(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

(v) Inverse of a Matrix

(vi) Solving of simultaneous equations using Matrix Method

2. Calculus

(i) Higher order derivatives (second order)

(ii) Increasing and Decreasing Functions

(iii) Maxima and Minima

3. Integration and its Applications

(i) Indefinite integrals of simple functions

(ii) Evaluation of indefinite integrals

(iii) Definite Integrals

(iv) Application of Integration as area under the curve (simple curve)

4. Differential Equations

(i) Order and degree of differential equations

(ii) Solving of differential equations with variable separable

5. Probability Distributions

(i) Random variable

6. Linear Programming

(i) Graphical method of solution for problems in two variables

(ii) Feasible and infeasible regions

(iii) Optimal feasible solution

Unit II: Algebra

Matrices: Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on Matrices: Addition, 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). Invertible Matrices: Proof of the uniqueness of inverse, if it exists. (All matrices will have real entries). Determinants: Determinant of a square matrix (up to 3×3 matrices), minors, co-factors, and applications of determinants in finding the area of a triangle. Adjoint and Inverse of a Square Matrix. System of Linear Equations: Consistency, inconsistency, and number of solutions of 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.

Unit III: Calculus

Continuity and Differentiability: Continuity and differentiability, chain rule, derivatives of inverse trigonometric functions, like sin⁻¹x, cos⁻¹x, and tan⁻¹x, derivative of implicit functions. Exponential and Logarithmic Functions: Concepts of exponential, logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second-Order Derivatives. Applications of Derivatives: Rate of change of quantities, increasing/decreasing functions, 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). Integrals: Integration as an inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions, and by parts. Evaluation of Simple Integrals: Including various standard forms and problems based on them.

\(\begin{gathered} \int \frac{d x}{x^2+a^2}, \int \frac{d x}{\sqrt{x^2 \pm a^2}}, \int \frac{d x}{a^2-x^2}, \int \frac{d x}{\sqrt{a^2-x^2}}, \int \frac{d x}{a x^2+b x+c}, \int \frac{d x}{\sqrt{a x^2+b x+c}}, \\ \int \frac{(p x+q) d x}{a x^2+b x+c}, \quad \int \frac{(p x+q) d x}{\sqrt{a x^2+b x+c}}, \int \sqrt{a^2 \pm x^2} d x, \int \sqrt{x^2-a^2} d x \end{gathered} \)

• 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).
• Differential Equations:

• Definition, order and degree, general and particular solutions of a differential equation.
• Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree.
• Solutions of linear differential equation of the type:

Unit IV

Vectors and Three Dimensional Geometry

Vectors: 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, 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. Three-dimensional Geometry: Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

Unit V

Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit VI

Probability

Conditional probability, Multiplications theorem on probability, independent events, total probability, Baye’s theorem. Random variable.

Unit I: Numbers, Quantification and Numerical Applications

Modulo Arithmetic

Define Modulus of an Integer, Apply Arithmetic Operations using Modular Arithmetic Rules

Congruence Modulo

Define Congruence Modulo, Apply the definition in various problems

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

Numerical Problems

Solve real-life problems mathematically

Boats and Streams

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

Pipes and Cisterns

Determine the time taken by two or more pipes to fill or empty the tank

Races and Games

Compare the performance of two players w.r.t. time

Numerical Inequalities

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

Unit II: Algebra

Matrices and Types of Matrices

Define matrix, Identify different kinds of matrices

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

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

Algebra of Matrices

Perform operations like addition & subtraction on matrices of same order, Perform multiplication of two matrices of appropriate order, Perform multiplication of a scalar with a matrix

Determinant of Matrices

Find determinant of a square matrix, Use elementary properties of determinants, Singular matrix, Non-singular matrix,

Inverse of a Matrix

Define the inverse of a square matrix, Apply properties of inverse of matrices, Inverse of a matrix using cofactors

If A and B are invertible square matrices of the same size: (AB)⁻¹ = B⁻¹A⁻¹, (A⁻¹)⁻¹ = A, (Aᵀ)⁻¹ = (A⁻¹)ᵀ

Solving system of simultaneous equations (upto three variables only (non-homogeneous equations))

Unit III: Calculus

Higher Order Derivatives

Determine second and higher-order derivatives, Understand the differentiation of parametric functions and implicit functions

Application of Derivatives

Determine the rate of change of various quantities, Understand the gradient of tangent and normal to a curve at a given point, Write the equations of tangents and normal to a curve at a given point

Marginal Cost and Marginal Revenue using Derivatives

Define marginal cost and marginal revenue, Find marginal cost and marginal revenue

Increasing/Decreasing Functions

Determine whether a function is increasing or decreasing, Determine the conditions for a function to be increasing or decreasing

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, Solve applied problems

Integration

Understand and determine indefinite integrals of simple functions as anti-derivative, Indefinite integrals as family of curves

Evaluation of Indefinite Integrals

Solve indefinite integrals using methods of Substitution, Partial Fraction, By Parts

Definite Integral as Area Under the Curve

Define definite integral as area under the curve, Understand fundamental theorem of integral calculus and apply it to evaluate the definite integral, Apply properties of definite integrals to solve problems

Application of Integration

Identify the region representing C.S. and P.S. graphically, Apply the definite integral to find consumer surplus-producer surplus

Differential Equations

Recognize a differential equation, Find the order and degree of a differential equation, Formulating and solving differential equations

Application of Differential Equations

Define growth and decay model, Apply the differential equations to solve growth and decay models

Unit IV: Probability Distributions

Probability Distribution

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

Mathematical Expectation

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

Variance

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

Binomial Distribution

Identify the Bernoulli Trials and apply Binomial Distribution, Evaluate Mean, Variance and S.D. of a Binomial Distribution

Poisson Distribution

Understand the conditions of Poisson Distribution, Evaluate the Mean and Variance of Poisson Distribution

Normal Distribution

Understand normal distribution as a continuous distribution, Evaluate value of Standard normal variate, Area relationship between Mean and Standard Deviation

Unit V: Index Numbers and Time-Based Data

Time Series

Identify time series as chronological data

Components of Time Series

Distinguish between different components of time series

Time Series Analysis for Univariate Data

Solve practical problems based on statistical data and interpret, Understand the long-term tendency

Methods of Measuring Trend

Demonstrate the techniques of finding trend by different methods

Unit VI: Inferential Statistics

Population and Sample

Define Population and Sample, Differentiate between population and sample, Define a representative sample from a population, Draw a representative sample using simple and systematic random sampling

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 limitations of Statistic, Interpret Statistical Significance and Inferences, State Central Limit Theorem

t-Test (one sample t-test and two independent groups t-test)

Define a hypothesis, Differentiate between Null and Alternate hypothesis, Define and calculate degree of freedom, Test Null hypothesis and make inferences using t-test statistic

Unit VII: Financial Mathematics

Perpetuity, Sinking Funds

Explain and calculate perpetuity, Differentiate between sinking fund and saving account

Calculation of EMI

Explain the concept of EMI, Calculate EMI using various methods

Calculation of Returns, Nominal Rate of Return

Explain and calculate rate of return and nominal rate of return

Compound Annual Growth Rate

Understand and calculate Compound Annual Growth Rate, Differentiate between CAGR and Annual Growth Rate

Linear Method of Depreciation

Define, interpret and calculate depreciation using the linear method

Unit VIII: Linear Programming

Introduction and Related Terminology

Familiarize with terms related to Linear Programming Problem

Mathematical Formulation of Linear Programming Problem

Formulate Linear Programming Problem, Identify and formulate different types of LPP

Graphical Method of Solution for Problems in Two Variables

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

Feasible and Infeasible Regions

Identify feasible, infeasible, and bounded regions, Understand feasible and infeasible solutions, Find optimal feasible solutions

Conducting Body

National Testing Agency (NTA)

Exam Mode

Computer-Based Test (CBT)

Exam Language Options

English, Hindi, Assamese, Bengali, Gujarati, Kannada, Malayalam, Marathi, Odia, Punjabi, Tamil, Telugu, and Urdu

Question Type

Multiple Choice Questions (MCQs)

Total Questions

50 (All compulsory)

Exam Duration

60 minutes

Maximum Marks

250

Negative Marking

Yes

Marking Scheme

+5 for each correct answer, -1 for each incorrect answer

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B.Sc. Mathematics, B.Sc. Computer Science, Bachelor of Business Administration

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