Welcome to Class 12 Mathematics
Thirteen chapters, one central character โ the function. How this year builds on Class 11, how each page works, and how to use it for the Board exam and for JEE.
1. From Class 11 to Class 12
Class 11 gave you the raw materials: sets and functions, trigonometric functions, straight lines and conics, a first look at limits and derivatives, and probability on simple sample spaces. Class 12 is where those materials are assembled into the tools that engineers, economists, doctors and data scientists actually use. You will learn to invert functions, to handle many equations at once with matrices, to measure rates of change and find the best possible design, to add up infinitely many tiny pieces with integrals, to describe change itself with differential equations, to work in three dimensions with vectors, to plan production with linear programming, and to update beliefs from evidence with Bayes' theorem.
The thread that ties the year together is the function. Chapter 1 asks when a function can be reversed; Chapter 2 reverses the trigonometric functions; Chapters 5โ6 measure how fast a function changes; Chapters 7โ9 run that process backwards; and even a matrix (Chapter 3) and a random variable (Chapter 13) are functions in disguise.
๐ฆ You already knowโฆ
how to differentiate a power:
$$\frac{d}{dx}\left(x^3\right)=3x^2$$
and what $\sin\tfrac{\pi}{6}=\tfrac12$ means.
๐ This year you willโฆ
run both of them backwards:
$$\int3x^2\,dx=x^3+C$$
and $\sin^{-1}\tfrac12=\tfrac{\pi}{6}$ โ once we know which angle to choose.
2. The year at a glance
The course follows the current CBSE syllabus chapter by chapter. Board papers group the chapters into six units; the marks shown are those of recent CBSE blueprints out of $80$ โ always confirm them in the current year's official sample paper.
Relations and Functions
Ch 1 Relations and Functions ยท Ch 2 Inverse Trigonometric Functions
about 8 marksAlgebra
Ch 3 Matrices ยท Ch 4 Determinants
about 10 marksCalculus
Ch 5 Continuity and Differentiability ยท Ch 6 Application of Derivatives ยท Ch 7 Integrals ยท Ch 8 Application of Integrals ยท Ch 9 Differential Equations
about 35 marks โ the heart of the paperVectors and 3-D Geometry
Ch 10 Vector Algebra ยท Ch 11 Three Dimensional Geometry
about 14 marksLinear Programming
Ch 12 Linear Programming
about 5 marksProbability
Ch 13 Probability
about 8 marks3. One function, four lenses
Before any theory, here is the whole year in one picture. A single function is looked at in four ways, and each way is a chapter (or several) of this course. Watch the readouts as the point slides along the curve.
4. How every page works
Each section page follows the same pattern, so you always know where you are:
- Why this idea? โ a short motivation with a real problem.
- Teaching in coloured boxes: ๐ definitions, ๐ theorems with all their conditions, ๐งญ step-by-step strategies, โ ๏ธ the mistakes students make most often when studying alone.
- โ๏ธ Worked examples, each with a full solution hidden behind โ๐ก Show solutionโ. Try first, then open it.
- Two interactive labs that show the idea moving. Change the controls and predict what will happen before you look.
- Exercises from routine to challenging, every part solved.
- ๐ฏ Exam corner โ original problems in the style of the Board paper, JEE Main and JEE Advanced, tagged by type and with full solutions.
- A ๐ฌ mini project that uses the idea on real data, and a ๐งช five-question quiz to check yourself.
Words with a dotted underline are in the Glossary โ hover or tap for the meaning. Every key result is collected in the Formula Sheet.
5. Board exam and JEE: what is different
The CBSE Class 12 mathematics theory paper is out of $80$ marks (plus $20$ for internal assessment). In recent years it has had five sections: one-mark MCQs and AssertionโReason questions, then $2$-, $3$- and $5$-mark questions, and finally case-study questions worth $4$ marks each. Marks are given for steps: a correct method with a slip still earns most of the marks, and a bare answer to a long question earns little. In the Exam corner these appear as Board ยท 1 mark โฆ Board ยท 5 marks and Board ยท Case study.
JEE Main asks single-correct MCQs and numerical-value questions, with negative marking for wrong MCQ answers, so speed and accuracy matter more than presentation. JEE Advanced mixes formats โ one or more correct options, numerical answers, matching and paragraph questions โ and rewards deep understanding of exactly the ideas in this course. JEE also uses a few topics that are no longer in the Board syllabus; where that happens, a page has one clearly marked box, ๐ Beyond the board syllabus, whose header names the exams (JEE Main, JEE Advanced) that list the topic; you can skip it if you are preparing only for the Board.
- Read with a pencil. Copy each definition and theorem into a notebook in your own words, with one example of your own.
- Cover the solutions. Attempt every worked example before opening it. If you get stuck, open only the first step.
- Do the exercises in order โ the early parts are the Board's bread and butter, the later ones prepare you for JEE.
- Write Board answers in full, even when you could do it in your head: state what you assume, what you prove and what you conclude.
- Revisit after a week. Take the quiz again and redo one Exam-corner problem per section without notes.
1. In this course, as in the standard Class 12 textbooks, $\log x$ means the natural logarithm $\log_e x$.
2. $\sin^{-1}x$ means the inverse sine (an angle), not $\dfrac{1}{\sin x}$. The reciprocal is written $(\sin x)^{-1}$ or $\operatorname{cosec}x$.
6. Where to begin
Start with Chapter 1 โ Relations and Functions. It is short, it uses only Class 11 ideas, and it sets up the question โ when can a function be reversed? โ that opens Chapter 2.
Relations and Functions
Types of relations, equivalence classes, one-one and onto functions.
Inverse Trigonometric Functions
Principal value branches and graphs.
Matrices
Operations, transpose, symmetric and skew-symmetric matrices, inverses.
Continuity and Differentiability
The calculus core begins here.