CUET-UG Complete Guide

1. Algebra

• Types of Matrices
• Equality of Matrices
• Transpose of a Matrix
• Symmetric and Skew-Symmetric Matrices
• Algebra of Matrices
• Determinants
• Inverse of a Matrix
• Solving Simultaneous Equations Using Matrix Method

2. Calculus

• Higher Order Derivatives up to Second Order
• Increasing and Decreasing Functions
• Maxima and Minima

3. Integration and Its Applications

• Indefinite Integrals of Simple Functions
• Evaluation of Indefinite Integrals
• Definite Integrals
• Application of Integration as Area Under the Curve

4. Differential Equations

• Order and Degree of Differential Equations
• Solving Differential Equations with Variable Separable

5. Probability Distributions

• Simple Probability

6. Linear Programming

• Graphical Method of Solution for Problems in Two Variables
• Feasible and Infeasible Regions
• Optimal Feasible Solution

UNIT I: Relations and Functions

1. Relations and Functions

• Types of Relations:
  • Reflexive Relations
  • Symmetric Relations
  • Transitive Relations
  • Equivalence Relations
• One-to-One Functions
• Onto Functions

2. Inverse Trigonometric Functions

• Definition
• Range
• Domain
• Principal Value Branches
• Graphs of Inverse Trigonometric Functions

UNIT II: Algebra

1. Matrices

• Concept and Notation
• Order of a Matrix
• Equality of Matrices
• Types of Matrices
• Zero Matrix
• Transpose of a Matrix
• Symmetric Matrices
• Skew-Symmetric Matrices
• Addition of Matrices
• Multiplication of Matrices
• Multiplication of a Matrix by a Scalar
• Properties of Addition, Multiplication and Scalar Multiplication
• Non-Commutativity of Matrix Multiplication
• Existence of Non-Zero Matrices Whose Product is the Zero Matrix
• Invertible Matrices
• Uniqueness of Inverse, If It Exists

2. Determinants

• Determinant of a Square Matrix up to 3 × 3 Matrices
• Minors
• Cofactors
• Applications of Determinants in Finding the Area of a Triangle
• Adjoint of a Square Matrix
• Inverse of a Square Matrix
• Consistency and Inconsistency of Linear Equations
• Number of Solutions of a System of Linear Equations
• Solving Linear Equations in Two or Three Variables
• Solving Linear Equations Using Inverse of a Matrix

UNIT III: Calculus

1. Continuity and Differentiability

• Continuity
• Differentiability
• Chain Rule
• Derivatives of Inverse Trigonometric Functions
• Derivative of sin^-1x
• Derivative of cos^-1x
• Derivative of tan^-1x
• Derivative of Implicit Functions
• Exponential Functions
• Logarithmic Functions
• Derivatives of Logarithmic Functions
• Derivatives of Exponential Functions
• Logarithmic Differentiation
• Derivative of Parametric Functions
• Second-Order Derivatives

2. Applications of Derivatives

• Rate of Change of Quantities
• Increasing Functions
• Decreasing Functions
• Maxima and Minima
• First Derivative Test
• Second Derivative Test
• Applications in Real-Life Situations

3. Integrals

• Integration as the Inverse Process of Differentiation
• Integration by Substitution
• Integration by Partial Fractions
• Integration by Parts
• Evaluation of Standard Types of Integrals
• Fundamental Theorem of Calculus
• Properties of Definite Integrals
• Evaluation of Definite Integrals

4. Applications of the Integrals

• Area Under Simple Curves
• Area Under Lines
• Area Under Circles
• Area Under Parabolas
• Area Under Ellipses in Standard Form

5. Differential Equations

• Definition of Differential Equation
• Order of a Differential Equation
• Degree of a Differential Equation
• General Solution
• Particular Solution
• Solution by Separation of Variables
• Homogeneous Differential Equations of First Order and First Degree
• Linear Differential Equations of the Form:

dy/dx + Py = Q

where P and Q are functions of x or constants.

Also:

dx/dy + Px = Q

where P and Q are functions of y or constants.

UNIT IV: Vectors and Three-Dimensional Geometry

1. Vectors

• Vectors and Scalars
• Magnitude and Direction of a Vector
• Direction Cosines
• Direction Ratios
• Types of Vectors
• Equal Vectors
• Unit Vectors
• Zero Vectors
• Parallel Vectors
• 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
• Geometrical Interpretation of Scalar Product
• Properties and Applications of Scalar Product
• Vector (Cross) Product of Vectors

2. Three-Dimensional Geometry

• Direction Cosines of a Line Joining Two Points
• Direction Ratios of a Line Joining Two Points
• Cartesian Equation of a Line
• Vector Equation of a Line
• Skew Lines
• Shortest Distance Between Two Lines
• Angle Between Two Lines

UNIT V: Linear Programming

• Introduction to Linear Programming
• Related Terminology
• Constraints
• Objective Function
• Optimization
• Graphical Method of Solution for Problems in Two Variables
• Feasible Regions
• Infeasible Regions
• Bounded Regions
• Unbounded Regions
• Feasible Solutions
• Infeasible Solutions
• Optimal Feasible Solutions
• Problems with up to Three Non-Trivial Constraints

UNIT VI: Probability

• Conditional Probability
• Multiplication Theorem on Probability
• Independent Events
• Total Probability
• Bayes' Theorem

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 the Rule of Allegation

D. Numerical Problems

• Solve Real-Life Problems Mathematically

E. Boats and Streams

• Distinguish Between Upstream and Downstream
• Express Problems in the Form of an Equation

F. Pipes and Cisterns

• Determine the Time Taken by Two or More Pipes to Fill or Empty a Tank

G. Races and Games

• Compare the Performance of Two Players with Respect to Time and Distance

H. 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, Transpose, Symmetric and Skew-Symmetric Matrices

• Determine Equality of Two Matrices
• Write the Transpose of a Given Matrix
• Define Symmetric Matrix
• Define Skew-Symmetric Matrix

C. Algebra of Matrices

• Addition of Matrices of the Same Order
• Subtraction of Matrices of the Same Order
• Multiplication of Two Matrices of Appropriate Order
• Multiplication of a Scalar with a Matrix

D. Determinant of Matrices

• Determinant of a Square Matrix
• Elementary Properties of Determinants
• Singular Matrix
• Non-Singular Matrix
• |AB| = |A||B|
• Finding Determinant Values

E. Inverse of a Matrix

• Definition of Inverse of a Square Matrix
• Properties of Inverse of Matrices
• Inverse of a Matrix Using Cofactors
• Properties of Inverse of Product Matrices

If A and B are invertible square matrices of the same size:

• (AB)^-1 = B^-1A^-1
• (A^-1)^-1 = A
• (A^T)^-1 = (A^-1)^T

F. Solving Systems of Simultaneous Equations

• Solving Simultaneous Equations up to Three Variables
• Non-Homogeneous Equations

UNIT III: Calculus

A. Higher Order Derivatives

• Second-Order Derivatives
• Higher Order Derivatives up to Second Order
• Differentiation of Parametric Functions
• Differentiation of Implicit Functions

B. Application of Derivatives

• Rate of Change of Various Quantities

C. Marginal Cost and Marginal Revenue Using Derivatives

• Define Marginal Cost
• Define Marginal Revenue
• Find Marginal Cost
• Find Marginal Revenue

D. Increasing / Decreasing Functions

• Determine Whether a Function is Increasing or Decreasing
• Determine Conditions for a Function to be Increasing or Decreasing

E. Maxima and Minima

• Determine Critical Points of a Function
• Find Local Maxima
• Find Local Minima
• Find Corresponding Maximum and Minimum Values
• Find Absolute Maximum Value
• Find Absolute Minimum Value
• Solve Applied Problems

F. Integration

• Understand Indefinite Integrals as Anti-Derivatives
• Determine Indefinite Integrals of Simple Functions

G. Indefinite Integrals as Family of Curves

• Integration by Substitution
• Integration by Partial Fractions
• Integration by Parts

H. Definite Integral as Area Under the Curve

• Definition of Definite Integral as Area Under the Curve
• Non-Trigonometric Functions
• Fundamental Theorem of Integral Calculus
• Properties of Definite Integrals
• Evaluation of Definite Integrals

I. Application of Integration

• Identify the Region Representing Consumer Surplus
• Identify the Region Representing Producer Surplus
• Find Consumer Surplus Using Definite Integration
• Find Producer Surplus Using Definite Integration

J. Differential Equations

• Recognize a Differential Equation
• Find the Order of a Differential Equation
• Find the Degree of a Differential Equation

K. Formulating and Solving Differential Equations

• Formulate Differential Equations
• Verify Solutions of Differential Equations
• Solve Simple Differential Equations

UNIT IV: Probability Distributions

A. Probability Distribution

• Concept of Random Variables
• Probability Distributions
• Probability Distribution of a Discrete Random Variable

B. Mathematical Expectation

• Arithmetic Mean of a Frequency Distribution
• Expected Value of a Random Variable

C. Variance

• Calculate Variance of a Random Variable
• Calculate Standard Deviation of a Random Variable

D. Binomial Distribution

• Bernoulli Trials
• Binomial Distribution
• Mean of Binomial Distribution
• Variance of Binomial Distribution
• Standard Deviation of Binomial Distribution

E. Poisson Distribution

• Conditions of Poisson Distribution
• Mean of Poisson Distribution
• Variance of Poisson Distribution

F. Normal Distribution

• Normal Distribution as a Continuous Distribution
• Standard Normal Variate
• Area Relationship Between Mean and Standard Deviation

UNIT V: Time Based Data

A. Time Series

• Time Series as Chronological Data

B. Components of Time Series

• Different Components of Time Series

C. Time Series Analysis for Univariate Data

• Solve Practical Problems Based on Statistical Data
• Interpret Statistical Data

D. Secular Trend

• Understand Long-Term Tendency

E. Methods of Measuring Trend

• Techniques for Finding Trends by Different Methods

UNIT VI: Inferential Statistics

A. Population and Sample

• Definition of Population
• Definition of Sample
• Difference Between Population and Sample
• Representative Sample
• Non-Representative Sample
• Simple Random Sampling
• Systematic Random Sampling

B. Parameter, Statistics and Statistical Inferences

• Define Parameter with Reference to Population
• Define Statistic with Reference to Sample
• Relationship Between Parameter and Statistic
• Limitations of Statistics in Generalizing Population Estimates
• Statistical Significance
• Statistical Inferences
• Central Limit Theorem
• Relationship Between Population, Sampling Distribution and Sample

C. t-Test

• One-Sample t-Test for a Small Group Sample
• Definition of Hypothesis
• Null Hypothesis
• Alternative Hypothesis
• Degree of Freedom
• Calculate Degree of Freedom
• Testing the Null Hypothesis
• Making Inferences Using the t-Test Statistic

UNIT VII: Financial Mathematics

A. Perpetuity and Sinking Funds

• Concept of Perpetuity
• Concept of Sinking Fund
• Calculate Perpetuity
• Difference Between Sinking Fund and Saving Account

B. Calculation of EMI

• Concept of EMI
• Calculation of EMI Using Various Methods

C. Calculation of Returns and Nominal Rate of Return

• Concept of Rate of Return
• Concept of Nominal Rate of Return
• Calculate Rate of Return
• Calculate Nominal Rate of Return

D. Compound Annual Growth Rate

• Concept of Compound Annual Growth Rate (CAGR)
• Difference Between CAGR and Annual Growth Rate
• Calculate Compound Annual Growth Rate

E. Linear Method of Depreciation

• Concept of Linear Method of Depreciation
• Interpret Cost of an Asset
• Interpret Residual Value
• Interpret Useful Life of an Asset
• Calculation of Depreciation

F. Valuation of Bonds

• Concept of Bond
• Related Terms
• Value of Bond Using Present Value Approach

UNIT VIII: Linear Programming

A. Introduction and Related Terminology

• Terms Related to Linear Programming Problems

B. Mathematical Formulation of Linear Programming Problems

• Formulation of Linear Programming Problems

C. Different Types of Linear Programming Problems

• Identify Different Types of LPP
• Formulate Different Types of LPP

D. Graphical Method of Solution for Problems in Two Variables

• Graphical Representation of Linear Inequalities
• Draw Graphs for Systems of Linear Inequalities
• Find Solutions Graphically

E. Feasible and Infeasible Regions

• Identify Feasible Regions
• Identify Infeasible Regions
• Identify Bounded Regions

F. Feasible and Infeasible Solutions

• Understand Feasible Solutions
• Understand Infeasible Solutions
• Find the Optimal Feasible Solution

Important Note

The syllabus above is based on the official CUET (UG) 2026 Mathematics / Applied Mathematics – Subject Code 319 syllabus published by the National Testing Agency.

Students preparing for CUET should refer to the latest official NTA notification and syllabus before beginning their preparation, as examination rules or syllabus details may be updated in subsequent years.

• Mathematics
• Relevant Science Subjects (Ex:Physics, Chemistry, and Biology)
• Language, depending on university requirements

• English / Language
• Accountancy
• Business Studies
• Economics
• Mathematics, where required

• Physics
• Chemistry
• Mathematics
• Language, depending on university requirements

Mathematics is an important requirement for several Economics programmes.

• Language
• Mathematics / Applied Mathematics
• General Aptitude Test
• Relevant Domain Subjects