Syllabus for written examination of TGT(Mathematics)

Real Number:
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 Principal, 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.

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 and Euclid. Equivalent versions of the fifth postulate. Relationship between axiom and theorem.Theorems and 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.

Trigonometry: Trigonometric ratios of an acute angle of a right – angled triangle. Relationships between the rations. 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.