KEAM brochure

ANNEXURE - I
SYLLABUS FOR THE ENTRANCE EXAMINATIONS,2023
(See Clause 9.5.1)

MATHEMATICS

UNIT I: 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; 1  2  1  2 ; 1.2  1 2 ; 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.




89 KEAM-2023, © CEE,
TVM

,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 finding the Area of a Triangle. Concept of elementary
row and column operations. 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 3X3 Determinants and Matrices should be
considered).
Linear Inequalities
Solutions of Linear Inequalities in one variable and its Graphical Representation; solution of
system of Linear Inequalities in one variable; Graphical solutions of Linear Inequalities in two
variables; solution of system of Linear Inequalities 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 II : 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
    3
functions of x for x = 0, 6, 4, 3, 2, , 2, 2 ; trigonometric functions of sum and
difference of numbers.
Tan x  Tan y
Tan x  y  
Sin x  y   Sin x Cos y  Cos x Sin y ; Cos x  y  Cos x Cos y  Sin x Sin y ; 1  Tan x Tan y ;


Sin 2  x   Sin x , Cos2  x  Cosx ; Cos  x   Cos x , Sin x   Sin x ; Cos  2  x   Sin x 

Sin 
2

 x  Cos x Cos   x  Cos x Sin   x   Sin x
; ,
Trigonometric functions of multiple and submultiples of numbers.
Sin 2 x  2 Sin x Cos x ; Sin3 x  3 Sinx - 4 Sin3 x ; Cos 2x  Cos 2 x  Sin 2 x  1  2 Sin 2 x  2 Cos 2 x  1 ;

Cos 3x  4 Cos 3 x  3 Cos x

3 Tan x - Tan 3 x x  y  x y xy x-y
Tan 3 x  Sin x  Sin y  2 Sin   Cos   Cos x  Cos y  2 Cos  Cos  
2  2   2 ;  2   2 
1  3Tan x ;
 x y x-y  x y x-y
Sin x  Sin y  2 Cos  Sin   Cos x  Cos y  - 2 Sin  Sin  
 2   2 ;  2   2 


90 KEAM-2023, © CEE,
TVM

, Conditional identities for the angles of a triangle, solution of trigonometric equations of the type
Sin x = Sin a ; Cos x = Cos a; Tan x = Tan a and equations reducible to these forms. Proofs and
simple application of sine and cosine formulae.Inverse Trigonometric functions. Range, domain,
principal value branch and graphs of inverse trigonometric functions.


(i) Sin -1 Sin x   x and other similar formula (ii)  x   Co sec
Sin -1 1 1
x
and other similar formula.
Sin 1  x    Sin 1 x , Tan  1  x   Tan  1 x ; Co sec 1  x   Co sec 1 x , Cos  1  x     Cos 1 ( x ) ;

Sec 1  x     Sec 1 ( x ) , Cot 1  x     Cot 1 ( x )

 x y 
Tan 1 x  Tan 1 y  Tan 1  , xy  -1
Sin 1 x  Cos 1 x   , Tan -1 x  Cot 1 x   Co sec 1 ( x)  Sec 1 ( x)    1  xy 
2 2; 2 ;
 x y  1 1 2 x 
 2
1 1  x   2x 
Tan 1x  Tan 1 y  Tan 1 ; xy  1 2 Tan x  Sin  
2
 Cos  Tan1 , x  1
1 x   2
 1  x2 
 1  xy  ; 1 x 


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 III: 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.


91 KEAM-2023, © CEE,
TVM