Mathematics can be a game-saver in your CBSE Class 12 board exams. A good score in this subject not only improves your overall percentage but also provides a solid foundation for entrance exams such as JEE and CUET. But with so many topics to study, students always ask where to put in the effort. The solution lies in intelligent preparation—knowing high-weightage chapters, practicing most critical questions, and polishing your problem-solving skills.

At Bhargav Tutorials, one of the best coaching centers for CBSE, JEE, NEET & CUET, we feel that success in maths is achieved through a systematic approach. Here in this blog, we are going to analyze the most important questions of each chapter, guiding you to plan your studies efficiently and score maximum in the CBSE Class 12 Mathematics Exam 2025.

Relations and Functions: The Backbone of Algebra

Mathematics is usually perceived as a puzzle, where each concept is a piece that slots into place perfectly if well understood. One such basic piece is Relations and Functions, a subject that not only provides the foundation of algebra but also plays an important part in advanced mathematical studies. Understanding this chapter can provide you with an advantage in CBSE Class XII board exams as well as competitive exams such as JEE. At Bhargav Tutorials, the topmost coaching institute for CBSE, JEE, NEET & CUET, we provide learning Relations and Functions in a way that is stimulating, organized, and productive.

In this blog, we will deconstruct the fundamental concepts of Relations and Functions, discuss crucial subtopics, and provide expert tips to enable you to perform well. Whether you are studying for CBSE board exams or targeting engineering entrance exams, this tutorial will make you understand this subject with ease.

Understanding Functions: One-One, Onto, and Bijective Functions

Functions are the core of mathematics, serving as a link between inputs and outputs. But all functions are not equal! You will encounter various types of functions, and knowing their properties will make it easy for you to solve intricate problems.

One-One (Injective) Function

A function is referred to as one-one or injective if every element in the domain is mapped to a distinct element in the codomain. In other words, no two elements in the domain are mapped to the same image. Think of distributing chocolates in which no two students receive the same flavor—this is a typical example of a one-one function.

Onto (Surjective) Function

A function is onto or surjective if and only if all elements of the codomain are pre-imaged at least once in the domain. In other words, no element in the codomain remains unpaired. If you invite five friends to a party and place five chairs for them in such a way that no chair remains vacant, you have an onto function.

Bijective Function: A Perfect Combination

A one-one and onto function is referred to as a bijective function. It has a one-to-one correspondence between the elements of the domain and codomain. This function is important in defining inverse functions, which is an essential concept in higher mathematics.

We make these concepts easy using interactive visual tools and problem-solving strategies at Bhargav Tutorials so that students have a strong conceptual base.

Understanding Relations: Reflexive, Symmetric, and Transitive Properties

Whereas functions provide a one-to-one relationship between elements, relations explain the interaction of elements within a set with one another. Knowledge about various types of relations is fundamental in understanding complex mathematical structures.

Reflexive Relations: Looking Back at Yourself in Math

A relation is reflexive when each element of a set relates to itself. Think about when you look into a mirror—what you see is a perfect illustration of a reflexive relation. Mathematically, if a is an element of a set A, then the relation (a, a) must always exist.

Symmetric Relations: The Two-Way Street

A relation is symmetric if whenever a is related to b, then b is related to a. For instance, the relation of friendship is symmetric: if you are friends with someone, then they too are your friend (hopefully!). Mathematically, if (a, b) is an element of a relation R, then (b, a) is also an element of R.

Transitive Relations: The Chain Reaction Logic

A relation is transitive when if a is related to b and b is related to c, a should also be related to c. Consider the chain reaction as an example—one domino will cause another one to fall. Mathematically speaking, if (a, b) and (b, c) are elements of a relation R, then (a, c) should also be an element in R.

At Bhargav Tutorials, we practice and use everyday analogies to enable students to visualize and master these abstract notions.

Equivalence Relations: The Gold Standard of Relations

An equivalence relation is one that holds all three properties—reflexivity, symmetry, and transitivity. This unique kind of relation assists us in dividing sets into separate classes, which comes in very handy in algebra and geometry.

For instance, the relationship “is congruent to” between figures in geometry is an equivalence relationship since:

• Any figure is congruent to itself (reflexive).
• One figure is congruent to a second, and the second congruent to a first, (symmetric).
• One figure is congruent to a second, and the second to a third, and hence the first is congruent to the third (transitive).

Students at Bhargav Tutorials, the best coaching center for CBSE, JEE, NEET & CUET, are aided by in-depth explanations and practice questions to enhance their grasp of equivalence relations.

Function Composition and Inverse Functions: Problem Solving Like a Pro

One of the most frequent question types in exams is function composition and inverse functions. These are powerful tools in problem-solving that help us compose and invert functions.

Function Composition: The Art of Combining Functions

If two functions are composed to generate a new function, then the operation is referred to as composition of functions. Mathematically, if two functions f(x) and g(x) are given, then their composition can be expressed as (f ∘ g)(x) = f(g(x)).

For instance, if g(x) = x + 2 and f(x) = x², then (f ∘ g)(x) = (x + 2)². This method is especially helpful in calculus, transformations, and applied mathematics.

Inverse Functions: Reversing the Effect

An inverse function basically undoes the action of a given function. If f(x) takes x to y, then its inverse function f⁻¹(y) takes it back to x. The function should be bijective (one-one and onto) if it is to have a definite inverse.

At Bhargav Tutorials, we simplify such problems into manageable, comprehensible steps so that students are made comfortable solving even the most challenging questions.

Mastering Calculus: Differentiation and Integration Made Easy

Calculus is among the most significant and score-fetching subjects in Mathematics Class 12. Whether you are trying to prepare for CBSE board examinations or JEE-level competitive examinations, a firm understanding of differentiation and integration is crucial. Although calculus can be a daunting subject for some students, the correct method can make it a strength. Bhargav Tutorials, a leading coaching academy for CBSE, JEE, NEET & CUET, makes calculus simpler with systematic classes, doubt-clearing sessions, and practice-learning.

This blog will make you familiar with the main concepts of continuity and differentiability, applications of derivatives, integral calculus, and differential equations. Let’s decompose these concepts and let’s analyze how you can excel at them!

Continuity and Differentiability: The Basics of Calculus

Before diving into differentiation and integration, you must know continuity and differentiability. These are the building blocks of solving problems in advanced calculus and are very important for board exams as well as JEE.

What is Continuity?

A function is continuous if you can plot its graph without lifting your pen. Simply put, a function is continuous if there are no breaks or jumps. Continuity is important to know when solving problems involving limits, which are common in CBSE and JEE exams.

What is Differentiability?

A function is said to be differentiable if you are able to find its derivative at each point. If a function is smooth and has no sharp corners, then it is differentiable. Bhargav Tutorials makes students proficient with this topic by teaching real-life applications and carrying out regular practice tests.

How to Solve Problems on Continuity & Differentiability?

Make use of the left-hand limit (LHL) and right-hand limit (RHL) method to test for continuity.

Keep in mind that a differentiable function is always continuous, but a continuous function is not necessarily differentiable.

Practice past years’ CBSE and JEE problems to reinforce your concept.

Applications of Derivatives: Maxima, Minima & Rate of Change

Differentiation is not only about solving equations; it finds applications in physics, economics, and engineering. CBSE and JEE regularly examine students on maxima and minima questions, rate of change, and increasing/decreasing functions.

Finding the Maxima and Minima of Functions

• The function attains its maximum point (maxima) when it ceases increasing and begins decreasing.
• The function attains its minimum point (minima) when it ceases decreasing and begins increasing.

Employ the first derivative test and second derivative test to determine the maxima and minima.

Rate of Change: Why It’s Important

The rate of change aids in problem-solving involving velocity, acceleration, and population growth. For instance, in physics, velocity is the time derivative of displacement. Knowing the way to calculate the rate of change renders differentiation meaningful and useful.

Secrets to Mastering Applications of Derivatives

• Determine the function and take its first derivative to test for increasing or decreasing behavior.
• Set the derivative equal to zero to determine critical points.
• Employ the second derivative test to identify maxima or minima.
• Solve real-life problems concerning motion, business, and physics to consolidate understanding.

We at Bhargav Tutorials simplify complicated derivative problems into simple, easy-to-follow steps, so students understand the concept deeply before they are asked to apply it in exams.

Integral Calculus: The Reverse of Differentiation

Integration is sometimes referred to as the reverse of differentiation. It assists in determining areas under curves, physics problems, and even probabilities in statistics. Class 12 Mathematics includes definite and indefinite integrals, which are the building blocks of integral calculus.

Types of Integrals: Knowing the Basics

• Indefinite Integrals: They do not contain limits and consist of a constant term (C) in the answer.
• Definite Integrals: They come with fixed upper and lower bounds and yield a numeric result.

Why is Integration Important?

Assists in computing areas of curves, which is helpful in physics and geometry.

Applied to find displacement and velocity in problems concerning motion.

Crucial in engineering and economics to solve practical problems.

Integration Tips for Solving Problems Quicker

Learn common integral formulas for polynomials, trigonometric, and exponential functions by heart.

Employ substitution techniques to reduce complicated integrals.

Practice integration through graphical approaches to build intuition.

We at Bhargav Tutorials provide shortcut methods for integration to enable students to solve problems quickly and correctly.

Differential Equations: Solving Real-World Problems

Differential equations can look challenging initially, but they are perhaps the most important tool in mathematics. They crop up all the time in physics, chemistry, and engineering, so they are a must-learn topic for CBSE as well as competitive exams.

Understanding Differential Equations

A differential equation consists of an unknown function and the derivatives of this function. One needs to solve for the function that makes the given equation valid.

Types of Differential Equations in Class 12

• First-order differential equations: Applied for the modeling of simple systems such as population growth and radioactive decay.
• Second-order differential equations: Used in problems in physics involving oscillations and wave motion.

Why Are Differential Equations Important?

Applied in Newton’s laws of motion for describing motion.

Critical in biological models such as the transmission of diseases.

Aids engineers in designing electrical circuits and mechanical devices.

How to Easily Solve Differential Equations?

• Determine the nature of the differential equation and use the suitable method.
• Employ separation of variables for first-order equations.
• Learn shortcut tips to solve difficult problems quickly.

With well-organized lessons and frequent mock tests at Bhargav Tutorials, students become confident in solving differential equations effectively.

Matrices and Determinants: The Spine of Linear Algebra

Mathematics is also regarded as a discipline of logic and patterns, and there is no better place to see this than in Matrices and Determinants. The basic ideas underlie the field of linear algebra, with applications in solving equations, transformations, and practical life. Whether one is studying for CBSE Class 12, JEE, or CUET, it is important to master matrices and determinants in order to excel. At Bhargav Tutorials, a top coaching institute for CBSE, JEE, NEET & CUET, we make these concepts easy to learn so that students not only grasp them but can also apply them efficiently.

Here, we will discuss:

• The basic operations of matrices
• The importance of determinants in solving equations
• The strength of Cramer’s Rule
• Applications of matrices in real life

If you’re searching for directed guidance, Bhargav Tutorials’ talented faculty and comprehensive test series will simplify learning linear algebra and make it more effective.

Matrix Operations and the Magic of Inverse Matrices

Understanding Matrix Basics

A matrix is simply a rectangular table of numbers in rows and columns. These numbers, referred to as elements, possess great potential in solving linear equations and working with transformations. In CBSE Class 12 Mathematics, matrices make complicated calculations easier and assist in engineering and physics applications.

Matrix Addition and Multiplication

Just like numbers, matrices can be added, subtracted, and multiplied, but with a twist—multiplication isn’t commutative! This means that A × B is not necessarily the same as B × A. Learning the correct order of operations helps avoid common mistakes in exams. At Bhargav Tutorials, our structured problem-solving approach ensures students grasp these operations intuitively.

The Inverse of a Matrix – Why Does It Matter?

The inverse of a matrix is similar to the reciprocal of a number. If A⁻¹ exists, then A times A⁻¹ equals the identity matrix. In practical applications, inverse matrices are applied in cryptography, engineering, and computer graphics. It is important to understand these properties to master higher-level problem-solving in CBSE, JEE, and CUET.

The Role of Determinants in Solving Equations

What is a Determinant?

A determinant is one number derived from a square matrix. But don’t be fooled by its simplicity—this number decides (from which the name is derived!) if a matrix has an inverse or not. In CBSE Class 12, JEE, and CUET, determinants are greatly examined in solving system of linear equations.

Important Properties of Determinants

Knowledge of determinant properties can save exam time:

If a determinant is zero, then the matrix is singular (has no inverse).

Interchanging two columns or rows reverses the sign of the determinant.

Two columns or rows being equal makes the determinant equal to zero.

At Bhargav Tutorials, our experienced faculty discusses these properties with shortcut tricks to enable students to approach determinant-based questions confidently.

Determinants in Real Life

Determinants are applied in engineering, physics, economics, and even artificial intelligence. Structural engineering, for instance, uses determinants to study stability. In economics, they are employed to solve market equilibrium equations. The moment students realize it is used in real life, they learn better—a philosophy we feel very strongly about in Bhargav Tutorials.

Cramer’s Rule – The Hidden Technique to Solve Linear Equations

How Does Cramer’s Rule Work?

Cramer’s Rule is a beautiful method of solving a system of linear equations by determinants. It offers a simple means to determine unknowns without the use of long elimination or substitution techniques.

We train students to use Cramer’s Rule in a practical way through systematic practice sessions and trick methods that assist in solving JEE and CUET problems with ease.

When to Use Cramer’s Rule?

• When the coefficient matrix is non-singular (determinant ≠ 0).
• When the number of unknowns is equal to the number of equations.
• When you need to solve equations at a fast pace in CBSE exams or JEE tests.

Most students find it hard to select the appropriate method of solving equations. That’s why Bhargav Tutorials offers detailed mock tests and doubt-clearing sessions so that students become proficient in all such methods.

Application-Based Problems on Matrices and Determinants

Matrices in Computer Graphics and AI

Each time you apply filters on social media or game a video game, matrices are doing the job behind the scenes. Matrix transformations are used to scale, rotate, and distort images. Even AI algorithms depend upon matrix operations to process large datasets.

Determinants in Robotics and Engineering

Robotic arms, drones, and even autonomous cars apply determinants to perform real-time calculations. This illustrates how mathematics lies at the heart of contemporary technology—something we like to highlight at Bhargav Tutorials.

Exam-Based Application Questions

CBSE and JEE exams frequently feature real-world problems in which students are required to apply matrices and determinants to physics and economics problems. At Bhargav Tutorials, our rigorous test series makes students well-equipped to tackle such questions.

Three-Dimensional Geometry and Vector Algebra: Space in Mind

Mathematics is all about understanding patterns, relationships, and spatial arrangements. When it comes to three-dimensional geometry and vector algebra, students often struggle with visualizing problems correctly. But once you master these concepts, solving complex problems becomes intuitive. At Bhargav Tutorials, a premier coaching institute for CBSE, JEE, NEET & CUET, we help students break down these topics into digestible concepts, ensuring they grasp the fundamentals and apply them effectively.

Understanding 3D Space: Equations of Lines and Planes

In three-dimensional geometry, every point, line, and plane has a defined equation. To understand spatial relationships, you must master the equation of a line and the equation of a plane in different forms.

• Equation of a Line in Space: Unlike in two-dimensional geometry, where a line is represented as y = mx + c, in 3D space, a line requires direction ratios and a point through which it passes. The parametric and symmetric forms of a line are commonly used to solve problems.
• Equation of a Plane: In 3D, a plane is expressed as Ax + By + Cz = D. Being able to work backwards from this equation to construct it from known conditions, i.e., normal vector and a point, is essential to solve problems.

We at Bhargav Tutorials focus on visualizing these equations using interactive diagrams, real-world illustrations, and conceptual shortcuts to make these intangible concepts tangible.

Shortest Distance Between Skew Lines: An Approach with Pragmatism

Unlike parallel or intersecting lines, skew lines are never intersecting lines and not even parallel lines. The shortest distance between skew lines is a recurrent question in competitive exams such as JEE.

• Vector Approach: The cross product of direction vectors is used to find the perpendicular distance between two skew lines. This approach facilitates otherwise long computations.
• Formula-Based Method: There is a specific formula to determine the shortest distance, but learning it by heart without knowing how it is derived can be dangerous. At Bhargav Tutorials, we take students through the reasoning behind such formulas so that conceptual clarity is maintained.

Mastering Dot Product and Cross Product in Vector Algebra

Vector algebra is the cornerstone of three-dimensional geometry. Two of the most significant operations—dot product and cross product—are useful in problem-solving involving angles, projection, and perpendicularity.

Dot Product: When Two Vectors Intersect

The dot product of two vectors is a scalar value that assists in finding the angle between them. If the dot product equals zero, the vectors are perpendicular. This feature is extensively used in physics and engineering problems.

Cross Product: Determining Perpendicular Vectors

The cross product of any two vectors yields a new vector perpendicular to the two given vectors. This operation is important to find normal planes and in the calculation of torque. We at Bhargav Tutorials explain these operations by simple analogies and step-by-step derivation so that problem-solving becomes easy. 

Angle Between Vectors and Conditions for Collinearity

Identifying vectors as collinear or at an angle is crucial and necessitates a deep concept of their attributes.

• Collinearity Condition: Two vectors are collinear if one is a scalar multiple of the other. This aspect is commonly checked in competitive examinations.
• Finding the Angle Between Vectors: The formula cos θ = (A · B) / (|A||B|) is used to determine the angle between two vectors. A solid understanding of this formula allows students to solve a great deal of geometry-related problems quickly.

Probability and Linear Programming: Real-World Applications

Mathematics isn’t all about numbers and formulas—it’s about solving real world problems in the most intelligent way possible. Probability and linear programming are two strong concepts that transcend books and are used in real-world applications in sectors such as finance, engineering, and data science. From studying risks to minimizing resources, both these concepts prove to be invaluable. At Bhargav Tutorials, which is a leading coaching center for CBSE, JEE, NEET & CUET, we simplify both these concepts using systematic learning, expert mentorship, and real-world examples.

Let’s see how probability and linear programming can assist you in refining your problem-solving skills and achieving high marks in exams!

Mastering Probability: Reading Between the Lines Like a Pro

Probability assists us in forecasting the chances of events, be it flipping a coin, forecasting the weather, or estimating insurance risks. In CBSE Class XII, probability is not merely about basic formulas—it involves logical reasoning and an in-depth grasp of concepts.

Bayes’ Theorem and Conditional Probability: Making Intelligent Predictions

Bayes’ Theorem is a probability game-changer. It allows you to update the probability of an event given new information. Here’s how it works: If a physician has a general probability of a disease and receives a positive test result, Bayes’ Theorem is used to calculate how probable the individual actually has the disease. This is used extensively in medical testing, spam filtering, and even AI-based recommendations.

At Bhargav Tutorials, we emphasize on real-life examples so that you learn how conditional probability and Bayes’ Theorem apply in reality. Our faculty, consisting of experienced tutors, explains these using real-life examples so that you never lose your way among abstract concepts.

Mean and Variance of a Random Variable: Interpreting Data Trends

Mean and variance are critical to data analysis. The mean (expected value) informs you about the average result of a random event, while variance indicates the degree to which results vary from the expected value.

For instance, when you’re rolling a die, the expected value can be used to find the long-run average outcome. In business, too, variance is employed to quantify financial risk. We employ common examples such as stock market variations and sports statistics at Bhargav Tutorials to make this subject more interesting.

Linear Programming: Maximizing Resources Like a Math Whiz

Linear programming is all about maximizing or minimizing a quantity under given constraints. It is widely used in business, logistics, and even sports management. If you’ve ever wondered how airlines schedule flights, how businesses maximize profits, or how dieticians create the best meal plans, linear programming is the answer!

Understanding Linear Programming Problems (LPP): Step-by-Step Approach

A Linear Programming Problem (LPP) consists of:

• Objective Function: The function that must be maximized or minimized (e.g., maximize profit or minimize cost).
• Constraints: The conditions or restrictions that limit potential solutions (e.g., budget constraints, availability of labor).
• Graphical Solution: The technique of graphing constraints and determining the optimal solution.

At Bhargav Tutorials, our systematic method makes it easier for you to visualize LPPs using graphical techniques so that you never get stuck on optimization problems in exams. We offer detailed solutions, solved examples, and mock test practice so that you become a master of this chapter.

Real-Life Applications of Probability and Linear Programming

These mathematical theories are not only for exams—there are many real-life applications!

1. Probability in Everyday Life

• Medicine: Physicians apply probability to forecast disease risks and rates of treatment success.
• Insurance: Firms compute probabilities to determine premium charges.
• Sports Analytics: Teams apply probability to plan and evaluate performance.

2. Linear Programming in Business and Industry

• Manufacturing: Plants maximize resources such as raw material and labor to ensure minimal waste.
• Logistics: Courier firms employ LPP to determine the shortest and least expensive routes.
• Marketing: Companies maximize profit by maximizing product pricing and advertising expenditures.

At Bhargav Tutorials, we clarify the practical usage of these subjects using real-life case studies, interactive problem-solving, and individualized doubt-clearing sessions.

Why Choose Bhargav Tutorials for CBSE Class XII Mathematics?

If you are appearing for CBSE, JEE, NEET & CUET, you require expert advice, systematic learning, and efficient problem-solving techniques to become proficient in probability and linear programming. At Bhargav Tutorials, we provide:

✔ Concept-Based Learning: We emphasize learning over memorization.

✔ Doubt-Clearing Sessions: Receive personalized attention to improve weak spots.

✔ Mock Tests & Practice Papers: Gain confidence with actual exam-type questions.

✔ Shortcut Techniques & Tricks: Master intelligent methods to solve problems quicker.

Learning probability and linear programming can enhance your problem-solving ability as well as your exam scores. With proper guidance and coaching from experts at Bhargav Tutorials, learning these subjects is not a challenge anymore.

So, are you ready to learn math the intelligent way? Enroll at Bhargav Tutorials, India’s best coaching institute for CBSE, JEE, NEET & CUET, and unlock new levels of preparation!

Conclusion

Achieving good scores in CBSE Class 12 Mathematics calls for a planned strategy, emphasizing high-weightage chapters and practicing crucial questions. By getting a command over concepts in relations & functions, calculus, matrices, 3D geometry, probability, and linear programming, the student can enhance confidence and enhance problem-solving abilities. Regular practice, time management, and working through last year’s question papers are keys to success.

Call to Action: Get a step ahead in your preparation with expert advice! Avail Bhargav Tutorials, the best coaching institute in Delhi and Gurugram, for individualized coaching, mock tests, and doubt-clearing sessions. Sign up now and crack your CBSE exams with ease!