Syllabus for written examination of TGT(Mathematics)
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
non-recurring / 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
written 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 variate. Curve Fitting. Correlation and Regression.
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