Section A

Section B1: Mathematics

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

   • Definition, 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 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).

2. 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.

UNIT III: CALCULUS

1. 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 x and e^x.
   • 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.

2. Applications of Derivatives

   • Applications of derivatives: Rate of change, increasing/decreasing functions, tangents 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).
   • Tangent and Normal.

3. Integrals

   • Integration as inverseprocess of differentiation.
   • Integration of avarietyof functions bysubstitution, bypartial fractions andbyparts, only simple integrals ofthetype –

   $$\int \frac{dx}{x^2 \pm a^2}, \quad \int \frac{dx}{\sqrt{x^2 \pm a^2}}, \quad \int \frac{dx}{\sqrt{a^2 - x^2}}, \quad \int \frac{dx}{ax^2 + bx + c}, \quad \int \frac{dx}{\sqrt{ax^2 + bx + c}},$$

   $$\int \frac{(px + q)}{ax^2 + bx + c} , dx, \quad \int \frac{(px + q)}{\sqrt{ax^2 + bx + c}} , dx, \quad \int \sqrt{a^2 \pm x^2} , dx \quad \text{and} \quad \int \sqrt{x^2 - a^2} , dx,$$

   $$\int \sqrt{ax^2 + bx + c} , dx \quad \text{and} \quad \int (px + q) \sqrt{ax^2 + bx + c} , dx $$

   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.

4. 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).

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 equation of the type –

   $$\frac{dy}{dx} + Py = Q, \text {where P and Q are functions of x or constant}$$

   $$\frac{dx}{dy} + Px = Q, \text{where P and Q are functions of y or constant.} $$

UNIT IV: VECTORS AND THREE-DIMENSIONAL GEOMETRY

1. 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.

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

Unit V: 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 feasible solutions (up to three non-trivial constraints).

Unit VI: Probability

• Multiplication theorem on probability.
• Conditional probability, independent events, total probability, Baye’s theorem.
• Random variable and its probability distribution, mean and variance of a random variable.
• Repeated independent (Bernoulli) trials and Binomial distribution.

Section B2: Applied Mathematics

Unit I: Numbers, Quantification and Numerical Applications

A. Modulo Arithmetic

• Define modulus of an integer
• Apply arithmetic operations using modular arithmetic rules

B. Congruence Modulo

• Define congruence modulo
• Apply the definition in various problems

C. Allegation and Mixture

• Understand the rule of allegation to produce a mixture at a given price
• Determine the mean price of a mixture
• Apply rule of allegation

D. Numerical Problems

• Solve real life problems mathematically

E. Boats and Streams

• Distinguish between upstream and downstream
• Express the problem in the form of an equation

F. Pipes and Cisterns

• Determine the time taken by two or more pipes to fill or drain

G. Races and Games

• Compare the performance of two players w.r.t. time, distance taken/distance covered/ work done from the given data

H. Partnership

• Differentiate between active partner and sleeping partner
• Determine the gain or loss to be divided among the partners in the ratio of their investment with due consideration of the time

I. Numerical Inequalities

• Describe the basic concepts of numerical inequalities
• Understand and write numerical inequalities

UNIT II: ALGEBRA

A. Matrices and Types of Matrices

• Define matrix
• Identify different kinds of matrices

B. Equality of Matrices, Transpose of a Matrix, Symmetric and Skew Symmetric Matrix

• Determine equality of two matrices
• Write transpose of given matrix
• Define symmetric and skew symmetric matrix

UNIT III: CALCULUS

A. Higher Order Derivatives

• Determine second and higher order derivatives
• Understand differentiation of parametric functions and implicit functions
• Identify dependent and independent variables

B. Marginal Cost and Marginal Revenue Using Derivatives

• Define marginal cost and marginal revenue
• Find marginal cost and marginal revenue

C. Maxima and Minima

• Determine critical points of the function
• Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
• Find the absolute maximum and absolute minimum value of a function

UNIT IV: PROBABILITY DISTRIBUTIONS

A. Probability Distribution

• Understand the concept of Random Variables and its Probability Distributions
• Find probability distribution of discrete random variable

B. Mathematical Expectation

• Apply arithmetic mean of frequency distribution to find the expected value of a random variable

C. Variance

• Calculate the Variance and S.D. of a random variable

UNIT V: INDEX NUMBERS AND TIME BASED DATA

A. Index Numbers

• Define Index numbers as a special type of average

B. Construction of Index Numbers

• Construct different types of index numbers

C. Test of Adequacy of Index Numbers

• Apply time reversal test

D. Time Series

• Identify time series as chronological data

E. Components of Time Series

• Distinguish between different components of time series

F. Time Series Analysis for Univariate Data

• Solve practical problems based on statistical data and interpret

UNIT VI: INFERENTIAL STATISTICS

A. Population and Sample

• Define Population and Sample
• Differentiate between population and sample
• Define a representative sample from a population

B. Parameter and Statistics and Statistical Inferences

• Define Parameter with reference to Population
• Define Statistics with reference to Sample
• Explain the relation between Parameter and Statistic
• Explain the limitation of Statistic to generalize the estimation for population
• Interpret the concept of Statistical Significance and Statistical Inferences
• State Central Limit Theorem
• Explain the relation between Population-Sampling Distribution-Sample

UNIT VII: FINANCIAL MATHEMATICS

A. Perpetuity, Sinking Funds

• Explain the concept of perpetuity and sinking fund
• Calculate perpetuity
• Differentiate between sinking fund and saving account

B. Valuation of Bonds

• Define the concept of valuation of bond and related terms
• Calculation of Bond Using Present Value Approach

C. Calculation of EMI

• Explain the concept of EMI
• Calculate EMI using various methods

D. Linear Method of Depreciation

• Define the concept of linear method of Depreciation
• Interpret cost, residual value and useful life of an asset from the given information
• Calculate depreciation

UNIT VIII: LINEAR PROGRAMMING

A. Introduction and Related Terminology

• Familiarize with terms related to Linear Programming Problem

B. Mathematical Formulation of Linear Programming Problem

• Formulate Linear Programming Problem

C. Different Types of Linear Programming Problems

• Identify and formulate different types of LPP

D. Graphical Method of Solution for Problems in Two Variables

• Draw the Graph for a system of linear inequalities involving two variables and to find its solution graphically

E. Feasible and Infeasible Regions

• Identify feasible, infeasible and bounded regions

F. Feasible and Infeasible Solutions, Optimal Feasible Solution

• Understand feasible and infeasible solutions
• Find optimal feasible solution