Welcome to the NCERT Solutions for Class 12 Mathematics. This page offers chapterwise solutions designed to help students grasp key concepts easily. With detailed answers and explanations for each chapter, students can strengthen their understanding and prepare confidently for exams. Ideal for CBSE and other board students, this resource will simplify your study experience.

Chapter 1 Relations and Functions
Fundamentals of relations and functions are already known to us such as their definitions, types, graphs, domain and range, ete. In this chapter, we will study these topics in detail. Once we come and over these topics further chapters of our curriculum will be easily understood. Topics of this chapter are  types of relations : reflexive, symmetric, transitive and equivalence relations, one to one and onto functions, inverse of a function, composite functions, Binary operations.

Chapter 2 Inverse Trigonometric Functions
In the previous chapter, types of functions are mentioned. If the function is not oneone, onto or both, then its inverse does not exist. This is the primary condition for inverse which must be fulfilled. Knowledge of the domain of range which we acquired earlier would help us in this chapter for graphing of inverse trigonometric functions. Trigonometric functions which are not oneone, onto or both will not be discussed. It has wide applications in engineering and other science related branches. This chapter consists of range, domain and principle value branches, graphs, elementary properties.

Chapter 3 Matrices
Matrix is one of the most fundamental chapters of Mathematics, which will help us not only in mathematics but also in other branches like mechanics, optics, computer science, etc. It solves tedious calculations very easily. Topics incorporated in this chapter are notation, order, types and equality of matrix, zero matrix, transpose, symmetric and skew symmetric matrices, addition and multiplication of matrices, properties, row and column operations, Invertible matrices and proof of uniqueness of inverse, if it exists.

Chapter 4 Determinants
Determinant is a continuation of the previous chapter matrix. Matrix and determinant are the core topics of algebra. These will help us to solve many algebraic linear equations very easily. Matrix and determinants both are interlinked. Topics which are covered in this chapter  determinant of a square matrix (upto 3x3), minors. cofactors, finding the area of triangle, adjoint and inverse of square matrix, consistency & inconsistency and number of solutions of system of linear equation, solving system of linear equation in two or three variables using inverse of a matrix.

Chapter 5 Continuity and Differentiability
In the previous class, introduction of limits and derivatives was given. That was basically a calculus introduction. This chapter is a continuation of it. We will study about differentiation of functions. New functions like exponential and logarithmic functions will be introduced. This chapter consists of continuity and differentiability. derivative of a composite function, chain rule, derivatives of inverse trigonometric functions and implicit functions, logarithmic differentiation, parametric forms of derivative of functions, second order derivatives, Rolle's and Lagrange's Mean value theorems.

Chapter 6 Application of Derivatives
In the previous chapter, we studied about differentiation of different types of functions. Now applications of derivatives in various disciplines like science will be discussed. This will also help us to find approximate values of certain quantities. Topics of discussion will be  rate of change of objects, increasing & decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima and some real life situations.

Chapter 7 Integrals
Integration is an inverse process of differentiation. In this chapter, we will learn how to find the integral of a function. Its knowledge in calculus is very much needed for finding the areas under curves, etc. This chapter consists of integration of a variety of functions by substitutions, by parts and by partial fractions. Definite integrals as a limit of a sum , fundamental theorem of calculus, properties of definite integrals.

Chapter 8 Application of Integrals
In this chapter, we will study applications of integrals to find areas bounded by curves and areas between conic sections like circles, parabola, ellipses, etc. Topics which include  areas under simple curves, especially lines, areas of conic sections in standard form only, the region should be clearly identifiable.

Chapter 9 Differential Equations
In this chapter, we will shidy about differential equations and their solutions. Concepts of differential equations will help us in this class and also in higher studies. It has many applications in other branches like science, algebra, etc. Key topics of this chapter are  definition, order and degree, general and particular solutions, formation of differential equaĊon whose general solution is given, method of separation of variables, homogeneous differential equations of first order and degree.

Chapter 10 Vector Algebra
Quantities which have only magnitude are called scalars. But quantities that involve magnitude and direction are called vectors. Discussion will be on algebra of vectors. Topics which covered in this chapter  vectors, scalars, direction cosines and direction ratios of a vector, Types of vectors, position vector of a point , negative of a vector, addition of vectors, multiplication of a vector by a scalar, dot and cross products, scalar triple product of vectors.

Chapter 11 Three Dimensional Geometry
In the previous class, we studied the introduction of 3d geometry. Now, we will go in depth of its concepts. This chapter consists of direction cosines and direction ratios of a line joining two points, coplanar and skew lines cartesian and vector equation of a line, shortest distance between two lines, cartesian and vector equation of a plane, angle between two lines; two planes; a line and a plane, distance of a point from a plane.

Chapter 12 Linear Programming
Linear programming is a method in which we represent problems with the help of graphs. It will enhance our practical understanding toward problems. This chapter consists of definitions of related terminology such as constraints, objective function, optimization, different types of linear programming problems, mathematical formulation graphical methods, feasible and infeasible regions and solutions, optimal feasible solutions.

Chapter 13 Probability
In this chapter, we will learn some new aspects of probability like conditional probability etc. All the concepts which we have studied in previous classes will help us in understanding these new topics in a better way. Topics which are included in this chapter  conditional probability, multiplication theorem on probability, independent events, Bayes theorem, total probability, random variable and its probability distribution, mean and variance, Binomial distribution.
Popular Questions of Class 12 Mathematics
 Q: Given an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.  Q: Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Relation R in the set A = {1, 2, 3,13, 14} defined as
R = {(x, y): 3x − y = 0}
(ii) Relation R in the set N of natural numbers defined as
R = {(x, y): y = x + 5 and x < 4}
(iii) Relation R in the set A = {1, 2, 3, 4, 5, 6} as
R = {(x, y): y is divisible by x}
(iv) Relation R in the set Z of all integers defined as
R = {(x, y): x − y is as integer}
(v) Relation R in the set A of human beings in a town at a particular time given by
(a) R = {(x, y): x and y work at the same place}
(b) R = {(x, y): x and y live in the same locality}
(c) R = {(x, y): x is exactly 7 cm taller than y}
(d) R = {(x, y): x is wife of y}
(e) R = {(x, y): x is father of y}  Q: Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
 Q: Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.  Q: Show that the relation R in the set R of real numbers, defined as R = {(a, b): a ≤ b^{2}} is neither reflexive nor symmetric nor transitive.
 Q:
In each of the following cases, state whether the function is oneone, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x^{2 }
 Q: Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1, L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.
 Q:
Show that the Modulus Function f : R → R, given by f(x) = x, is neither oneone nor onto, where  x  is x, if x is positive or 0 and x is – x, if x is negative.
 Q:
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither oneone nor onto, where [x] denotes the greatest integer less than or equal to x.
 Q: Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
Recently Viewed Questions of Class 12 Mathematics
 Q:
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
(A) f is oneone onto
(B) f is manyone onto
(C) f is oneone but not onto
(D) f is neither oneone nor onto.
 Q:
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
 Q: Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
 Q:
Maximise Z = 3x + 4y
Subject to the constraints:x + y ≤ 4, x ≥ 0, y ≥ 0
 Q:
Show that the Signum Function f : R → R, given by
is neither oneone nor onto.
 Q:
Determine order and degree(if defined) of differential equation
\begin{align}\left(\frac{d^2y}{dx^2}\right)^2\;+\;cos\left(\frac{dy}{dx}\right)\;=\;0\end{align}
 Q: Check whether the relation R in R defined as R = {(a, b): a ≤ b^{3}} is reflexive, symmetric or transitive.
 Q:
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
 Q: Show that each of the relation R in the set A = { x ∈Z: 0≤x≤12}, A={x} given by
(i) R = { (a,b) : a  b is a multiple of 4}
(ii) R = {(a,b):a = b} is an equivalence relation.
Find the set of all elements related to 1 in each case.  Q:
The order of the differential equation
\begin{align}2x^2\frac{d^2y}{dx^2}\; \;3\frac{dy}{dx}\;+ y=\;0\end{align}
is (A) 2 (B) 1 (C) 0 (D) not defined