sets, and their algebraic properties,
Relations, equivalence relations, mappings, one-one, into and onto mappings,
composition of mappings.
Principle of Mathematical Induction:
Processes of the proof by induction. The principle of
mathematical induction.
Permutations & Combinations:
Fundamental principle of counting. Factorial n. Permutations and combinations,
derivation of formulae and their connections, simple applications.
Complex Numbers:
Complex numbers, Algebraic properties of complex numbers,
Argand plane and polar representation of complex numbers, Statement of
Fundamental Theorem of Algebra, solution of quadratic equations in the complex
number system. Modulus and Argument of a complex number, square root of a
complex number. Cube roots of unity, triangle inequality.
Linear Inequalities:
Linear inequalities. Algebraic solutions of linear
inequalities in one variable and their representation on the number line.
Graphical solution of linear inequalities in two variables. Solution of system
of linear inequalities in two variables- graphically. Absolute value,
Inequality of means, Cauchy-Schwarz Inequality, Tchebychef’s Inequality.
Binomial Theorem:
Statement and proof of the binomial theorem for positive
integral indices. Pascal's triangle, general and middle term in binomial
expansion, simple applications. Binomial Theorem for any index. Properties of
Binomial Co-efficients. Simple applications for approximations.
Sequence and Series:
Sequence and Series. Arithmetic, Geometric and Harmonic
progressions (G.P.), General terms and sum to n terms of A.P., G.P. and H.P. Arithmetic Mean (A.M.), Geometric
Mean (G.M.), and Harmonic Mean (H.M.), Relation between A.M., G.M. and H.M.
Insertion of Arithmetic, Geometric and Harmonic means between two given
numbers. Special series, Sum to n terms
of the special series. . Arithmetico-Geometric Series, Exponential and
Logarithmic series.
Elementary Number Theory:
Peano’s Axioms, Principle of Induction; First Principle,
Second Principle, Third Principle, Basis
Representation Theorem, Greatest Integer Function Test of
Divisibility, Euclid’s algorithm, The Unique Factorisation Theorem, Congruence,
Sum of divisors of a number . Euler’s totient function, Theorems of Fermat and
Wilson.
Quadratic Equations:
Quadratic equations in real and complex number system and
their solutions. Relation between roots and co-efficients, nature of roots,
formation of quadratic equations with given roots; Symmetric functions of
roots, equations reducible to quadratic equations – application to practical
problems.
Polynomial functions, Remainder & Factor Theorems and
their converse, Relation between roots and coefficients, Symmetric functions of
the roots of an equation. Common roots.
Matrices and Determinants:
Determinants and matrices of order two and three,
properties of determinants, Evaluation of determinants. Area of triangles using
determinants, Addition and multiplication of matrices, adjoint and inverse of
matrix. Test of consistency and solution of simultaneous linear equations using
determinants and matrices.
Two dimensional Geometry:
Cartesian system of rectangular co-ordinates in a plane,
distance formula, section formula, area of a triangle, condition for the
collinearity of three points, centroid and in-centre of a triangle, locus and
its equation, translation of axes, slope of a line, parallel and perpendicular
lines, intercepts of a line on the coordinate axes.
Various forms of equations of a line, intersection of
lines, angles between two lines, conditions for concurrence of three lines,
distance of a point from a line, Equations of internal and external bisectors
of angles between two lines, coordinates of centroid, orthocentre and
circumcentre of a triangle, equation of family of lines passing through the
point of intersection of two lines, homogeneous equation of second degree in x
and y, angle between pair of lines through the origin, combined equation of the
bisectors of the angles between a pair of lines, condition for the general
second degree equation to represent a pair of lines, point of intersection and
angle between two lines.
Standard form of equation of a circle, general form of
the equation of a circle, its radius and centre, equation of a circle in the
parametric form, equation of a circle when the end points of a diameter are
given, points of intersection of a line and a circle with the centre at the
origin and condition for a line to be tangent to the circle, length of the
tangent, equation of the tangent, equation of a family of circles through the
intersection of two circles, condition for two intersecting circles to be
orthogonal.
Sections of cones, equations of conic sections (parabola,
ellipse and hyperbola) in standard forms, condition for y = mx + c to be a
tangent and point(s) of tangency.
Trigonometric Functions:
Positive and negative angles. Measuring angles in radians
& in degrees and conversion from one measure to another. Definition of
trigonometric functions with the help of unit circle. Graphs of trigonometric
functions. Expressing sin (x+y) and cos (x+y) in terms of sinx, siny, cosx & cosy. Identities related to sin2x, cos2x,
tan 2x, sin3x, cos3x and tan3x. Solution of trigonometric equations, Proofs and
simple applications of sine and cosine formulae. Solution of triangles. Heights
and Distances.
Inverse Trigonometric Functions:
Definition, range, domain, principal value branches. Graphs
of inverse trigonometric functions.
Elementary properties of inverse trigonometric functions.
Differential Calculus:
Polynomials, rational, trigonometric, logarithmic and
exponential functions, Inverse functions. Graphs of simple functions. Limits,
Continuity and differentiability; Derivative, Geometrical interpretation of the
derivative, Derivative of sum, difference, product and quotient of functions.
Derivatives of polynomial and trigonometric functions, Derivative of composite
functions; chain rule, derivatives of inverse trigonometric functions,
derivative of implicit function. Exponential and logarithmic functions and
their derivatives. Logarithmic differentiation. Derivative of functions
expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's
Mean Value Theorems and their geometric interpretations.
Applications of Derivatives:
Applications of derivatives: rate of change, increasing /
decreasing functions, tangents & normals, approximation, maxima and minima.
Integral Calculus:
Integral as an anti-derivative. Fundamental integrals
involving algebraic, trigonometric, exponential and logarithmic functions.
Integration by substitution, by parts and by partial fractions. Integration
using trigonometric identities. Definite integrals as a limit of a sum,
Fundamental Theorem of Calculus. Basic Properties of definite integrals and
evaluation of definite integrals; Applications of definite integrals in finding
the area under simple curves, especially lines, areas of circles / Parabolas /
ellipses, area between the two curves.
Differential Equations:
Definition, order and degree, general and particular
solutions of a differential equation. Formation of differential equation whose
general solution is given. Solution of differential equations by method of
separation of variables, homogeneous differential equations of first order and
first degree. Solutions of linear differential equation.
Vectors:
Vectors and scalars, magnitude and direction of a
vector. Direction cosines / ratios of vectors. Types of vectors (equal, unit,
zero, parallel and collinear vectors), position vector of a point, negative of
a vector, components of a vector, addition of vectors, multiplication of a
vector by a scalar, position vector of a point dividing a line segment in a
given ratio. Scalar (dot) product of vectors, projection of a vector on a line.
Vector (cross) product of vectors.
Three dimensional Geometry:
Coordinates of a point in space, distance between two
points; Section formula, Direction cosines / ratios of a line joining two
points. Cartesian and vector equationof a line, coplanar and skew lines,
shortest distance between two lines. Cartesian and vector equation of a plane.
Angle between (i) two lines, (ii) two planes. (iii) a line and a plane.
Distance of a point from a plane. Scalar and vector triple product. Application
of vectors to plane geometry. Equation of a
sphere, its centre and radius. Diameter form of the
equation of a sphere.
Statistics:
Calculation of Mean, median and mode of grouped and
ungrouped data. Measures of dispersion; mean deviation, variance and standard deviation
of ungrouped / grouped data. Analysis of frequency distributions with equal
means but different variances.
Probability:
Random experiments: outcomes, sample spaces. Events:
occurrence of events, exhaustive events, mutually exclusive events, Probability
of an event, probability of 'not', 'and' & 'or' events., Multiplication
theorem on probability. Conditional probability, independent events,, Baye's
theorem, Random variable and its probability distribution, Binomial and Poisson
distributions and their properties.
Linear Algebra
Examples of vector spaces, vector
spaces and subspace, independence in vector spaces, existence of a Basis, the
row and column spaces of a matrix, sum and intersection of subspaces. Linear
Transformations and Matrices, Kernel, Image, and Isomorphism, change of bases,
Similarity, Rank and Nullity. Inner Product spaces, orthonormal sets and the
Gram- Schmidt Process, the Method of Least Squares. Basic theory of
Eigenvectors and Eigenvalues, algebraic and geometric multiplicity of eigen
value, diagonalization of matrices, application to system of linear
differential equations. Generalized Inverses of matrices, Moore-Penrose
generalized inverse.
Real quadratic forms, reduction and classification of
quadratic forms, index and signature, triangular reduction of a pair of forms,
singular value decomposition, extrema of quadratic forms. Jordan canonical
form, vector and matrix decomposition.
Analysis
Monotone functions and functions of bounded variation. Real
valued functions, continuous functions, Absolute continuity of functions,
standard properties. Uniform continuity, sequence of functions, uniform
convergence, power series and radius of convergence. Riemann-Stieltjes
integration, standard properties, multiple integrals and their evaluation by
repeated integration, change of variable in multiple integration. Uniform
convergence in improper integrals, differentiation under the sign of integral -
Leibnitz rule.
Dirichlet integral, Liouville’s extension. Introduction to
n-dimensional Euclidean space, open and closed intervals (rectangles), compact
sets, Bolzano-Weierstrass theorem, Heine-Borel theorem. Maxima-minima of
functions of several variables, constrained maxima-minima of functions.
Analytic function, Cauchy-Riemann equations, singularities, Statement of Cauchy
theorem and of Cauchy integral formula with applications, Residue and contour
integration. Fourier and Laplace transforms, Mellin’s inversion theorem.
