Chapter 1: Relations and Functions
Relations and Functions opens Class 12 Mathematics by sharpening ideas you first met a year earlier. You will revisit relations and study the kinds that matter most: reflexive, symmetric and transitive relations, and the equivalence relations they combine to form. On the functions side you will learn to test whether a function is one-one (injective), onto (surjective) or both, which makes it bijective and therefore invertible. The chapter then builds composition of functions and shows how to find an inverse step by step. Worked examples, three proofs and graded practice take you from definitions to confident problem solving, with clear mapping diagrams and a bijective graph to picture every idea, free on SchoolRevise.com.
Chapter 2: Inverse Trigonometric Functions
Inverse Trigonometric Functions explains how each trigonometric ratio is turned back into an angle once its domain is sensibly restricted. You will learn why the restriction is needed, then meet the principal value branches of the six inverses, sin inverse, cos inverse, tan inverse, cot inverse, sec inverse and cosec inverse, with their exact domains and ranges. The heart of the chapter is finding principal values accurately, including the careful cases where the angle inside lies outside its branch. You will also use elementary properties such as sin inverse x plus cos inverse x making a right angle. Clear graphs, three proofs and graded practice help you read every value correctly, free on SchoolRevise.com.
Chapter 3: Matrices
Matrices introduces a compact way to store and work with rows and columns of numbers that runs through the whole of higher mathematics. You will learn the order of a matrix, the common types, and how to add, subtract and multiply matrices and multiply by a scalar, taking care that multiplication is not commutative. The chapter covers the transpose and the families it defines, symmetric and skew-symmetric matrices, and shows that every square matrix splits neatly into one of each. You will also meet the inverse of a matrix and verify it by multiplication. Bracketed matrix figures, three proofs and graded practice build the fluency you will rely on later, free on SchoolRevise.com.
Chapter 4: Determinants
Class 12 Determinants attaches a single decisive number to every square matrix. You will evaluate two by two and three by three determinants, expand along any row or column using minors and cofactors, and apply properties that simplify the work. The chapter shows how the determinant gives the area of a triangle from coordinates, tests points for collinearity, and builds the adjoint and inverse of a matrix. These tools then solve systems of linear equations by the matrix method. With twelve worked examples and sixteen graded practice questions, this chapter links algebra to geometry and prepares students thoroughly for board exams and competitive foundations.
Chapter 5: Continuity and Differentiability
Class 12 Continuity and Differentiability sharpens two ideas at the heart of calculus. You will test whether a function is continuous using limits, learn which functions are automatically continuous, and understand why every differentiable function is continuous though not the reverse. The chapter develops the chain rule for composite functions, implicit and logarithmic differentiation, the derivatives of inverse trigonometric, exponential and logarithmic functions, and parametric and second order derivatives. Worked examples move one logical step at a time, and graded practice spans direct recall to higher order thinking. Mastery here is essential for the application of derivatives, integration and every later topic that depends on smooth change.
Chapter 6: Application of Derivatives
Class 12 Application of Derivatives turns the slope of a curve into a decision making tool. You will compute rates of change and related rates, find where functions increase or decrease using the sign of the derivative, and write the equations of tangents and normals. The chapter locates maxima and minima through the first and second derivative tests and solves optimisation problems on closed intervals, where the best value sits at a stationary point or an endpoint. Twelve worked examples and sixteen graded practice questions build confidence from basics to higher order thinking. These methods power optimisation across business, engineering, physics and everyday problem solving.
Chapter 7: Integrals
Class 12 Integrals presents integration as the reverse of differentiation and as the exact area under a curve. You will build the standard integrals, then master the three main techniques: substitution, integration by parts and partial fractions. The chapter introduces the definite integral, the Fundamental Theorem of Calculus that links area to antiderivatives, and the symmetry properties that crack otherwise hard problems. Twelve worked examples proceed one careful step at a time, with sixteen graded practice questions from basics to higher order thinking. Integration is indispensable for areas, volumes, probability and physics, making this one of the most important and frequently examined chapters of the year.
Chapter 8: Application of Integrals
Class 12 Application of Integrals uses the definite integral to measure the exact area of curved regions. You will find the area under a single curve, handle parts that dip below the axis with absolute values, and compute the area trapped between two curves by integrating the difference of the top and bottom boundaries. The chapter shows how to slice with vertical or horizontal strips and how symmetry shortens the work, deriving the areas of circles and ellipses. Twelve worked examples and sixteen graded practice questions build the sketch, set up and evaluate routine. These skills connect calculus to geometry, physics and real measurement problems.
Chapter 9: Differential Equations
Class 12 Differential Equations studies equations that involve a derivative, the natural language of change. You will classify equations by order and degree, distinguish general solutions with arbitrary constants from particular solutions fixed by conditions, and form a differential equation from a family of curves. The chapter solves the three standard first order types: variable separable, homogeneous using the substitution y equals vx, and linear using the integrating factor. Twelve worked examples verify each method step by step, with sixteen graded practice questions. Differential equations model population growth, cooling, circuits and finance, so this chapter bridges pure mathematics and the sciences for board and entrance preparation.
Chapter 10: Vector Algebra
Class 12 Vector Algebra handles quantities that carry both size and direction. You will represent vectors in component form, recognize the special types, add and scale them, and compute magnitudes and unit vectors. The chapter develops the two products that unlock geometry in space: the dot product, which gives angles and projections and tests perpendicularity, and the cross product, which yields a perpendicular vector whose length is an area. Twelve worked examples and sixteen graded practice questions build fluency from basics to higher order thinking. Vectors are the working language of physics, engineering and computer graphics, and they lead directly into three dimensional geometry.
Chapter 11: Three Dimensional Geometry
Class 12 Three Dimensional Geometry extends coordinate geometry into space. You will describe directions with direction cosines and ratios, write the equation of a line in vector and cartesian form, and find the angle between two lines. The chapter covers the equation of a plane, the angle between two planes and between a line and a plane, and the perpendicular distance of a point from a plane. Twelve worked examples work through each formula one step at a time, with sixteen graded practice questions. These techniques describe the real three dimensional world of architecture, computer graphics, aviation and robotics, and they build directly on vector algebra.
Chapter 12: Linear Programming
Class 12 Linear Programming finds the best possible outcome when both the goal and the limits are linear. You will write a linear programming problem with an objective function and constraints, shade the feasible region formed by the constraints, and locate its corner points by solving boundary equations. The corner point method then evaluates the objective at each vertex to read off the maximum or minimum, with care taken over bounded and unbounded regions. Twelve worked examples and sixteen graded practice questions include formulating word problems into a model. Linear programming guides decisions in manufacturing, diet planning, logistics, agriculture and finance, making it a practical and high scoring chapter.
Chapter 13: Probability
Class 12 Probability teaches reasoning under uncertainty, with a focus on how new information updates the chances. You will use conditional probability, the multiplication theorem and the test for independent events, then combine cases with the total probability theorem. The chapter builds to Bayes theorem, which reverses conditioning to find a cause from an observed effect, and introduces random variables with their probability distribution and mean. Twelve worked examples and sixteen graded practice questions move from basics to higher order thinking. Probability underpins medical testing, spam filtering, insurance and machine learning, so this closing chapter ties school mathematics to powerful real world and competitive applications.
