**MU OET 2021 Mathematics Syllabus**

**SETS, RELATIONS AND FUNCTIONS**

**Sets :** 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.

**Relations and 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 × R × 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 of a function, domain, co-domain and 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

Relations and functions : Types of relations: Reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.Partial fractions, Logarithms and its related properties. 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. Truth of the identity sin 2x + cos2x x = 1 , for all x. Signs of trigonometric functions and sketch of their graphs.

Expressing sin ( x+y ) and cos ( x+y ) in terms of sin x, sin y, cos x and cos y. Deducing the identities like following :

Identities related to sin 2x ,cos2x , tan 2x ,sin3x ,cos3x and tan3x. General solution of trigonometric equations of the type sinθ= sinα , cosθ = cosα and tanθ = tanα. Proofs and simple applications of sine and cosine formulae.

Inverse trigonometric functions: Definition, range, domain, principal value branches. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

**ALGEBRA**

**Principle of mathematical induction :** Process of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications.

**Complex numbers and quadratic equations :** Introduction, Complex numbers, Algebra of complex numbers, Brief description of algebraic properties of complex numbers, The modulus and the conjugate of complex number, Argand plane and polar representation, Fundamental theorem of algebra. Solution of quadratic equations in the complex number system, Square root of a complex number.

**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.

**Permutations and combinations :** Fundamental principle of counting. Factorial n. Permutations and combinations derivation of formulae and their connections, simple applications.

**Binomial theorem :** History, statement and proof of the binomial theorem for positive integral indices. Pascal’s triangle, general and middle term in binomial expansion, simple applications.

**Sequence and series :** 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. Sumto n terms of the special series :

**Matrices :** Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non-commutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

**Determinants :** Determinant of a square matrix (up to 3 × 3 matrices), 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 (having unique solution) using inverse of a matrix.

**COORDINATE GEOMETRY, VECTORS AND THREE-DIMENSIONAL GEOMETRY**

**Straight lines :** Brief recall of 2-D from earlier classes, 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, twopoint form, intercepts form and normal form. General equation of a line. Equation of family of lines passing through the point of intersection of two lines. Distance of a point from a line.

**Conic sections :** Sections of a cone: Circles, ellipse, parabola, hyperbola, a point, a straight line and pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.

**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.

**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.

**CALCULUS**

**Limits and derivatives :** Derivatives introduced as rate of change both as that of distance function and

geometrically. Intuitive idea of limit of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial of trigonometric functions

**Continuity and differentiability :** Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concepts of exponential, logarithmic functions. Derivatives of log_{e} z and z^{e} . Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Rolle’s and Lagrange’s Mean Value Theorems

(without proof) and their geometric interpretations.

**Applications of derivatives :** Applications of derivatives: Rate of change, increasing / decreasing functions, tangents and normal, approximation, maxima and minima ( first derivative test motivated geometrically and second derivative test given as a provable tool ). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations ).

**Integrals :** Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type to be evaluated.

Definite integrals as a limit of a sum. Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

**Applications of the integrals :** Applications in finding the area under simple curves, especially lines, arcs of circles / parabolas / ellipses ( in standard form only ), area between the two above said curves ( the region should be clearly identifiable ).

**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 and are functions of x or constant

**dx /****dy + Px = Q** where and are functions of y or constant

**MATHEMATICAL REASONING**

Mathematically acceptable statements. Connecting words/phrases – consolidating the understanding of “if and only if (necessary and sufficient) condition”, “implies”, “and/or”, “implied by”, “and”, “or”, “there exists” and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words – difference between contradiction, converse and contrapositive.

**STATISTICS AND PROBABILITY**

**Statistics :** Measure 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 (set representation). Events: Occurrence of events, ‘not’, ‘and’ & ‘or’ events, exhaustive events, mutually exclusive events. Axiomatic (set theoretic) probability, connections with the theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’, & ‘or’ events.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.

**LINEAR PROGRAMMING**

Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming ( L.P. ) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasiblesolutions ( up to three non-trivial constrains ).

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