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# CBSE Class 12 Maths Notes for All Chapters

CBSE Class 12 Maths Notes: Preparation for CBSE Class 12 Maths requires consistent practice. To score good marks effortlessly, students are required to prepare notes for every subject and every chapter. This can help students refer to the notes at the last-minute to have a thorough revision. While preparing the notes, it is important that students focus on the important topics and increase their scope of acquiring maximum marks. In this article, we have provided the Class 12 Maths notes for every chapter.

Mathematics is one of those subjects for which marks can be easily scored. Having the basic understanding of the concepts and application of the right formula can make the students easily score a couple of marks. Students must practice various types of questions from different chapters to handle any type of question asked in the exam. Having CBSE Class 12 Maths notes is highly useful in the revision of various topics. Therefore, to help the students acquire good marks easily, we have provided CBSE Class 12 notes.

## CBSE Class 12 Maths Notes: Overview

The Central Board of Secondary Education conducts Maths exams every year. Before knowing the CBSE Class 12 Maths notes, let us have an overview of the Maths exam. The entire paper is set for 100 marks with 30 marks for practical exams and 70 marks for theory exams. Students are required to score 33% in the subject to pass the exam. Below mentioned are all the chapters of CBSE Class 12.

### CBSE Class 12 Maths Notes

The below-mentioned are the notes for all the chapters of CBSE Class 12 Mathematics. Students can first study their syllabus and then go through the notes to have a proper revision of all the topics.

#### Chapter 1: Relations and Functions

Relations

A relation can be defined as the interconnection between any two quantities or objects. Some examples of relation have been mentioned below.

• Empty Relation – A relation R in a set A is called empty relation, if no element of A is related to any element of A
• Universal Relation – If each element of A is related to every element of A, A relation R in a set A is called universal relation
• Reflexive Relation – If R is reflexive, symmetric and transitive, A relation R in a set A is said to be an equivalence relation
• Symmetric relation – R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.
• Transitive relation – R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.
• Equivalence relation – R in X is a relation which is reflexive, symmetric and transitive.

Functions

Functions are known as the special kind of relations. Different types of functions have been mentioned below.

• One-one Function – A function f:X →Y is one-one (or injective) if f(x1) = f(x2) ⇒ x1 = x2 ∀ x1 , x2 ∈ X
• Onto Function – A function f:X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y
• One-One and Onto Function – A function f:X → Y is one-one and onto (or bijective), if f is both one-one and onto

#### Chapter 2: Inverse Trigonometric Functions

Students can refer to the below mentioned Inverse Trigonometric Formulas and make a note of them for future reference.

Domain and Range Of Inverse Functions

sin(sin−1x) = x, if -1 ≤ x ≤ 1

cos(cos−1x) = x, if -1 ≤ x ≤ 1

tan(tan−1x) = x, if -∞ ≤ x ≤∞

cot(cot−1x) = x, if -∞≤ x ≤∞

sec(sec−1x) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞

cosec(cosec−1x) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞

The below-mentioned formulas are for Inverse trigonometric functions

sin−1(sin y) = y, if -π/2 ≤ y ≤ π/2

cos−1(cos y) =y, if 0 ≤ y ≤ π

tan−1(tan y) = y, if -π/2 <y< π/2

cot−1(cot y) = y if 0<y< π

sec−1(sec y) = y, if 0 ≤ y ≤ π, y ≠ π/2

cosec−1(cosec y) = y if -π/2 ≤ y ≤ π/2, y ≠ 0

#### Chapter 3: Matrices

Definition: A matrix can be defined as an array of numbers or functions arranged in a rectangular order.

Types of Matrices

• Column Matrix: A matrix which consists of a singular column can be defined as a column matrix
• Row Matrix: If a matrix has only a singular row, then it is termed as a row matrix
• Square Matrix:- When the number of rows is equal to the number of columns, then the matrix is defined as a square matrix.
• Diagonal Matrix:- If the diagonal elements are zero in a square matrix, then it can be defined as a diagonal matrix.
• Scalar Matrix:- When the diagonal elements are equal in a diagonal matrix, then this matrix is assumed to be a Scalar matrix.
• Identity Matrix:- An identity matrix can be defined as a square matrix, whose elements in the diagonal are 1, while the other elements are zero.
• Zero Matrix:- In a matrix, when all the elements are zero, it can simply be defined as a null matrix or a zero matrix.

#### Chapter 4: Determinants

When a square matrix “A” of an order “n” is associated with a number, then it is titled as a determinant of the aforementioned matrix. The number involved in this square matrix can be a real number or a complex number.

In the field of mathematics, Determinants can be used for a myriad of different calculations as mentioned below:

• Finding the area of a Triangle.
• Obtaining the solutions of linear equations in two or three variables.
• Solve Linear equations by using the inverse of a matrix
• Getting the adjoint and the inverse of a square matrix

#### Chapter 5: Continuity And Differentiability

Continuous Function

When in a function, the real value at a point is said to be continuous when at that point, the function of that point is equal to the limit of the function at that point. The continuity exists when all of the domain is continuous.

Chain Rule

The composite of the functions can be differentiated with the help of chain rule. If f=v, t=u(x)

can be seen, then,

Logarithmic Differentiation

When the differential equation is in the form

. Here, the positive values of f(x) and u(x) are considered.

Rolle’s Theorem

Consider, a continuous function f:[a,b]

—–>

R which is continuous on the point [a,b] and differentiable on the point (a,b) then , f(a)=f(b) and some external point exists such as c in (a,b) such that f'(c)=0.

Mean Value Theorem

Let us consider, a continuous function f:[a,b]

——>

R which is continuous on the point [a,b] and differentiable on the point (a,b), some external point exists such as c in (a,b) such that

#### Chapter 6: Application of Derivatives

Derivatives are referred to the situation when a quantity p varies with respect to another quantity q, fulfilling a condition p =f (q) and dp/dq (or f ‘ (q)) represents the change of rate of p with respect to q and dp/dq where q =

#### Chapter 7: Integrals

Integration can be defined as the process that is inverse to that of differentiation. In the integration, we can find the function for which the differential is provided. Integrals are those functions that satisfy a given differential equation.

Indefinite Integrals

If integration is inverse of differentiation, then

They are known to be indefinite or general integrals and all these differ by a constant term. From a geometrical perspective, an indefinite integral is a compilation of curves which are obtained by translation of one of the curves parallel to itself downward and upward along with y-axis

#### Chapter 8: Application of Integrals

Integrals are the functions which satisfy a given differential equation for finding the area of a curvy region y=f(x), the x-axis and the line x=a and x=b(b>a) is represented through this formula:

Area=

If curvy region is x=

, y-axis and the line y=c, y=d is represented through the formula:

Area =

#### Chapter 9: Differential Equations

• Differential Equations: Equations involving the derivative of the variable that is dependent with respect to the independent variable are known as differential equations.
• Solutions: The function that is able to satisfy the given differential equation is known as the solution. A general solution can be reffered to the presence of many arbitrary constants as their order. If the solution is free from these arbitrary constants, it is called a particular solution. To form a differential equation from a function which is given, the function should be differentiated several times as the number of arbitrary constants present in the function and then eliminating them at last.
• Variable Separation Method: The method is used to solve specific types of equations in which the variables can be separated entirely. This means that the terms which contain y should have with dy and the terms which contain x should have with dx.
• Homogeneous Differential Equation: When the zero degree homogeneous functions can be expressed in the form, dy/dx=f(x,y) or dx/dy=g(x,y) where, f(x,y) and g(x,y) are the mentioned functions then it is called homogeneous differential equations.

#### Chapter 10: Vector Algebra

Position of a Vector

If we are provided with a point Q(x,y,z) and

and the magnitude is given by

The direction ratio for a vector is its scalar components and is responsible for its projections along the respective axes.

The relation between magnitude, direction ratios, direction cosines of a vector

If a vector has been given with dimensions such as magnitude(p), direction ratios (x,y,z) and direction cosines (l,m,n) then the relation between them is:

l=x/p, m=y/p,n=z/p

The order taken for the vector sum of the three sides of the triangle is

—–>

0

And the vector sum of coinitial vectors is the diagonal of the parallelogram which has the vectors as its adjacent sides.

When multiplying a vector by a scalar λ, the magnitude of the vector changes by the multiple |λ|, and the direction remains same (or makes it opposite) according to as the value of λ is positive (or negative).

#### Chapter 11: Three Dimensional Geometry

Direction Cosine of Line

This can be known as the cosine of the angles subtended by a line on the positive direction of the coordinate axis. If we are given a line whose direction cosines are p,q,r, then

If we have line joining two points such as R

are represented as

And RS=

If a line has p,q,r are the direction cosines and a,b,c are the direction ratios then,

#### Chapter 12: Linear Programming

Variables are non-negative and satisfy a set of linear inequalities also known as linear constraints and the problems have the goal to find the optimal value (maximum or minimum) of a linear function of several variables (called objective function) with respect to the conditions. Variables are sometimes called decision variables and are non-negative in nature.

Feasible Region: Feasible region (or solution region) refers to the common region represented by all the boundaries including the non-negative boundaries x ≥ 0, y ≥ 0.

Optimal Solution: In an objective function, the optimal solution of any point in the feasible region gives the optimal value (maximum or minimum).

Theorems of Linear Programming: There are theorems which help in solving problems of linear programming and they include:

• If P be is the feasible region and S =ax + by be the objective function. The optimal value must occur at a corner point of the feasible region when S has an optimal value and the variables x and y are subject to boundaries described by linear inequalities.
• If P is the feasible region and S = ax + by be the objective function. If P is constrained, then the objective function S has both minimum and maximum value of P and each of these occurs at a corner point (vertex) of P

#### Chapter 13: Linear Programming

The sample space of an experiment of tossing three coins is S = {TTT, HHH, TTH, HHT, THT, HTH, HTT, THH}. As the sample space consists of 8 elements, the probability of occurring each sample point is 1/8. Let A and B be the events of displaying 2 heads and 1st coin showing tail respectively.

Then, A = {HHT, HHH, THH, HTH} and B = {THT, THH, TTT, TTH}.

Therefore P(A) = P({HHT}) + P ({HHH}) + P ({THH}) + P ({HTH})  =1/8 +1/8 +1/8 +1/8 =1/2.

Similarly, P(B) = P({THT}) + P({THH}) + P ({TTT}) + P ({TTH}) =1/8 +1/8 +1/8 +1/8 =1/2.

Also, A ∩ B = {THH} and P({THH}) = P(A ∩ B) = 1/8

The sample point of B which is favourable to event A is THH. Thus, P(A) considering B as the sample space (S) = 1/4. This P(A) is known as the conditional probability of A provided B has already occurred. The conditional probability of an event is denoted by P (A|B). Thus, from the above case P(A|B) = 1/4.

### FAQs on CBSE Class 12 Maths Notes

Below-mentioned are some frequently asked questions related to CBSE Class 12 Maths Notes

Q.1: How many chapters are there in CBSE Class 12 Maths?
Ans: There are a total of 13 chapters in CBSE Class 12 Maths

Q.2: What does Chapter 3 Matrices of CBSE Class 12 Maths talk about?
Ans: Chapter 3 Matrices talks about Column Matrix, Row Matrix, Square Matrix, Diagonal Matrix, Scalar Matrix, Identity Matrix and Zero Matrix.

Q.3: For how many marks is the CBSE Class 12 Maths exam conducted?
Ans: CBSE Class 12 Maths exam is conducted for 100 marks.

Q.4: What is the time duration to complete CBSE Class 12 Maths?
Ans: Students are required to complete the CBSE Class 12 Maths exam within a duration of 3 hours.

Q.5: What is the definition of “Relation”?
Ans:
A relation can be defined as the interconnection between any two quantities or objects.

Students can download the previous year question papers from CBSE official website.

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