Even though the university never publishes a formal syllabus for its CUSAT CAT for B.Tech or B.E programmes, it follows the pattern for other graduate level admission entrances.
Students appearing for the entrance examination need to have a good command over the topics studied Class 10 – 12.
A detailed Mathematics syllabus to study for CUSAT CAT is this:
|Relations and Functions||Types of relations: reflexive, symmetric, transitive and equivalence relations. |
One to one and onto functions, composite functions, the inverse of a function. Binary operations.
Inverse Trignometric functions: Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.
Elementary properties of inverse trigonometric functions.
Matrices: concept, notation, order, equality, types of matrices, zero and identity matrix, the transpose of a matrix, symmetric and skew-symmetric matrices.
Operation on matrices: Addition and multiplication and multiplication with a scalar. 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).The 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 x 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 the number of solutions of the system of linear equations by examples, solving system of linear equations in two or three variables (having a unique solution) using the inverse of a matrix.
|Calculus||Continuity and Differentiability|
Continuity and differentiability, a derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. The concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. 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 interpretation.
Applications of Derivatives
Applications of derivatives: the rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in 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).
Integration as the inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.
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, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).
Definition, order, and degree, general and particular solutions of a differential equation. Formation of the differential equation whose general solution is given. A solution of differential equations by the method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of the linear differential equation of the type:
dy/dx + py = q, where p and q are functions of x or constants.
dx/dy + px = q, where p and q are functions of y or constants.
|Vectors and three-dimensional geometry||Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, the addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties, and application of scalar (dot) product of vectors, vector (cross) product of vectors, the scalar triple product of vectors.|
Three -Dimensional Geometry
Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, the shortest distance between two lines. Cartesian and vector equation of a plane. The angle between (i) two lines, (ii) two planes, (iii) a line and a plane. The distance of a point from a plane.