Syllabus for the Exam of Teacher-Mathematics tobe Conducted by HPRCA
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Syllabus for the Exam of Teacher-Mathematics to
be Conducted by HPRCA
- MATHEMATICS OF POST GRADUATION LEVEL AS PER
ESSENTIAL QUALIFICATION:
a) NUMBER SYSTEMS: Review of representation of natural numbers, integers,
and rational numbers on the number line, Rational and irrational numbers as
recurring / terminating decimals. Operations on real numbers, definition of nth
root of a real number, Fundamental Theorem of Arithmetic.
b) POLYNOMIALS AND EQUATIONS: Definition of a polynomial in one
variable, Coefficients of a polynomial, terms of a polynomial and zero
polynomial, Degree of a polynomial, Constant, linear, quadratic and cubic
polynomials, Monomials, binomials, trinomials, Factors and multiples, Zeros of a
polynomial, Relationship between zeros and coefficients of quadratic
polynomials, Quadratic equation, Cubic equation, biquadratic equations, roots
and relations between roots and coefficients of these equations, Relationship
between discriminant and nature of roots.
c) SETS, RELATION AND FUNCTIONS: Sets and their representations, Empty
set, Finite and Infinite sets, Equal sets, Subsets, Subsets of the set of real
numbers especially intervals (with notations), Power set, Universal set, Venn
diagrams, Union and intersection of sets, Difference of sets, Complement of a
set, Properties of Complement sets, Definition of relation, Function as a special
kind of relation from one set to another, 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.
d) SEQUENCE AND SERIES: Arithmetic Progression (A.P.), Arithmetic Mean
(A.M.), Geometric Progression (G.P.), general term of a G.P., sum of n terms of a
G.P. Arithmetic and geometric series, infinite G.P. and its sum, geometric mean
(G.M.). Relation between A.M. and G.M. Sum to n terms of the special
series: 𝑛, 𝑛
2
and 𝑛
3
.
e) REAL ANALYSIS: Definition of Limit, different formulae of limits, Continuity,
discontinuity and Types of discontinuities, Real Sequence, Bounded sequence,
Cauchy convergence criterion for sequences, Cauchy’s theorem on limits, order
preservation and squeeze theorem, monotone sequences and their convergence
(monotone convergence theorem without proof). Cauchy convergence criterion
for series, positive term series, geometric series, comparison test, convergence
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of p-series, Root test, Ratio test, alternating series, Leibnitz’s test, Definition and
examples of absolute and conditional convergence.
f) DERIVATIVES AND THEIR APPLICATIONS: Derivative of a function,
derivative of composite functions, chain rule, derivatives of trigonometric and
inverse trigonometric functions, derivative of implicit function., Concepts of
exponential, logarithmic functions. Logarithmic differentiation. Derivative of
functions expressed in parametric forms. Second order derivatives. Successive
differentiation, Leibnitz’s theorem. Indeterminate forms, Rolle’s theorem,
Lagrange’s & Cauchy Mean Value theorems, Taylor’s theorem with Lagrange’s
and Cauchy’s forms of remainder, Taylor’s series. Maclaurin’s series. Concavity,
Convexity & Points of Inflexion, Curvature, Radius of curvature, center of
curvature, Asymptotes, Singular points, Double point, Functions of several
variables (upto three variables): Limit and Continuity of these functions Partial
differentiation, Euler’s theorem on homogeneous functions, Maxima and
Minima with Lagrange Multipliers Method (two variables), Jacobian (upto three
variables).Applications of derivatives: Rate of change, increasing/decreasing
functions, tangents and normals, approximation, maxima and minima (first
derivative test motivated geometrically and second derivative test. Problems
(that illustrate basic principles and understanding of the subject as well as reallife situations).
g) INTEGRALS AND THEIR APPLICATIONS: Integration of a variety of
functions by substitution, by partial fractions and by parts, Definite integrals as a
limit of a sum, Fundamental Theorem of Integral Calculus, Properties of definite
integrals and evaluation of definite integrals, Applications in finding the area
under simple curves, especially lines, arcs of circles/parabolas/ellipses (in
standard form only), area between the two curves and length of curves.
h) 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 of the type –
𝑑𝑦
𝑑𝑥
- 𝑃𝑦 = 𝑄,
where P and Q are functions of x or constant. Basic theory of linear differential
equations, Wronskian, and its properties. First order exact differential equations.
Integrating factors, rules to find an integrating factor. First order higher degree
equations solvable for x, y, p. Clairut’s form, Methods for solving higher-order
differential equations. Solving a differential equation by reducing its order. Linear
homogenous equations with constant coefficients, Linear non-homogenous equations.
The Cauchy-Euler equation and Legendre equation.
i) PARTIAL DIFFERENTIAL EQUATION: Order and degree of partial
differential equations, Concept of linear and non-linear partial differential
equations and their solution, Classification of second order partial differential
equations into elliptic, parabolic and hyperbolic through illustrations only.
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j) COMPLEX NUMBERS: Importance of complex numbers, especially −1, to
be motivated by inability to solve every quadraticequation. Algebraic properties
of complex numbers. Polarrepresentation of complex numbers. Fundamental
Theorem of Algebra, solution of quadratic equations in the complex number
system, Square-root of a Complex number. Applications of Complex numbers.
k) PERMUTATIONS AND COMBINATIONS: Fundamental principle of
counting. Factorial n. Permutations and combinations derivation of formulae and
their connections, applications of Permutations and Combinations.
l) BINOMIAL THEOREM: Binomial theorem for positive integral indices and
rational index. Pascal’s triangle, general and middle term in binomial expansion.
m) MATRICES AND DETERMINANTS: Concept, order, equality, types of
matrices, Addition, multiplication and scalar multiplication of matrices,
properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices
whose product is the zero matrix. Concept of elementary row and column
operations. Invertible matrices and uniqueness of inverse, if it exists,
Determinant of a square matrix properties of determinants, minors, cofactors
and applications of determinants in finding the area of a triangle. Adjoint and
inverse of a square matrix. Consistency, inconsistency and number of solutions
of system of linear equations by examples, solving system of linear equations in
two or three variables.
n) STATISTICS : Mean, Median, Mode, Measure of dispersion; mean deviation,
variance and standard deviation of ungrouped/groupeddata. Analysis of
frequency distributions with equal means but different variances.
o) PROBABILITY : Random experiments: outcomes, sample spaces (set
representation). Events: Occurrence of events, exhaustive events, mutually
exclusive events, Axiomatic (settheoretic) probability, Probability of an event,
Multiplications theorem on probability, Conditional probability, independent
events, total probability, Baye’s theorem. Random variable and its probability
distribution, mean and variance of haphazard variable, Repeated independent
(Bernoulli) trials and Binomial distribution.
p) TRIGONOMETRIC FUNCTIONS: Positive and negative angles. Measuring
angles in radians and in degrees and conversion from one measure to another.
Definition of trigonometric functions with the help of unit circle. Identities of
trigonometric and inverse trigonometric functions. Identities related to sin2x,
cos2x,tan2x,sin3x,cos3x and tan3xetc.,Inverse Trigonometric Functions,
Definition, range, domain, principal value branches, Elementary properties of
inverse trigonometric functions. Height and Distance.
q) TWO-DIMENSIONAL GEOMETRY (COORDINATE GEOMETRY): Shifting
of origin. Slope of a line and angle between two lines. Various forms of equations
of a line: parallel to axes, point-slope form, slope-intercept form, two-point
form, intercepts form and normal form. General equation of a line, Equation of
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family of lines passing through the point of intersection of two lines, Distance of
a point from aline, Circle, ellipse, parabola, hyperbola, a point, a straight line and
pair of intersecting lines as a degenerated case of a conic section. Equations and
properties of Circle, parabola, ellipse and hyperbola.
r) 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, scalar triple product, Gradient of a scalar point function. Divergence and
curl of a vector point function, Character of divergence and curl of a vector point
function, Gradient, Divergence and Curl of sums and products and their related vector
identities, Laplacian operator.
s) THREE-DIMENSIONAL GEOMETRY: Introduction to three-dimensional
Geometry, Coordinate axes and coordinate planes in three dimensions,
Coordinates of a point, Distance between two points and section formula,
Direction cosines/ratios of a line joining two points, Cartesian and vector
equation of 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.
t) MODERN ALGEBRA: Definition and examples of groups, abelian and nonabelian groups, Cyclic groups from number systems, groups of symmetries of (i)
an isosceles triangle, (ii) an equilateral triangle, (iii) a rectangle, and (iv) a square,
the permutation group, Subgroups, cyclic subgroups, examples of subgroups
including the centre of a group, Cosets, Index of subgroup, Lagrange’s theorem,
order of an element, Normal subgroups: their definition, examples, and
characterizations.
u) NUMERICAL METHOD: Numerical solutions of algebraic equations, Bisection
method, False position method, Fixed point iteration method, Newton’s method,
Secant method, Gauss-Jacobi, Gauss-Siedel iterative methods, Integration:
Trapezoidal rule, Simpson’s rule, Euler’s method.
- SUBJECTS OF 01 YEAR B.Ed. – Learning and Teaching, Assessment for
Learning, Teaching of Mathematics, Lesson Planning, Assessment and Evaluation,
ICT in Teaching-Learning Process, Guidance and Counseling. - GENERAL AWARENESS
(a) General knowledge: General Knowledge including General knowledge of
Himachal Pradesh.
(b) Current Affairs .
(c) Everyday Science .
(d) Logical Reasoning .
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(e) Social Science (10th standard).
(f) General English (10th standard).
(g) General Hindi (10th standard).