# KEAM Maths Syllabus

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## Kerala Engineering Architecture Medical 2021 Maths Syllabus

#### Unit 1 : Algebra

Sets, Relations and Functions

Sets and their representations: Finite and Infinite sets; Empty set; Equal sets; Subsets; Power set; Universal set; Venn Diagrams; Complement of a set; Operations on Sets (Union, Intersection and Difference of Set); Applications of sets: Ordered Pairs, Cartesian Product of Two sets; Relations, reflexive, symmetric, transitive and equivalence relations. Domain, Co-domain and Range: Functions: into, onto, one – one into, one-one onto Functions; Constant Function; Identity Function; composition of Functions; Invertible Functions; Binary Operations.

Complex Numbers

Complex Numbers in the form a + i b ; Real and Imaginary Parts of a complex Number; Complex Conjugate, Argand Diagram, Representation of Complex Number as a point in the plane; Modulus and Argument of a Complex Number; Algebra of Complex Numbers; Triangle Inequality; | Ζ1 + Ζ2 | ≤ | Ζ1 | + | Ζ|; | Ζ12 | = | Ζ1 | | Ζ2 | ; Polar Representation of a Complex Number and square root of a complex number. Solution of a Quadratic Equation in the Complex Number System.

Sequences and Series

Sequence and Examples of Finite and Infinite Sequences; Arithmetic Progression (A..P): First Term, Common Difference, nth Term and sum of n terms of an A.P.; Arithmetic Mean (A.M); Insertion of Arithmetic Means between any Two given Numbers; Geometric Progression (G.P): first Term, Common Ratio and nth term, Sum to n Terms, infinite GP and its sum. Geometric Mean (G.M); Insertion of Geometric Means, Relation between AM and GM. between any two given numbers . Formula for finding the sum of first n natural numbers, sum of the squares of first n natural numbers and sum of the cubes of first n natural numbers.

Permutations, Combinations, Binomial Theorem and Mathematical Induction

Fundamental Principle of Counting; The Factorial Notation; Permutation as an Arrangement; Meaning of P ( n, r ); Combination : Meaning of C ( n, r ); Applications of Permutations and Combinations. Statement of Binomial Theorem; Proof of Binomial Theorem for positive integral Exponent using Principle of Mathematical Induction and also by combinatorial Method; General and Middle Terms in Binomial Expansions; Properties of Binomial Coefficients; Binomial Theorem for any Index ( without proof ); Application of Binomial Theorem. The Principle of Mathematical Induction, simple Applications.

Matrices and Determinants

Concept of a Matrix; Types of Matrices; Equality of Matrices ( only real entries may be considered ) : Operations of Addition, Scalar Multiplication and Multiplication of Matrices; Statement of Important Results on operations of Matrices and their Verifications by Numerical Problem only; Determinant of a Square Matrix; Minors and Cofactors; singular and non-singular Matrices; Applications of Determinants in ( i ) finding the Area of a Triangle ( ii ) solving a system of Linear Equations ( Cramer’s Rule ); Transpose, Adjoint and Inverse of a Matrix; Consistency and Inconsistency of a system of Linear Equations; Solving System of Linear Equations in Two or Three variables using Inverse of a Matrix ( only up to 3×3 Determinants and Matrices should be considered ).

Linear In-equations

Solutions of Linear Inequation in one variable and its Graphical Representation; solution of system of Linear Inequations in one variable; Graphical solutions of Linear inequations in two variables; solutions of system of Linear Inequations in two variables.

Mathematical Reasoning

Mathematically acceptable statements and their Negation. Connecting words /phrases consolidating the understanding of if and only if condition, implies, and/or, implied by, there exists. Validating the statements involving the connecting words, difference among contradiction, converse and contrapositive.

#### Unit 2 : Trigonometry

Trigonometric functions and Inverse Trigonometric functions : Degree measures and Radian measure of positive and negative angles; relation between degree measure and radian measure, definition of trigonometric functions with the help of a unit circle, periodic functions, concept of periodicity of trigonometric functions, value of trigonometric functions of X for X = 0, π/6, π/4, π/3, π/2, π, 3π/2, 2π; trigonometric functions of sum and difference of numbers.  Simple problems

Graph of the following trigonometric functions;

y = Sin x ; y = Cos x ; y = Tan x ; y = a Sin x ;y = a Cos x, y = a Sin bx ; y = a Cos bx;

#### Unit 3 : Geometry

Lines and Family of lines

Cartesian system of coordinates in a plane, shifting of origin. Distance formula, Slope of line, parallel and perpendicular lines. Various forms of equations of a line parallel to axes, slope-intercept form, The Slope point form, Intercept form, Normal form, General form, Intersection of lines. Equation of bisectors of angle between two lines, Angles between two lines, condition for concurrency of three lines, Distance of a point from a line, Equations of family of lines through the intersection of two lines.

Conic sections

Sections of a cone. Circles, standard form of the equation of a circle, its radius and centre. Equations of conic sections [Parabola, Ellipse and Hyperbola] in standard form and simple properties.

Vectors

Vectors and scalars, Magnitude and Direction of a vector, Types of vectors (Equal vectors, unit vector, Zero vector). Position vector of a point, Localized and free vectors, parallel and collinear vectors, Negative of a vector, components of a vector, Addition of vectors, multiplication of a vector by a scalar, position vector of point dividing a line segment in a given ratio, Application of vectors in geometry. Scalar product of two vectors, projection of a vector on a line, vector product of two vectors.

Three-Dimensional Geometry

Coordinate axes and coordinate planes in three dimensional space, coordinate of a point in space,
distance between two points, section formula, direction cosines, and direction ratios of a line joining two points, projection of the join of two points on a given line, Angle between two lines whose direction ratios are given, Cartesian and vector equation of a line through

(i) a point and parallel to a given vector
(ii) through two points, Collinearity of three points, coplanar and skew lines, Shortest distance between two lines, Condition for the intersection of two lines,

Cartesian and vector equation of a plane

(i) When the normal vector and the distance of the plane from the origin is given

(ii) passing through a point and perpendicular to a given vector

(iii) Passing through a point and parallel to two given lines through the intersection of two other planes

(iv) containing two lines (v) passing through three points, Angle
between (i) two lines (ii) two planes (iii) a line and a plane, Condition of coplanarity of two lines in vector and Cartesian form, length of perpendicular of a point from a plane by both vector and Cartesian methods.

#### Unit 4 : Statistics

Statistics and probability

Mean deviation, variance, standard deviation for grouped an ungrouped data. Analysis of frequency distributions with equal means but different variances. Random experiments and sample space, Events as subset of a sample space, occurrence of an event, sure and impossible events, Exhaustive events, Algebra of events, Meaning of equality likely outcomes, mutually exclusive events. Probability of an event; Theorems on probability; Addition rule, Multiplication rule, Independent experiments and events. Finding P (A or B), P (A and B), Bayes’ theorem, random variables, Probability distribution of a random variable and its mean and variance. Repeated independent (Bernoulli) trials and Binomial distribution.

#### Unit 5 : Calculus

Functions, Limits and continuity

Concept of a real function; its domain and range; Modulus Function, Greatest integer function : Signum functions; Trigonometric functions and inverse trigonometric functions and their graphs; composite functions, Inverse of a function. and quotient of continuous functions; Continuity of special functions-Polynomial, Trigonometric, exponential, Logarithmic and Inverse trigonometric functions.

Differentiation

Derivative of a function; its geometrical and physical significance; Relationship between continuity and differentiability; Derivatives of polynomial, basic trigonometric, exponential, logarithmic and inverse trigonometric functions from first principles; derivatives of sum, difference, product and quotient of functions; derivatives of polynomial, trigonometric, exponential, logarithmic, inverse trigonometric and implicit functions; Logarithmic differentiation; derivatives of functions expressed in parametric form; chain rule and differentiation by substitution; Derivatives of Second order.

Application of Derivatives
Rate of change of quantities; Tangents and Normals; increasing and decreasing functions and sign of the derivatives; maxima and minima; Greatest and least values; Rolle’s theorem and Mean value theorem; Approximation by differentials. Simple problems.

Indefinite Integrals

Integration as inverse of differentiation; properties of integrals; Integrals involving algebraic,
trigonometric, exponential and logarithmic functions; Integration by substitution; Integration by
parts; Integrals of the type: Integration of rational functions; Partial fractions and their use in integration; Integrals of the type Definite Integrals
Definite integral as limit of a sum; Fundamental theorems of integral calculus without proof;
Evaluation of definite integrals by substitution and by using the following properties. Application of definite integrals in finding areas bounded by a curve, circle, parabola and ellipse in standard form between two ordinates and x-axis; Area between two curves, line and circle; line and parabola: line and ellipse.

Differential Equations
Definition; order and degree; general and particular solutions of a differential equation; formation of differential equations whose general solution is given; solution of differential equations by method of Separation of variables; Homogeneous differential equations of first order and their solutions; Solution of linear differential equations of the type (dy/dx)+P(x)y=Q(x) where P (x), Q (x) are functions of x or constants.

Linear Programming

Introduction, related terminology such as constraints, 0bjective function, optimisation, different types of linear programming problems, mathematical formulation of Linear Programming Problems, graphical method of solution for problems in two variables, feasible and infeasible regions, feasible and infeasible solutions, optimal feasible solutions ( up to three non-trivial constraints).

#### JEE Main

Application Form Submission 16 Dec 2020 to 16 Jan 2021.