Examination Syllabus For TGT (Math)
Real Numbers: Representation
of natural numbers, integers, rational numbers on the number line.
Representation of terminating / non-terminating recurring decimals, on the
number line through successive magnification. Rational numbers as recurring /
terminating decimals. Examples of nonrecurring / non terminating decimals.
Existence of non-rational numbers (irrational numbers) and their representation
on the number line. Explaining that every real number is represented by a
unique point on the number line and conversely, every point on the number line
represents a unique real number. Laws of exponents with integral powers.
Rational exponents with positive real bases. Rationalization of real numbers.
Euclid's division lemma, Fundamental Theorem of Arithmetic. Expansions of
rational numbers in terms of terminating / non-terminating recurring decimals.
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, Chinese Remainder Theorem, Sum of divisors of a number .
Euler’s totient function, Theorems of Fermat and Wilson.
Matrices: R, R2,
R3 as vector spaces over R and concept of Rn. Standard basis for each of them.
Linear Independence and examples of different bases. Subspaces of R2, R3.
Translation, Dilation, Rotation, Reflection in a point, line and plane. Matrix
form of basic geometric transformations. Interpretation of eigenvalues and
eigenvectors for such transformations and eigenspaces as invariant subspaces.
Matrices in diagonal form. Reduction to diagonal form upto matrices of order 3.
Computation of matrix inverses using elementary row operations. Rank of matrix.
Solutions of a system of linear equations using matrices.
Polynomials: Definition
of a polynomial in one variable, its coefficients, with examples and counter
examples, its terms, zero polynomial. Degree of a polynomial. Constant, linear,
quadratic, cubic polynomials; monomials, binomials, trinomials. Factors and
multiples. Zeros / roots of a polynomial / equation. Remainder Theorem with
examples and analogy to integers. Statement and proof of the Factor Theorem.
Factorization of quadratic and of cubic polynomials using the Factor Theorem.
Algebraic expressions and identities and their use in factorization of
polymonials. Simple expressions reducible to these polynomials.
Linear Equations in
two variables: Introduction to the equation in two variables. Proof that a
linear equation in two variables has infinitely many solutions and justify
their being as ordered pairs of real
numbers, Algebraic and graphical solutions.
Pair of Linear
Equations in two variables: Pair of linear equations in two variables.
Geometric representation of different possibilities of solutions /
inconsistency. Algebraic conditions for number of solutions. Solution of pair
of linear equations in two variables algebraically - by substitution, by
elimination and by cross multiplication.
Quadratic Equations:
Standard form of a quadratic equation. Solution of the quadratic equations
(only real roots) by factorization and by completing the square, i.e. by using
quadratic formula. Relationship between discriminant and nature of roots.
Relation between roots and coefficients, Symmetric functions of the roots of an
equation. Common roots.
Arithmetic
Progressions: Derivation of standard results of finding the nth term and
sum of first n terms.
Inequalities: Elementary
Inequalities, Absolute value, Inequality of means, Cauchy-Schwarz Inequality,
Tchebychef’s Inequality.
Combinatorics: Principle
of Inclusion and Exclusion, Pigeon Hole Principle, Recurrence Relations,
Binomial Coefficients.
Calculus: Sets.
Functions and their graphs: polynomial, sine, cosine, exponential and
logarithmic functions. Step function. Limits and continuity. Differentiation.
Methods of differentiation like Chain rule, Product rule and Quotient rule.
Second order derivatives of above functions. Integration as reverse process of
differentiation. Integrals of the functions introduced above.
Euclidean Geometry: Axioms
/ postulates and theorems. The five postulates of Euclid. Equivalent versions
of the fifth postulate. Relationship between axiom and theorem. Theorems on
lines and angles, triangles and quadrilaterals, Theorems on areas of
parallelograms and triangles, Circles, theorems on circles, Similar triangles,
Theorem on similar triangles. Constructions. Ceva’s Theorem, Menalus Theorem,
Nine Point Circle, Simson’s Line, Centres of Similitude of Two Circles, Lehmus
Steiner Theorem, Ptolemy’s Theorem.
Coordinate Geometry:
The Cartesian plane, coordinates of a point, Distance between two points
and section formula, Area of a triangle.
Areas and Volumes: Area
of a triangle using Hero's formula and its application in finding the area of a
quadrilateral. Surface areas and volumes of cubes, cuboids, spheres (including
hemispheres) and right circular cylinders / cones. Frustum of a cone. Area of a
circle; area of sectors and segments of a circle.
Trigonometry: Trigonometric
ratios of an acute angle of a right-angled triangle. Relationships between the
ratios. Trigonometric identities. Trigonometric ratios of complementary angles.
Heights and distances.
Statistics: Introduction
to Statistics: Collection of data, presentation of data, tabular form,
ungrouped / grouped, bar graphs, histograms, frequency polygons, qualitative
analysis of data to choose the correct form of presentation for the collected
data. Mean, median, mode of ungrouped data. Mean, median and mode of grouped
data. Cumulative frequency graph. Probability:
Elementary Probability and basic laws. Discrete and Continuous Random
variable, Mathematical Expectation, Mean and Variance of Binomial, Poisson and
Normal distribution. Sample mean and Sampling Variance. Hypothesis testing
using standard normal variety. Curve Fitting. Correlation and Regression.
Teaching
Methodology:
· Learning by
Discovery: Nature and purpose of learning by discovery; guided discovery
strategies in teaching Mathematical concepts.
Teaching for
Understanding Proof: Proof by induction and deduction; proof by analysis
and synthesis,
· Problem Solving in
Mathematics: Importance of problem solving in
Mathematics, Steps of problem solving in Mathematics,
Problem Posing, Generating and solving real life problems using Mathematical
principles, Situation model for solving word problems.
· Constructivist
approaches: Self-learning and peer learning strategies,
Collaborative strategies; 5E and ICON Models,
· Preparation of Lesson Plans (Traditional, Activity and
Constructivist Approaches),
· Activities in Mathematics: Mathematics Quiz, Mathematics
Club activities, Mathematics Exhibition, Planning and organizing Mathematics
laboratory activities, Mathematics outside the classroom.
· Learning Materials in Mathematics: Types, functions,
preparation and utilization of learning materials - Textbook, Models,
Calculators and computers, Graphic calculators, Maintaining portfolio in
Mathematics
· Key Learning Resources in Mathematics: Assessing progress
and performances, Monitoring and giving feedback, Local and community
resources, Using pair work, Using group work, Using questioning (both by
teacher and learners) to promote thinking, Talk for learning and Involving all
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