## AMUEE 2015 Mathematics Syllabus

**AMUEE Mathematics Syllabus 2015 – Aligarh Muslim University Entrance Exam Mathematics Syllabus 2015**

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**AMUEE 2015 Mathematics Syllabus :**

**UNIT – I : SETS AND FUNCTIONS**

**1. SETS **

Sets and their representations. Empty set. Finite & 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.

**2. RELATIONS & FUNCTIONS**

Orders 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 x R x R). Definition of relation, pictorial diagrams, domain, codomain 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.

**3. TRIGNOMETRIC FUCNTIONS :**

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 sin2 x + cos2 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 the following :

Identities related to sin 2 x, cos 2 x, tan 2 x, sin 3 x, cos 3 x and tan 3 x. General solution of trigonometric equiations of the type sin q = sin a

**UNIT II : ALGEBRA**

**1. 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.

**2. COMPLEX NUMBERS AND QUADRATIC EQUATIONS **

Need for complex numbers, especially -1 , to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, Solution of quadratic equations in the complex number system.

**3. 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 mof linear inequalities in two variables graphically.

**4. PERMUTATIONS AND COMBINATIONS**

Fundamental principle of counting. Factorial n (n1) Permutations and combinations, derivation of formulae and their connections, simple applications.

**5. BINOMIAL THEOREM : **

History, statement and proof of the binomial theorem for positive integral indices. Pascals triangle, General and middle term in binomial expansion, simple applications.

**6. 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., geometric mean (G > M), relation between A.M. and G.M. Sum to a terms of the

**UNIT III : COORDINATE GEOMETRY :**

**1. STRAIGHT LINES**

Brief recall of 2 D from earlier classes. Slope of a line and angel between two lines. Various forms of eqwuations 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. Distance of a point from a line.

**2. CONIC SECTION**

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

**3. INTRODUCTION TO THREE DIMENSIONAL GEOMETRY**

Coordinate axes and coordinate planes in three dimensions. Coordinatoes of a point. Distance between two points and section formula.

**UNIT IV : CALCULUS**

**1. LIMITS AND DERIVATIVES :**

Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit. Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

**UNIT V : MATHEMATICAL REASONING**

**1. 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”, “three 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.

**2. PROBABILITY :**

Random experiments : Outcomes, sample, spaces (set representation). Events : occurrence of events, `not, `and 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 and `or events.

**UNIT – I RELATIONS AND FUNCTIONS**

**1. 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.

**2. Inverse Trigonometric Functions :**

Defintion, range, domain, principal value branches, Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions

**UNIT II : ALGEBRA**

**1. Matrices :**

Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and kew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of edition, multiplication and scalar multiplication. Non commutativity of multiplication of matrices and existence of non zero matrices whose product is the zero amt4rix (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).

**2. Determinants :**

Determinant of a square matrix (upto 3 x 3 matrices), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle. A joint 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.

**UNIT III : CALCULUS**

**1. Continuity and Differentiability :**

Continuity and Differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions, derivative of implicit function. Concept of exponential and logarithmic functions and their derivative. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Rolles and Lagranges Mean Value Theorems (without proof) and their geometric interoperations.

**2. Applications of Derivatives :**

Applications of derivatives : rate of change, increasing / decreasing functions, tangets and normals, 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).

**3. 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.

Define integrals as a limit of sum, Fundamental Theorem of calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

**4. Applications of the Integrals :**

Applications in findings the area under simple curves, especially lines, areas of circles / parabolas / ellipse (in standard form only), area between the two above said curves (the region should be clearly identifiable).

**5. 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 equations of the type :

### UNIT IV : VECTORS AND THREE DIMENSIONAL GEOMETRY :

**1. Vectors : **

Vectors and scales, 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.

**2. Three – dimensional Geometry :**

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 (1) two lines, (ii) two planes, (iii) a line and plane. Distance of a point from a plane.

**UNIT V : LINEAR PROGRAMMING**

**1. Linear programming**

Introduction, definition of related terminology such as constraints, objective function, optimization, different types of linear programming (LP) problems, mathematical formulation of LP, problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optional feasible solutions (upto three non trivial constraints).

**UNIT VI : PROBABILITY**

**1. Probability :**

Multiplication theorem on probability, Conditional probability, independent events, total probability, Bayes theorem, Random Variable and its probability distribution, mean and variance of haphazard variable. Repeated independent (Bernoulli) trials

and Binomial distribution.