Mathematics
Section A
Topic | Subtopic |
---|---|
1. Algebra | |
(i) Matrices and Types of Matrices | |
(ii) Equality of Matrices | |
- Transpose of a Matrix | |
- Symmetric and Skew Symmetric Matrix | |
(iii) Algebra of Matrices | |
(iv) Determinants | |
- Inverse of a Matrix | |
- Solving Simultaneous Equations using Matrix Method | |
2. Calculus | |
(i) Higher Order Derivatives | |
(ii) Tangents and Normals | |
(iii) Increasing and Decreasing Functions | |
(iv) Maxima and Minima | |
3. Integration and its Applications | |
(i) Indefinite Integrals of Simple Functions | |
(ii) Evaluation of Indefinite Integrals | |
(iii) Definite Integrals | |
4. Differential Equations | |
(i) Order and Degree of Differential Equations | |
(ii) Formulating and Solving Differential Equations | |
- Variable Separable | |
5. Probability Distributions | |
(i) Random Variables and its Probability Distribution | |
(ii) Expected Value of a Random Variable | |
(iii) Variance and Standard Deviation of a Random Variable | |
(iv) Binomial Distribution | |
6. Linear Programming | |
(i) Mathematical Formulation of Linear Programming Problem | |
(ii) Graphical Method of Solution for Problems in Two Variables | |
(iii) Feasible and Infeasible Regions | |
(iv) Optimal Feasible Solution |
Section B1: Mathematics
UNIT I: RELATIONS AND FUNCTIONS
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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.
-
Inverse Trigonometric Functions
- Definition, range, domain, principal value branches.
- Graphs of inverse trigonometric functions.
- Elementary properties of inverse trigonometric functions.
UNIT II: ALGEBRA
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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).
-
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
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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.
-
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.
-
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.
-
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).
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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
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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.
-
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