# CBSE Class 11 Maths Notes

When life gives you choices – tell us what you chose after boards? We’re guessing it’s either Commerce or Science *(given that you’re here to revise our GeeksforGeeks’ CBSE Class 11 Maths Notes)*. Our CBSE Class 11 Maths Revision Notes have been designed in the most basic and detailed format possible, covering nearly all domains such as differential calculus, arithmetic, trigonometry, and coordinate geometry. We know how hard it gets when you shift to an altogether new grade where subjects are no longer the same, especially with maths.

Our *CBSE Class 11 Maths NCERT Notes* are curated for students who want to achieve high marks in their 11th grade as well as competitive exams such as JEE Mains and JEE Advanced. These Class 11 Maths NCERT notes provided by GeeksforGeeks would assist students in easily grasping every idea and properly revising before the exams. These notes were written by subject experts which has a significant benefit in that students would be well qualified to answer any kind of question that could be posed in the exams.

Our experts developed these notes, which are available for free at GeeksforGeeks. The CBSE Class 8 Maths Notes include all of the important chapters from the improved NCERT textbooks, including Trigonometric Functions, Relation and Functions, Principles of mathematical induction, and more.

Other important topics covered in the Class 11 Maths curriculum are Complex Numbers and Quadratic Equations, Linear Inequalities, Limits and Derivatives, Statistics and Probability, etc. NCERT Solutions for Class 11 and RD Sharma Solutions for Class 11 are also covered by our experts for Class 11 Students.

This doesn’t end here GeeksforGeeks also covered some important resources for all the students studying maths are 1500+ Most asked Questions of Mathematics, Chapterwise Important Formulas for Class 11, and many more.

These subject-specific revision notes include all of the essential topics that are necessary for CBSE Board Class 11 students. Simplify your mathematics problems with more up-to-date math revision notes available for free on the internet.

## Chapter 1: Sets

Let’s start with **Class 11 Maths Sets Notes**. The chapter explains the concept of sets along with their representation. The **Class 11 Maths Notes** cover topics such as writing numbers in the form of sets, verifying empty, finite, infinite, and equal sets, identifying subsets, performing various operations on sets, Venn Diagrams, and finding union and intersection of sets.

**Some Important formulas learned in CBSE Class 11 Chapter 1- Sets,**

- A – A = Ø
- B – A = B⋂ A’
- B – A = B – (A⋂B)
- (A – B) = A if A⋂B = Ø
- (A – B) ⋂ C = (A⋂ C) – (B⋂C)
- A ΔB = (A-B) U (B- A)
- n(A∪B) = n(A) + n(B) – n(A⋂B)
- n(A∪B∪C)= n(A) +n(B) + n(C) – n(B⋂C) – n (A⋂ B)- n (A⋂C) + n(A⋂B⋂C)
- n(A – B) = n(A∪B) – n(B)
- n(A – B) = n(A) – n(A⋂B)
- n(A’) = n(∪) – n(A)
- n(U) = n(A) + n(B) + – n(A⋂B) + n((A∪B)’)
- n((A∪B)’) = n(U) + n(A⋂B) – n(A) – n(B)

**Chapter 1 of CBSE Class 11 Maths Notes covers the following topics:**

- Sets and their representations
- Different kinds of Sets
- Subsets, Power Sets, and Universal Sets
- Venn Diagrams
- Operations on Sets
- Union and Intersection of Sets

**More resources for CBSE Class 11 Maths Chapter 1**

## Chapter 2: Relations & Functions

The chapter **Relations & Functions** explains whether or not a relation is a function, determining different types of functions, adding, subtracting, multiplying functions, and determining their range.

The chapter is divided into two sections, Relation, and Functions. The topics covered in the first part are the Cartesian product of sets, which includes subtopics like the Number of elements in the Cartesian product of two finite sets and the Cartesian product of the set of reals with itself. Further, the concept of relation, graphical diagrams, domain, co-domain and range of a relation are discussed.

The next section of this chapter consists of topics like Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic, and greatest integer functions, with their graphs.

**Important formulas used in CBSE Class 11 Chapter 2- Relations & Functions are,**

Inverse of Relation:A and B are any two non-empty sets. Let R be a relationship between two sets A and B. The inverse of relation R, indicated as R^{-1}, is a relationship that connects B and A and is defined by

R^{-1}={(b, a) : (a, b) ∈ R}where, Domain of R = Range of R

^{-1}and Range of R = Domain of R^{-1}.

- A
cartesian productA × B of two sets A and B is given by:A × B = { (a,b) : a ϵ A, b ϵ B}

- If (a, b) = (x, y); then a = x and b = y
- If n(A) = x and n(B) = y, then n(A × B) = xy and A × ϕϕ = ϕϕ
- The cartesian product: A × B ≠ B × A.
Algebra of functions:If the function f : X → R and g : X → R; we have:

- (f + g)(x) = f(x) + g(x) ; x ϵ X
- (f – g)(x) = f(x) – g(x)
- (f . g)(x) = f(x).g(x)
- (kf)(x) = k(f(x)) where k is a real number
- {f/g}(x) = f(x)/g(x), g(x)≠0

**Chapter 2 of CBSE Class 11 Maths Notes covers the following topics:**

- Cartesian Product of Sets
- Relations and Functions
- Introduction to Domain and Range
- Piecewise Function
- Range of a Function

**More resources for CBSE Class 11 Maths Chapter 2**

- Class 11 NCERT Solutions Maths Chapter 2
- Class 11 RD Sharma Solutions Relations and Functions Chapter 1 and Chapter 2
- All important formulas for Chapter 2

## Chapter 3: Trigonometric Functions

The chapter Trigonometric Functions mainly focuses on how to measure angles in radians and degrees, and how to convert between the two. The chapter also covers the use of a unit circle to define trigonometric functions, the general solution of trigonometric equations, the signs, domain, and range of trigonometric functions, as well as their graphs.

The chapter introduces students to the process of expressing sin (xy) and cos (xy) in terms of sinx, siny, cosx, and cosy, as well as their simple applications and deducing identities, for sin 2x, cos 2x, tan 2x, sin 3x, cos 3x, and tan 3x, respectively.

**Useful Important formulas in CBSE Class 11 Chapter 3: Trigonometric Functions are,**

Reciprocal Trigonometric Ratios:

- sin θ = 1 / (cosec θ)
- cosec θ = 1 / (sin θ)
- cos θ = 1 / (sec θ)
- sec θ = 1 / (cos θ)
- tan θ = 1 / (cot θ)
- cot θ = 1 / (tan θ)
Trigonometric Ratios of Complementary Angles:

- sin (90
°– θ) = cos θ- cos (90
°– θ) = sin θ- tan (90
°– θ) = cot θ- cot (90
°– θ) = tan θ- sec (90
°– θ) = cosec θ- cosec (90
°– θ) = sec θPeriodic Trigonometric Ratios

- sin(π/2-θ) = cos θ
- cos(π/2-θ) = sin θ
- sin(π-θ) = sin θ
- cos(π-θ) = -cos θ
- sin(π+θ)=-sin θ
- cos(π+θ)=-cos θ
- sin(2π-θ) = -sin θ
- cos(2π-θ) = cos θ
Trigonometric Identities

sin⇒ sin^{2}θ + cos^{2}θ = 1^{2}θ = 1 – cos^{2}θ ⇒ cos^{2}θ = 1 – sin^{2}θcosec⇒ cosec^{2}θ – cot^{2}θ = 1^{2}θ = 1 + cot^{2}θ ⇒ cot^{2}θ = cosec^{2}θ – 1sec⇒ sec^{2}θ – tan^{2}θ = 1^{2}θ = 1 + tan^{2}θ ⇒ tan^{2}θ = sec^{2}θ – 1Product to Sum Formulas

- sin x sin y = 1/2 [cos(x–y) − cos(x+y)]
- cos x cos y = 1/2[cos(x–y) + cos(x+y)]
- sin x cos y = 1/2[sin(x+y) + sin(x−y)]
- cos x sin y = 1/2[sin(x+y) – sin(x−y)]
Sum to Product Formulas

- sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]
- sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]
- cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]
- cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]
General Trigonometric Formulas:

- sin (x+y) = sin x × cos y + cos x × sin y
- cos(x+y)=cosx×cosy−sinx×siny
- cos(x–y)=cosx×cosy+sinx×siny

sin(x–y)=sinx×cosy−cosx×siny- If there are no angles x, y and (x ± y) is an odd multiple of (π / 2); then:

- tan (x+y) = tan x + tan y / 1 − tan x tan y
- tan (x−y) = tan x − tan y / 1 + tan x tan y
- If there are no angles x, y and (x ± y) is an odd multiple of π; then:

- cot (x+y) = cot x cot y−1 / cot y + cot x
- cot (x−y) = cot x cot y+1 / cot y − cot x
Formulas for twice of the angles:

- sin2θ = 2sinθ cosθ = [2tan θ /(1+tan2θ)]
- cos2θ = cos2θ–sin2θ = 1–2sin2θ = 2cos2θ–1= [(1-tan2θ)/(1+tan2θ)]
- tan 2θ = (2 tan θ)/(1-tan2θ)
Formulas for thrice of the angles:

- sin 3θ = 3sin θ – 4sin 3θ
- cos 3θ = 4cos 3θ – 3cos θ
- tan 3θ = [3tan θ–tan 3θ]/[1−3tan 2θ]

**Chapter 3 of CBSE Class 11 Maths Notes covers the following topics:**

- Angle and its Measurement
- Trigonometric Functions
- Trigonometric Functions of Sum and Difference of Two Angles

**More resources for CBSE Class 11 Maths Chapter 3**

- Class 11 NCERT Solutions Maths Chapter 3
- Class 11 RD Sharma Solution Trigonometric Functions Chapter 1, Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7, and Chapter 8.
- All important formulas for Chapter 3

## Chapter 4: Principle of Mathematical Induction

As the name suggests, the chapter explains the concept of the** Principle of Mathematical Induction**. The chapter Principle of Mathematical Induction covers a variety of topics, including verifying the induction and justifying the application by considering natural numbers as the least inductive subset of real numbers. The chapter’s exercise covers problems relating to the Principle of Mathematical Induction, as well as its basic applications.

The topics discussed are the process to prove the induction and motivating the application taking natural numbers as the least inductive subset of real numbers.

**Major points covered in CBSE Class 11 Chapter 4: Principle of Mathematical Induction are,**

Principle of Mathematical Induction:The principle of mathematical induction is one such tool that can be used to prove a wide variety of mathematical statements.Working Rule:

Step 1:Show that the given statement is true for n = 1.Step 2:Assume that the statement is true for n = k.Step 3:Using the assumption made in step 2, show that the statement is true for n = k + 1. We have proved the statement is true for n = k. According to step 3, it is also true for k + 1 (i.e., 1 + 1 = 2). By repeating the above logic, it is true for every natural number.

**Chapter 4 of CBSE Class 11 Maths Notes covers the following topic:**

**More resources for CBSE Class 11 Maths Chapter 4**

## Chapter 5: Complex Numbers and Quadratic Equations

As the name of the chapter suggests, the **Complex Numbers and Quadratic Equations** this chapter explains the concept of complex numbers and quadratic equations and their properties. The topics discussed are the square root, algebraic properties, argand plane and polar representation of complex numbers, and solutions of quadratic equations in the complex number system.

The major topics covered in this chapter are determining the modulus and conjugate of a complex number, representing a complex number in the polar form on the argand plane. Solving a quadratic equation, and analyzing the discriminant of a quadratic equation are also explained in this chapter.

**Useful Important information covered in CBSE Class 11 Chapter 5-Complex Numbers and Quadratic Equations are,**

Imaginary Numbers:The square root of a negative real number is called an imaginary number, e.g. √-2, √-5 etc. The quantity √-1 is an imaginary unit and it is denoted by‘i’called iota.

i = √-1, i^{2}= -1, i^{3}= -i, i^{4}= 1

Equality of Complex Number:Two complex numbers z_{1}= x_{1}+ iy_{1}and z_{2}= x_{2}+ iy_{2}are equal, if x_{1}= x_{2}and y_{1}= y_{2}i.e. Re(z_{1}) = Re(z_{2}) and Im(z_{1}) = Im(z_{2})

Addition:Consider z_{1}= x_{1}+ iy_{1}and z_{2}= x_{2}+ iy_{2}are any two complex numbers, then their sum is defined as

z_{1}+ z_{2}= (x_{1}+ iy_{1}) + (x_{2}+ iy_{2}) = (x_{1}+ x_{2}) + i (y_{1}+ y_{2})

Subtraction:Consider z_{1}= (x_{1}+ iy_{1}) and z_{2}= (x_{2}+ iy_{2}) are any two complex numbers, then their difference is defined as

z_{1}– z_{2}= (x_{1}+ iy_{1}) – (x_{2}+ iy_{2}) = (x_{1}– x_{2}) + i(y_{1}– y_{2})

Multiplication:Consider z_{1}= (x_{1}+ iy_{1}) and z_{2}= (x_{2}+ iy_{2}) be any two complex numbers, then their multiplication is defined as

z_{1}z_{2}= (x_{1}+ iy_{1}) (x_{2}+ iy_{2}) = (x_{1}x_{2}– y_{1}y_{2}) + i (x_{1}y_{2}+ x_{2}y_{1})

Division:Consider z_{1}= x_{1}+ iy_{1}and z_{2}= x_{2}+ iy_{2}be any two complex numbers, then their division is defined as

Conjugate of Complex Number:Consider z = x + iy, if ‘i’ is replaced by (-i), then it is called to be conjugate of the complex number z and it is denoted by z¯, i.e.

Modulus of a Complex Number:Consider z = x + iy be a complex number. So, the positive square root of the sum of square of real part and square of imaginary part is called modulus (absolute values) of z and it is denoted by |z| i.e.

|z| = √x_{2}+y_{2}

Argand Plane:Any complex number z = x + iy can be represented geometrically by a point (x, y) in a plane, called argand plane or gaussian plane.

Argument of a complex Number:The angle made by line joining point z to the origin, with the positive direction of X-axis in an anti-clockwise sense is called argument or amplitude of complex number. It is denoted by the symbol arg(z) or amp(z).

arg(z) = θ = tan^{-1}(y/x)

Principal Value of Argument

- When x > 0 and y > 0 ⇒ arg(z) = θ
- When x < 0 and y > 0 ⇒ arg(z) = π – θ
- When x < 0 and y < 0 ⇒ arg(z) = -(π – θ)
- When x > 0 and y < 0 ⇒ arg(z) = -θ

Polar Form of a Complex Number:When z = x + iy is a complex number, so z can be written as,

z = |z| (cosθ + isinθ), where θ = arg(z).which is known as the polar form.

Now, when the general value of the argument is θ, so the polar form of z is written as,

z = |z| [cos (2nπ + θ) + isin(2nπ + θ)], where n is an integer.

**Chapter 5 of CBSE Class 11 Maths Notes covers the following topics:**

- Complex Numbers
- Algebra of Real Functions
- Algebraic Operations on Complex Numbers
- Argand plane and polar representation
- Absolute Value of a Complex Number
- Conjugate of a Complex Number
- Imaginary Numbers

**More resources for CBSE Class 11 Maths Chapter 5**

- Class 11 NCERT Solutions Maths Chapter 5
- Class 11 RD Sharma Solutions Complex Numbers and Quadratic Equations Chapter 1 and Chapter 2
- All important formulas for Chapter 5

## Chapter 6: Linear Inequalities

Chapter 6 of Class 11 Maths NCERT notes explains the concept of **Linear Inequalities**. Linear inequalities deal with the graphical meaning of the algebraic solutions to linear equations in one and two variables illustrated by linear inequalities. The notes of this chapter can help learners develop their visualization abilities. The following notes cover solving linear inequalities, finding the graphical solution to linear equations in two variables, and translating word problems to convert them to mathematical equations.

**Useful important information provided in CBSE Class 11 Chapter 6- Linear Inequalities are,**

Symbols used in inequalities

- The symbol < means less than.
- The symbol > means greater than.
- The symbol < with a bar underneath ≤ means less than or equal to.
- The symbol > with a bar underneath ≥ means greater than or equal to.
- The symbol ≠ means the quantities on the left and right sides are not equal to.

**Chapter 6 of CBSE Class 11 Maths Notes covers the following topics:**

- Compound Inequalities
- Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
- Graphical Solution of Linear Inequalities in Two variables
- Word Problems on Linear Inequalities

**More resources for CBSE Class 11 Maths Chapter 6**

## Chapter 7: Permutations and Combinations

In chapter 7 of Class 11 Maths NCERT notes it is explained that the concepts of **permutation **(an arrangement of a number of objects in a definite order) and **combination **(a collection of the objects irrespective of the order). The topics discussed are the fundamental principle of counting, factorial, permutations, combinations and their applications.

**Important formulas used in CBSE Class 11 Chapter 7- Permutations and Combinations are,**

Factorial:The continued product of first n natural number is called factorial ‘n’. It is denoted by n! which is given by,

n! = n(n – 1)(n – 2)… 3 × 2 × 1and0! = 1! = 1

Permutations:Permutation refers to the various arrangements that can be constructed by taking some or all of a set of things. The number of an arrangement of n objects taken r at a time, where 0 < r ≤ n, denoted by^{n}P_{r}is given by

^{n}P_{r}= n! / (n−r)!

- The number of permutation of n objects of which p
_{1}are of one kind, p_{2}are of second kind,… pk are of kth kind such that p_{1}+ p_{2}+ p_{3}+ … + p_{k}= n is

n! / p_{1}! p_{2}! p_{3}! ….. p_{k}!

Combinations:Combinations are any of the various selections formed by taking some or all of a number of objects, regardless of their arrangement. The number of r objects chosen from a set of n objects is indicated by^{n}C_{r}, and it is given by

^{n}C_{r}= n! / r!(n−r)!

Relation Between Permutation and combination:The relationship between the two concepts is given by two theorems as,

^{n}P_{r}=^{n}C_{r }r! when 0 < r ≤ n.+^{n}C_{r}=^{n}C_{r-1 }^{n+1}C_{r}

**Chapter 7 of CBSE Class 11 Maths Notes covers the following topics:**

**More resources for CBSE Class 11 Maths Chapter 7**

- Class 11 NCERT Solutions Maths Chapter 7
- Class 11 RD Sharma Solutions Permutations and Combinations Chapter 1 and Chapter 2
- All important formulas for Chapter 7

## Chapter 8: Binomial Theorem

The** ****binomial theorem** is a principle that can be used to answer and simplify a variety of problems in not only the previous chapter but also in related topics like probability. As a result, students must be familiar with the binomial theorem and how to use it to expand expressions.

Chapter 8 of Class 11 Maths NCERT notes discusses the binomial theorem for positive integers used to solve complex calculations. The topics discussed are the history, statement, and proof of the binomial theorem and its expansion along with Pascal’s triangle.

**Important conclusions from CBSE Class 11 Chapter 8- Binomial Theorem are,**

Binomial Theorem:The expansion of a binomial for any positive integer n is given by Binomial Theorem, which is

(a + b)^{n}=^{n}C_{0}a_{n}+^{n}C_{1}a_{n-1}b +^{n}C2 a_{n-2}b_{2}+ … +^{n}C_{n-1}a b_{n-1}+^{n}C_{n}b_{n}

- Some special cases from the binomial theorem can be written as:

- (x – y)
^{n}=^{n}C_{0}x_{n}–^{n}C_{1}x_{n-1}y +^{n}C_{2}x_{n-2}y_{2}+ … + (-1)^{n}^{n}C_{n}x_{n}- (1 – x)
^{n}=^{n}C_{0}–^{n}C_{1}x +^{n}C_{2}x^{2}– …. (-1)^{n}^{n}C_{n}x_{n}^{n}C_{0}=^{n}C_{n}= 1Pascal’s triangle:The coefficients of the expansions are arranged in an array called Pascal’s triangle.General Termof following expansions are:

- (a + b)
^{n}is T_{r+1 }=^{n}C_{r}a^{n−r}.b^{r}- (a – b)
^{n}is (-1)^{r}^{n}C_{r}a^{n−r}.b^{r}- (1 + x)
_{n}=^{n}C_{r}. x_{r}- (1 – x)
_{n}= (-1)^{r}^{n}C_{n}x^{r}Middle Terms:In the expansion(a + b), if n is even, then the middle term is the (n/2 + 1)^{n}^{th}term. If n is odd, then the middle terms are (n/2 + 1)^{th}and ((n+1)/2+1)^{th}terms.

**Chapter 8 of CBSE Class 11 Maths Notes covers the following topics:**

**More resources for CBSE Class 11 Maths Chapter 8**

## Chapter 9: Sequences and Series

Students will learn about arithmetic and geometric progressions, as well as how they are related to one another, through sequences and series. This lesson also includes a step-by-step guide to working with special series.

The chapter of Class 11 Maths NCERT notes – **Sequences and Series** discusses the concepts of a sequence (an ordered list of numbers) and series (the sum of all the terms of a sequence). The topics discussed are sequence and series, arithmetic and geometric progression, and arithmetic and geometric mean.

**Some Important formulas covered in CBSE Class 11 Chapter 9- Sequences and Series are,**

- For an Arithmetic Series:
a, a+d, a+2d, a+3d, a+4d, …….a +(n-1)d

- The first term: a
_{1}= a,- The second term: a
_{2}= a + d,- The third term: a
_{3}= a + 2d,- The nth term: a
_{n}= a + (n – 1)d- nth term of an AP from the last term is a’
_{n}=a_{n}– (n – 1)d.- a
_{n}+ a’_{n}= constant- Common difference of an AP i.e. d = a
_{n}– a_{n-1}, ∀ n>1.Sum of n Terms of an AP:S_{n}= n/2 [2a + (n – 1)d] = n/2 (a_{1}+ a_{n})

- A sequence is an AP If the sum of n terms is of the form An
^{2}+ Bn, where A and B are constant and A = half of common difference i.e. 2A = d.

a_{n}=S_{n}– S_{n-1}

Arithmetic Mean:If a, A and b are in A.P then A = (a+b)/2 is called the arithmetic mean of a and b. If a_{1}, a_{2}, a_{3},……a_{n}are n numbers, then their arithmetic mean is given by:

- The common difference is given as,
d = (b – a)/(n + 1)- The Sum of n arithmetic mean between a and b is,
n (a+b/2).Geometric Progression (GP):A sequence in which the ratio of two consecutive terms is constant is called geometric progression.

- The constant ratio is called common ratio (r).

i.e. r = a_{n}+1/a_{n}, ∀ n>1- The general term or nth term of GP is a
_{n}=ar^{n-1}- nth term of a GP from the end is a’
_{n}= 1/r^{n-1}, l = last term- If a, b and c are three consecutive terms of a GP then b
^{2}= ac.Geometric Mean (GM):If a, G and b are in GR then G is called the geometric mean of a and b and is given by G = √(ab).

- If a,G
_{1}, G_{2}, G_{3},….. G_{n}, b are in GP then G_{1}, G_{2}, G_{3},……G_{n}are in GM’s between a and b, then

common ratio is:- If a
_{1}, a_{2}, a_{3},…, a_{n}are n numbers are non-zero and non-negative, then their GM is given by

GM = (a_{1}. a_{2}. a_{3}…a_{n})^{1/n}- Product of n GM is G
_{1}× G_{2}× G_{3}×… × G_{n}= G_{n}= (ab)^{n/2}- Sum of first n natural numbers is: Σn = 1 + 2 + 3 +… + n = n(n+1)/2
- Sum of squares of first n natural numbers is: Σn
^{2}= 1^{2}+ 2^{2}+ 3^{2}+ … + n^{2}= n(n+1)(2n+1)/6- Sum of cubes of first n natural numbers is: Σn
^{3}= 1^{3}+ 2^{3}+ 3^{3}+ .. + n^{3}= (n(n+1)(2n+1)/6)^{2}

**Chapter 9 of CBSE Class 11 Maths Notes covers the following topics:**

- Introduction to Sequences and Series
- General and Middle Terms – Binomial Theorem
- Arithmetic Series
- Arithmetic Sequences
- Geometric Sequence
- Geometric Series
- Arithmetic and Geometric Progressions Word Problems
- Special Series

**More resources for CBSE Class 11 Maths Chapter 9**

- Class 11 NCERT Solutions Maths Chapter 9
- Class 11 RD Sharma Solutions Chapter 1, Chapter 2, and Chapter 3
- All important formulas for Chapter 9

## Chapter 10: Straight Lines

Chapter 10 **Straight lines** in Class 11 is an easy lesson but quite confusing due to a large number of formulas. So, finds it difficult for some students to understand. Therefore, our experts would recommend students first understand the derivation and concept behind these formulas. Then do the constant practice by solving multiple questions on each of them.

Straight lines defined the concept of the line, its angle, slope, and general equation. The topics discussed are the slope of a line, the angle between two lines, various forms of line equations, the general equation of a line, and the family of lines respectively.

**Important formulas covered in CBSE Class 11 Chapter 10- Straight Lines**

Distance Formula:The distance between two points A(x_{1}, y_{1}) and B (x_{2}, y_{2}) is given by,

Section Formula:The coordinates of the point which divides the joint of (x_{1}, y_{1}) and (x_{2}, y_{2}) in the ratio m:n internally, isAnd externally is:

Mid-Pointof the joint of (x_{1}, y_{1}) and (x_{2}, y_{2}) is: .

- X-axis divides the line segment joining (x
_{1}, y_{1}) and (x_{2}, y_{2}) in the ratio -y_{1}: y_{2}.- Y-axis divides the line segment joining (x
_{1}, y_{1}) and (x_{2}, y_{2}) in the ratio -x_{1}: x_{2}.Coordinates of Centroid of a Trianglewith vertices (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is

Area of Triangle:The area of the triangle, the coordinates of whose vertices are (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is,

Slope or Gradient of Line:The inclination of angle θ to a line with a positive direction of X-axis in the anti-clockwise direction, the tangent of angle θ is said to be slope or gradient of the line and is denoted by m. i.e.

m = tan θ

Angle between Two Lines:The angle θ between two lines having slope m_{1}and m_{2}is,

- If two lines are parallel, their slopes are equal i.e. m
_{1}= m_{2}.- If two lines are perpendicular to each other, then their product of slopes is -1 i.e. m
_{1}m_{2}= -1.Point of intersection of two lines:Let equation of lines be ax_{1}+ by_{1}+ c_{1}= 0 and a_{2}x + b_{2}y + c_{2}= 0, then their point of intersection is

Distance of a Point from a Line:The perpendicular distanced of a point P(x_{1}, y_{1})from the line Ax + By + C = 0 is given by,

Distance Between Two Parallel Lines:The distance d between two parallel lines y = mx + c_{1}and y = mx + c_{2}is given by,

Different forms of Equation of a line:

General Equation of a Line:Any equation of the formAx + By + C = 0, where A and B are simultaneously not zero is called the general equation of a lineNormal form:The equation of a straight line upon which the length of the perpendicular from the origin is p and angle made by this perpendicular to the x-axis is α, is given by:x cos α + y sin α = p.Intercept form:The equation of a line that cuts off intercepts a and b respectively on the x and y-axes is given by:x/a + y/b = 1.Slope-intercept form:The equation of the line with slope m and making an intercept c on the y-axis, is y = mx + c.

- One point-slope form: The equation of a line that passes through the point (x
_{1}, y_{1}) and has the slope of m is given by y – y_{1}= m (x – x_{1}).- Two points form: The equation of a line passing through the points (x
_{1}, y_{1}) and (x_{2}, y_{2}) is given by

**Chapter 10 of CBSE Class 11 Maths Notes covers the following topics:**

- Slope of a Straight Line
- Introduction to Two-Variable Linear Equations in Straight Lines
- Forms of Two-Variable Linear Equations of a line
- Point-slope Form
- Slope-Intercept Form of Straight Lines
- Slope-Intercept Equations
- Standard Form of a Straight Line
- x-intercepts and y-intercepts of a Line
- Graphing slope-intercept equations

**More resources for CBSE Class 11 Maths Chapter 10**

- Class 11 NCERT Solutions Maths Chapter 10
- Class 11 RD Sharma Solutions Chapter 1 and Chapter 2
- All important formulas for Chapter 10

## Chapter 11: Conic Sections

**Conic sections** go further into a number of figures, including **cone****, ****circle****, ****hyperbola**, and **parabola**, as well as the many characteristics of each. The various components of these figures are explained to the students, as well as how to determine their measurements.

The topics discussed in the present chapter are the sections of a cone, the degenerate case of a conic section along with the equations and properties of conic sections.

**Some Important formulas learned in CBSE Class 11 Chapter 11- Conic Sections are,**

Equation of a circlewith radius r having a centre (h, k) is given by(x – h)^{2}+ (y – k)^{2}= r^{2}.

- The general equation of the circle is given by
x, where, g, f and c are constants.^{2}+ y^{2}+ 2gx + 2fy + c = 0- The centre of the circle is
(-g, -f).- The radius of the circle is
r = √(g^{2 }+ f^{2 }− c)- The parametric equation of the circle x
^{2}+ y^{2}= r^{2}are given byx = r cos θ, y = r sin θ, where θ is the parametre.- And the parametric equation of the circle (x – h)
^{2}+ (y – k)^{2}= r^{2}are given byx = h + r cos θ, y = k + r sin θ.Parabola:A parabola is the set of points P whose distances from a fixed point F in the plane are equal to their distance from a fixed-line l in the plane. The fixed point F is called focus and the fixed-line l is the directrix of the parabola.

Different forms of parabola

y^{2}= 4ax

y^{2}= -4ax

x^{2}= 4ay

x^{2}= -4ay

Axis of parabolay = 0

y = 0

x = 0

x = 0

Directrix of parabolax = -a

x = a

y = -a

y = a

Vertex(0, 0)

(0, 0)

(0, 0)

(0, 0)

Focus(a, 0)

(-a, 0)

(0, a)

(0, -a)

Length of latus rectum4a

4a

4a

4a

Focal length|x + a|

|x – a|

|y + a|

|y – a|

Ellipse:An ellipse is the set of all points in the plane whose distances from a fixed point in the plane bears a constant ratio, less than to their distance from a fixed point in the plane. The fixed point is called focus, the fixed line a directrix and the constant ratio (e) the eccentricity of the ellipse. The two standard forms of ellipse with their terminologies are mentioned below in the table:

Different forms of Ellipse

x^{2}/a^{2 }+ y^{2}/b^{2}= 1, a > b

x^{2}/b^{2 }+ y^{2}/a^{2}= 1, a > b

Equation of Major Axisy = 0

x = 0

Length of Major Axis2a

2a

Equation of Minor Axisx = 0

y = 0

Length of Minor Axis2b

2b

Equation of Directricesx = ±a/e

y = ±a/e

Vertex(±a, 0)

(0, ±a)

Focus(±ae, 0)

(0, ±ae)

Length of latus rectum2b

^{2}/a2b

^{2}/a

Hyperbola:A hyperbola is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant which is always greater than unity. The fixed point is called the focus, the fixed line is called the directrix and the constant ratio, generally denoted bye, is known as the eccentricity of the hyperbola. The two standard forms of hyperbola with their terminologies are mentioned below in the table:

Different forms of Hyperbola

x^{2}/a^{2 }– y^{2}/b^{2}= 1

x^{2}/a^{2 }– y^{2}/b^{2}= 1

Coordinates of centre(0, 0)

(0, 0)

Coordinates of vertices(±a, 0)

(0, ±a)

Coordinates of foci(±ae, 0)

(0, ±ae)

Length of Conjugate axis2b

2b

Length of Transverse axis2a

2a

Equation of Conjugate axisx = 0

y = 0

Equation of Transverse axisy = 0

x = 0

Equation of Directricesx = ±a/e

y = ±a/e

Eccentricity (e)√(a

^{2}+b^{2})/a^{2}√(a

^{2}+b^{2})/a^{2}

Length of latus rectum2b

^{2}/a2b

^{2}/a

**Chapter 11 of CBSE Class 11 Maths Notes covers the following topics:**

- Introduction to Conic Sections
- Circle
- Parabola
- Ellipse
- Hyperbola
- Identifying Conic Sections from their Equation

**More resources for CBSE Class 11 Maths Chapter 11**

- Class 11 NCERT Solutions Maths Chapter 11
- Class 11 RD Sharma Solutions Conic Sections Chapter 1, Chapter 2, Chapter 3 and Chapter 4
- All important formulas for Chapter 11

## Chapter 12: Introduction to Three-dimensional Geometry

In this chapter** Introduction to Three-dimensional Geometry** of Class 11 Maths NCERT notes, it is explained that the concepts of geometry in three-dimensional space. The topics discussed are the coordinate axes and planes respectively, points coordinate, distance, and a section for points.

Students learn geometrical principles such as the distance formula and the section formula through an introduction to three-dimensional geometry. It helps students in understanding how to effectively apply these formulas to solve problems.

**Important points covered in CBSE Class 11 Chapter 12- Introduction to Three-dimensional Geometry,**

Coordinate Axes:In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are three mutually perpendicular lines. These axes are called the X, Y and Z axes.Coordinate Planes:The three planes determined by the pair of axes are the coordinate planes. These planes are called XY, YZ and ZX planes and they divide the space into eight regions known as octants.Coordinates of a Point in Space:The coordinates of a point in the space are the perpendicular distances from P on three mutually perpendicular coordinate planes YZ, ZX, and XY respectively. The coordinates of a point P are written in the form of triplet like (x, y, z). The coordinates of any point on:

- X-axis is of the form (x, 0,0)
- Y-axis is of the form (0, y, 0)
- Z-axis is of the form (0, 0, z)
- XY-plane are of the form (x, y, 0)
- YZ-plane is of the form (0, y, z)
- ZX-plane are of the form (x, 0, z)

**Chapter 12 of CBSE Class 11 Maths Notes covers the following topics:**

**More resources for CBSE Class 11 Maths Chapter 12**

## Chapter 13: Limits and Derivatives

Chapter 13 of Class 11 Maths NCERT notes explains the concept of **calculus **that deals with the study of change in the value of a function when the change occurs in the domain points. The topics discussed are the** ****definition **and **algebraic operations of limits**** **and **derivatives** respectively.

The Chapter Limits and derivatives comprise topics such as determining the limit of a function at a point, algebra of limits, limits of trigonometric functions, using the limit formula to find the derivative of a function, and algebra of derivatives.

**Some Important formulas covered in CBSE Class 11 Chapter 13- Limits and Derivatives,**

Left Hand and Right-Hand Limits:If values of the function at the point which are very near to a on the left tends to a definite unique number as x tends to a, then the unique number so obtained is called the left-hand limit of f(x) at x = a, we write it as

- Similarly, right hand limit is given as,

- A limit exists when:
and both exists or,

Some Important Properties of Limits:Consider f and g be two functions such that both \lim_{x\to a}f(x) and \lim_{x\to a}g(x) exists, then:

SomeStandard Limitsare given as:

Derivatives:Consider a real-valued function f, such that:

is known as the Derivative of function f at x if and only if,

exists finitely.

Some Important Properties of Derivatives:Consider f and g be two functions such that their derivatives can be defined in a common domain as:

SomeStandard Derivativesare given as:

**Chapter 13 of CBSE Class 11 Maths Notes covers the following topics:**

- Introduction to Limits
- Formal Definition of Limits
- Strategy in Finding Limits
- Determining Limits using Algebraic Manipulation
- Limits of Trigonometric Functions
- Properties of Limits
- Limits by Direct Substitution
- Estimating Limits from Graphs
- Estimating Limits from Tables
- Squeeze Theorem
- Introduction to Derivatives
- Average and Instantaneous Rate of Change
- Algebra of Derivative of Functions
- Product Rule – Derivatives
- Quotient Rule
- Derivatives of Polynomial Functions
- Derivatives of Trigonometric Functions
- Power Rule in Derivatives
- Application of Derivatives
- Applications of Power Rule

**More resources for CBSE Class 11 Maths Chapter 13**

- Class 11 NCERT Solutions Maths Chapter 13
- Class 11 RD Sharma Solutions Limits and Derivatives Chapter 1 and Chapter 2
- All important formulas for Chapter 13

## Chapter 14: Mathematical Reasoning

Mathematical problem-solving skills help students in developing and enhancing their reasoning abilities. Students will read sentences and make logical conclusions from them in this lesson. As the name suggests, the chapter explains the concepts of **mathematical reasoning** (a critical skill to analyze any given hypothesis in the context of mathematics).

The topics explained in detail are compound statements, the negation, and implication of statements, how to validate statements as well as contrapositive and converse statements.

**Important points learned in CBSE Class 11 Chapter 14- Mathematical Reasoning are,**

Compound statement: A statement is a compound statement if it is made up of two or more smaller statements. The smaller statements are called component statements of the compound statement. The Compound statements are made by:

Connectives: “AND”, “OR”Quantifiers: “there exists”, “For every”Implications:The meaning of implications “If ”, “only if ”, “ if and only if ”.“p ⇒ q” :

- p is a sufficient condition for q or p implies q.
- q is necessary to condition for p. The converse of a statement p ⇒ q is the statement q ⇒ p.
- p⇒ q together with its converse gives p if and only if q.
“p ⇔ q”:

- p implies q (denoted by p ⇒ q)
- p is a sufficient condition for q
- q is a necessary condition for p
- p only if q
- ∼q implies ∼p

**Chapter 14 of CBSE Class 11 Maths Notes covers the following topics:**

**More resources for CBSE Class 11 Maths Chapter 14**

## Chapter 15: Statistics

**Chapter 15 of Class 11 Maths** is a very crucial lesson from an examination perspective. A student must revise **Statistics **from previous classes, and understand and memorize all the statistics formulas. Also, use these formulas to practice all the **NCERT questions from Chapter 15 Statistics**.

Chapter 15 of** Class 11 Maths NCERT notes **explains the concepts of **statistics **(data collected for specific purposes), dispersion, and methods of calculation for ungrouped and grouped data. The topics discussed are range, mean deviation, variance and standard deviation, and analysis of frequency distributions.

**Some Important formulas covered in CBSE Class 11 Chapter 15- Statistics,**

Range:The measure of dispersion which is easiest to understand and easiest to calculate is the range. Range is defined as the difference between two extreme observation of the distribution.

Range of distribution = Largest observation – Smallest observation.

Mean Deviation:Mean deviation for ungrouped data- For n observations x_{1}, x_{2}, x_{3},…, x_{n}, the mean deviation about their mean x¯ is given by:And, the Mean deviation about its median M is given by,

Mean deviation for discrete frequency distribution-

Variance:Variance is the arithmetic mean of the square of the deviation about mean x¯. Let x_{1}, x_{2}, ……x_{n}be n observations with x¯ as the mean, then the variance denoted by σ^{2}, is given by

Standard deviation:If σ^{2}is the variance, then σ is called the standard deviation is given byStandard deviation of a discrete frequency distribution is given by

Coefficient of variation:In order to compare two or more frequency distributions, we compare their coefficient of variations. The coefficient of variation is defined asCoefficient of variation = (Standard deviation / Mean) × 100

**Chapter 15 of CBSE Class 11 Maths Notes covers the following topics:**

- Measures of Spread
- Mean Absolute Deviation
- Measures of Central Tendency
- Difference Between Mean, Median, and Mode with Examples
- Analysis of Frequency Distribution
- Variance and Standard Deviation

**More resources for CBSE Class 11 Maths Chapter 15**

## Chapter 16: Probability

**Class 11 Maths ****Probability** builds on previous classes by introducing students to probability concepts such as **random experiments****, ****outcomes****, ****sample space****, ****different sorts of events****, **and other related principles that make up the chapter’s backbone.

**Chapter 16 of Class 11 Maths NCERT notes** discusses the concept of probability (a measure of uncertainty of various phenomena or a chance of occurrence of an event). The topics discussed are the random experiments, outcomes, sample spaces, event, and their type.

**Major points covered in CBSE Class 11 Chapter 16- Probability,**

- For any random experiment, let S be the sample space. The probability P is a real-valued function whose domain is the power set of S and [0, 1] is the range interval. For any event E:
P(E) ≥ 0andP(S) = 1Mutually exclusive events:If E and F are mutually exclusive events, then:P(E ∪ F) = P(E) + P(F)Equally likely outcomes:All outcomes with equal probability are called equally likely outcomes. Let S be a finite sample space with equally likely outcomes and A be the event. Therefore, the probability of an event A is:P(A) = n(A) / n(S),where n(A) is the number of elements on the set A and n(S) is the Total number of outcomes or the number of elements in the sample space S

- Let P and Q be any two events, then the following formulas can be derived.

- Event P or Q: The set P ∪ Q
- Event P and Q: The set P ∩ Q
- Event P and not Q: The set P – Q
- P and Q are mutually exclusive if P ∩ Q = φ
- Events P
_{1}, P_{2}, . . . . . , P_{n}are exhaustive and mutually exclusive if P_{1}∪ P_{2}∪ . . . . . ∪ P_{n}= S and E_{i}∩ E_{j}= φ for all i ≠ j.

**Chapter 16 of CBSE Class 11 Maths Notes covers the following topics:**

- Random Experiments
- Events
- Types of Events in Probability
- Axiomatic Approach to Probability
- Analysis of Frequency distribution
- Variance and Standard Deviation
- Measure of dispersion and mean deviation

**More resources for CBSE Class 11 Maths Chapter 16**

**Important Resources for CBSE Class 11th provided by GeeksforGeeks:-**

**NCERT Solutions Maths Class 11****RD Sharma Solutions Maths Class 11****CBSE Class 11 Maths Formulas****CBSE Physics Class 11 Notes****CBSE Class 11 Chemistry Notes**

### Frequently Asked Questions (FAQs)

**Question 1: What is the key importance of Class 11 Maths NCERT Notes?**

**Answer:**

Students in class 11 must study for their home examinations as well as competitive tests (IIT-JEE, BITS, etc.). These NCERT Notes explanations give a step-by-step interpretation of all solutions as well as the concepts, ensure that students learn how to write an exam and achieve the highest possible result. It also includes tips and tactics for solving complex problems in seconds, which are very useful for competitive exams.

**Question 2: What are some important topics from Class 11 Maths NCERT for CBSE final year exams?**

Following is the list of some important chapters/topics from Class 11 Maths NCERT:

- Set Theory
- Trigonometry
- Algebra
- Arithmetic
- Calculus
- Geometry
- Probability and Statistics
- Number System

**Question 3: What are the Best Ways to Learn NCERT Class 11th Maths Concepts?**

**Answer: **

The easiest approach to study the topics contained in NCERT answers class 11th mathematics is to understand a topic and practise questions on it on a regular basis. Students may get the most out of these answers by studying the subject on a regular basis and ensuring that they obtain good grades in their exams.

**Question 4: What are the Important Formulas for Class 11 Maths?**

**Answer:**

Algebra, trigonometry, quadratic equations, statistics, and probability are among topics covered in class 11 math formulae. The following are some of the most important Class 11 math formulae from these topics:

- A U B = {x: x ∈ A (or) x ∈ B}
- A ∩ B = {x: x ∈ A (and) x ∈ B}
- A × B = {(a, b): a ∈ A, b ∈ B}
- sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
- cos(x + y) = cos(x) cos(y) – sin(x) sin(y)
- tan(x + y) = (tan(x) + tan(y)) / (1- tan(x) × tan(y))
- sin(x – y) = sin(x) cos(y) – cos(x) sin(y)
- cos(x – y) = cos(x) cos(y) + sin(x) sin(y)
- tan(x – y) = (tan(x) – tan(y)) / (1 + tan(x) × tan (y))
- f ´( x ) = d f(x)/dx
- x̅ = ∑|xi – x̅| / n
- σ
^{2}= ∑fi(xi – x̅)^{2}/ N

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