GeeksforGeeks brings a Formula sheet for Class 11 Maths students, which is strictly based on the NCERT Syllabus to ease out the preparation and revision process of the students for school exams as well as various competitive exams like JEE and NEET. This article not only includes the list of formulae but offer students a summary of the chapters, important points to remember, a brief explanation of important concepts and derivations of formulae for better comprehension and retaining of the chapters. Hence, these Chapter-wise CBSE Class 11 Maths Formulae are prepared to ensure maximum preparation and good marks in any examination.

### Chapter 1: Sets

The chapter explains the concept of sets along with their representation. The topics discussed are empty sets, equal sets, subsets, finite and infinite sets, power sets, and universal sets. A set is a well-collaborated collection of objects. A set consisting of definite elements is a finite set. Otherwise, it is an infinite set. Below are the important terms and properties used in Sets are listed as:

- The union of two sets A and B are said to be contained elements that are either in set A and set B. The union of A and B is denoted as,
A∪B.- The intersection of two sets A and B are said to be contained elements that are common in both sets. The intersection of A and B is denoted as,
A∩B.- The complement of a set A is the set of all elements given in the universal set U that are not contained in A. The complement of A is denoted as,
A’.

- For any two sets A and B, the following holds true:(DE Morgan’s Theorem)

- (A∪B)′=A′∩B′
- (A∩B)′=A′∪B′
- If the finite sets A and B are given such that,
, then:(A∩B)=ϕ

n(A∪B)=n(A)+n(B)

- If
, then:(A∪B)=ϕ

n(A∪B)=n(A)+n(B)−n(A∩B)

- Some other important formulas of Sets for any three sets A, B, and C are as follows:

- 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 2: Relations & Functions

The chapter Relations & Functions explains the advanced concepts of sets theory using the concept of ordered and unordered pairs of elements. An ordered pair is a pair of elements grouped together in a certain order. A relation R towards a set A to a set B can be described as a subset of the cartesian product A × B which is obtained by describing a relationship between the first of its element x and the second of its element y given in the ordered pairs of A × B.

The below-mentioned properties will surely assist students to solve various maths problems:

A relation R is the subset of the cartesian product of A × B, where A and B are two non-empty elements. It is derived by stating the relationship between the first element and second element of the ordered pair of A × B.Relations: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 RInverse of Relation:^{-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 relation f from a set A to set B is said to be function, if every element of set A has one and only image in set B.Functions:- A cartesian product A × 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.
- A function f from set A to set B considers a specific relation type where every element x in set A has one and only one image in set B. A function can be denoted as
f : A → B, where f(x) = y.If the function f : X → R and g : X → R; we have:Algebra of functions:

- (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 3: Trigonometric Functions

In Mathematics, trigonometric functions are the real functions that relate to an angle of a right-angled triangle forming some finite ratios of two side lengths. Find the important Maths formulas for Class 11 related to trigonometric functions below.

- If in a circle of radius r, an arc of length l subtends an angle of θ radians, then
.l = r×θ

- Radian Measure = π/180 × Degree Measure
- Degree Measure = 180/π × Radian Measure
Trigonometric ratios:

- sin θ = (Perpendicular (P)) / (Hypotenuse (H)).
- cos θ = (Base (B)) / ( Hypotenuse (H)).
- tan θ = (Perpendicular (P)) / (Base (B)).
- cosec θ = (Hypotenuse (H)) / (Perpendicular (P)).
- sec θ = (Hypotenuse (H)) / (Base (B)).
- cot θ = (Base (B)) / (Perpendicular (P)).
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 θ°Periodic 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^{2}θ + cos^{2}⇒ sinθ = 1^{2}θ = 1 – cos^{2}θ ⇒ cos^{2}θ = 1 – sin^{2}θcosec^{2}θ – cot^{2}⇒ cosecθ = 1^{2}θ = 1 + cot^{2}θ ⇒ cot^{2}θ = cosec^{2}θ – 1sec^{2}θ – tan^{2}⇒ secθ = 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+tan
^{2}θ)]- cos2θ = cos
^{2}θ–sin^{2}θ = 1–2sin2θ = 2cos^{2}θ–1= [(1-tan^{2}θ)/(1+tan^{2}θ)]- tan 2θ = (2 tan θ)/(1-tan
^{2}θ)Formulas for thrice of the angles:

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

### Chapter 4: Principle of Mathematical Induction

As the name suggests, the chapter explains the concept of the Principle of Mathematical Induction. 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. One key basis for mathematical thinking is deductive reasoning. In contrast to deduction, inductive reasoning depends on working with different cases and developing a conjecture by observing incidences till we have observed each and every case. Thus, in simple language we can say the word ‘induction’ means the generalisation from particular cases or facts.

Below mentioned is the list of some important terms and steps used in the chapter mentioned above:

A sentence is called a statement if it is either true or false.Statement:Motivation is tending to initiate an action. Here Basis step motivate us for mathematical induction.Motivation:The principle of mathematical induction is one such tool that can be used to prove a wide variety of mathematical statements. Each such statement is assumed as P(n) associated with positive integer n, for which the correctness for the case n = 1 is examined. Then assuming the truth of P(k) for some positive integer k, the truth of P (k+1) is established.Principle of Mathematical Induction:Working Rule:

Show that the given statement is true for n = 1.Step 1:Assume that the statement is true for n = k.Step 2: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.Step 3:

### Chapter 5: Complex Numbers and Quadratic Equations

As the name of the chapter suggests, therefore, 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, solutions of quadratic equations in the complex number system. A few important points related to the Complex Numbers and Quadratic Equations are as follows:

Complex NumbersA number that can be expressed in the form a + b is known as the complex number; where a and b are the real numbers and i is the imaginary part of the complex number.:Imaginary NumbersThe 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:called iota.‘i’

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

Two complex numbers zEquality of Complex Number:_{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})

Consider zAddition:_{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})

Consider zSubtraction:_{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})

Consider zMultiplication:_{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})

Consider zDivision:_{1}= x_{1}+ iy_{1}and z_{2}= x_{2}+ iy_{2}be any two complex numbers, then their division is defined as[Tex]\dfrac{z_1}{z_2}=\dfrac{x_1+iy_1}{x_2+iy_2}=\dfrac{(x_1x_2+y_1y_2)+i(x_2y_1-x_1y_2)}{x_2^2+y_2^2}\,\,\,\,\,\text{where}\,z_2\neq0.[/Tex]

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.

[Tex]\bar{z} = x – iy[/Tex]

Modulus of a Complex NumberConsider z = x + y 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}Any complex number z = x + y can be represented geometrically by a point (x, y) in a plane, called argand plane or gaussian plane.

- A pure number x, i.e. (x + 0i) is represented by the point (x, 0) on X-axis. Therefore, X-axis is called real axis.
- A purely imaginary number y i.e. (0 + y) is represented by the point (0, y) on the y-axis. Therefore, the y-axis is called the imaginary axis.

Argument of a complex NumberThe 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 NumberWhen z = x + iy is a complex number, so z can be written as,

, where θ = arg(z).z = |z| (cosθ + isinθ)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,

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

### Chapter 6: Linear Inequalities

The chapter explains the concept of Linear Inequalities. The topics discussed are algebraic solutions and graphical representation of Linear Inequalities in one variable and two variables respectively. In mathematics, an inequality is a relation that holds between two values when they are different, Solving linear inequalities is very similar to solving linear equations, except for one small but important detail: you flip the inequality sign whenever you multiply or divide the inequality by a negative.

An inequation or inequality is a statement involving variables and the sign of inequality like >, <, ≥ or ≤.Inequation:

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.

Algebraic Solutions for Linear Inequalities in One Variableandits Graphical RepresentationUsing the trial-and-error method, the solution to the linear inequality can be determined. However, this method isn’t always possible, and computing the solution takes longer. So, using a numerical approach, the linear inequality can be solved. When solving linear inequalities, remember to follow these rules:

Don’t change the sign of an inequality by adding or subtracting the same integer on both sides of an equation.Rule 1:

Add or subtract the same positive integer from both sides of an inequality equation.Rule 2:

### Chapter 7: Permutations and Combinations

The present chapter explains 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 along with the concept of restricted permutation. If a certain event occurs in ‘m’ different ways followed by an event that occurs in ‘n’ different ways, then the total number of occurrences of the events can be given in m × n order. Find the important Maths formulas for class 11 Permutations and Combinations are as under:

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

andn! = n(n – 1)(n – 2)… 3 × 2 × 10! = 1! = 1

PermutationsPermutation 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}!

CombinationsCombinations 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)!

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

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

### Chapter 8: Binomial Theorem

This chapter 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. A Binomial Theorem helps to expand a binomial given for any positive integer n.

Binomial TheoremThe 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 triangleThe coefficients of the expansions are arranged in an array called Pascal’s triangle.:of following expansions are:General Term

- (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 TermsIn 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 9: Sequences and Series

The chapter Sequences and Series discuss 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, arithmetic, and geometric mean. Here is the list of some important terms used in Sequence and Series are as listed below:

A sequence whose terms follow certain patterns is known as progression.Progression:Arithmetic Progression (AP)An arithmetic progression (A.P.) is a sequence where the terms either increase or decrease regularly by the same constant. This constant is called the common difference (d). The first term is denoted by a and the last term of an AP is denoted by l.:

- 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 APS:_{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 MeanIf 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: [Tex]A = \dfrac{a_1+a_2+a_3+…+a_n}{n}[/Tex]

- 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.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).Geometric Mean (GM):

- 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: [Tex]r = \left(\dfrac{b}{a}\right)^{\dfrac{1}{n+1}}[/Tex]- 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 10: Straight Lines

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, general equation of a line, and family of lines respectively. Here are some important formulas used in the Chapter Straight lines:

Distance FormulaThe distance between two points A(x:_{1}, y_{1}) and B (x_{2}, y_{2}) is given by,[Tex]AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/Tex]

- The distance of a point A(x, y) from the origin 0 (0, 0) is given by OA = √(x
^{2}+ y^{2}).Section FormulaThe coordinates of the point which divides the joint of (x:_{1}, y_{1}) and (x_{2}, y_{2}) in the ratio m:n internally, is[Tex]\left(\dfrac{mx_2+nx_1}{m+n},\,\dfrac{my_2+ny_1}{m+n}\right)[/Tex]

And externally is:

[Tex]\left(\dfrac{mx_2-nx_1}{m-n},\,\dfrac{my_2-ny_1}{m-n}\right)[/Tex]

of the joint of (xMid-Point_{1}, y_{1}) and (x_{2}, y_{2}) is: [Tex]\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right) [/Tex].

- 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}.with vertices (xCoordinates of Centroid of a Triangle_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is[Tex]\left(\dfrac{x_1+x_2+x_3}{3},\,\dfrac{y_1+y_2+y_3}{3}\right)[/Tex]

The area of the triangle, the coordinates of whose vertices are (xArea of Triangle:_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is,[Tex]\begin{aligned}\text{Area of Triangle}&=\dfrac{1}{2}\begin{vmatrix}x_1&x_2&1\\x_2&y_2&1\\x_3&x_2&1\end{vmatrix}\\&=\dfrac{1}{2}\left[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right]\end{aligned}[/Tex]

- If the points (x
_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) are collinear, then x_{1}(y_{2}– y_{3}) + x_{2}(y_{3}– y_{1}) + x_{3}(y_{1}– y_{2}).= 0Slope or Gradient of LineThe 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

- The slope of a line passing through points P(x
_{1}, y_{1}) and Q(x_{2}, y_{2}) is given by,[Tex]m = \text{tan}\theta=\dfrac{y_2-y_1}{x_2-x_1}[/Tex]

The angle θ between two lines having slope mAngle between Two Lines:_{1}and m_{2}is, [Tex]\text{tan}\theta=\left|\dfrac{m_2-m_1}{1+m_1m_2}\right|[/Tex]

- 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.Let equation of lines be axPoint of intersection of two lines:_{1}+ by_{1}+ c_{1}= 0 and a_{2}x + b_{2}y + c_{2}= 0, then their point of intersection is[Tex]\left(\dfrac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1},\,\dfrac{a_2c_1-a_1c_2}{a_1b_2-a_2b_1}\right)[/Tex]

The perpendicular distanced of a point P(xDistance of a Point from a Line:_{1}, y_{1})from the line Ax + By + C = 0 is given by,[Tex]d=\left|\dfrac{Ax_1+By_1+C}{\sqrt{A^2+B^2}}\right|[/Tex]

The distance d between two parallel lines y = mx + cDistance Between Two Parallel Lines:_{1}and y = mx + c_{2}is given by,[Tex]d=\dfrac{\left|c_1-c_2\right|}{\sqrt{1+m^2}}[/Tex]

Different forms of Equation of a line:

- If a line is at a distance k and parallel to X-axis, then the equation of the line is
y = ± k.- If a line is parallel to Y-axis at a distance of c from the Y-axis, then its equation is
x = ± c.Any equation of the formGeneral Equation of a Line:, where A and B are simultaneously not zero is called the general equation of a lineAx + By + C = 0The 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:Normal form: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 formThe 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[Tex]y-y_1=\left(\dfrac{y_2-y_1}{x_2-x_1}\right)(x-x_1)[/Tex]

### Chapter 11: Conic Sections

The topics discussed in the chapter Conic Sections are the sections of a cone, the degenerate case of a conic section along the equations and properties of conic sections. A circle is a geometrical figure where all the points in a plane are located equidistant from the fixed point on a given plane. Following are the list of some important formulas discussed in the chapter Conic Sections as,

with radius r having a centre (h, k) is given byEquation of a circle(x – h)^{2}+ (y – k)^{2}= r^{2}.

- The general equation of the circle is given by
x^{2}+ y^{2}, where, g, f and c are constants.+ 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 by, where θ is the parameter.x = r cos θ, y = r sin θ- 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.ParabolaA 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|

EllipseAn 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 Directrixx = ±a/e

y = ±a/e

Vertex(±a, 0)

(0, ±a)

Focus(±ae, 0)

(0, ±ae)

Length of latus rectum2b

^{2}/a2b

^{2}/a

HyperbolaA 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 Directrixx = ±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 12: Introduction to Three-dimensional Geometry

As the name suggests, the chapter explains the concepts of geometry in three-dimensional space. The topics discussed are the coordinate axes and planes respectively, points coordinate, distance and section for points. The three planes determined by the pair of axes are known as coordinate planes with XY, YZ and ZX planes. Find below the important Maths formulas for Class 11 Introduction to Three-dimensional Geometry as:

Coordinate AxesIn 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 PlanesThe 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.: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:Coordinates of a Point in Space:

- 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)
Distance Formula:The distance between two points A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) is given by,[Tex]AB=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}[/Tex]

- While, the distance between two points A(x, y, z) from the origin O(0, 0, 0) is given by,
[Tex]OA=\sqrt{x^2+y^2+z^2}[/Tex]

Section FormulaThe coordinates of the point R which divides the line segment joining two points P(x:_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) internally or externally in the ratio m : n are given by,[Tex]\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n},\dfrac{mz_2+nz_1}{m+n}\right)\,\text{and}\left(\dfrac{mx_2-nx_1}{m-n},\dfrac{my_2-ny_1}{m-n},\dfrac{mz_2-nz_1}{m-n}\right)[/Tex]

of the joint of (xMid-Point_{1}, y_{1}) and (x_{2}, y_{2}) is:[Tex]\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right) [/Tex]

with vertices (xCoordinates of Centroid of a Triangle_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is[Tex]\left(\dfrac{x_1+x_2+x_3}{3},\,\dfrac{y_1+y_2+y_3}{3}\right)[/Tex]

### Chapter 13: Limits and Derivatives

The chapter 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.

A limit of a function at a certain point holds a common value of the left as well as the right-hand limits if they coincide with each other. Here are the list of some important formulas used to solve problems on Limits and Derivatives as,

LimitLet y = f(x) be a function of x. If at x = a, f(x) takes indeterminate form, then we consider the values of the function which is very near to a. If these value tend to a definite unique number as x tends to a, then the unique number so obtained is called the limit of f(x) at x = a and we write it as [Tex]\lim_{x\to a}f(x) [/Tex].: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 asLeft Hand and Right-Hand Limits:[Tex]f(a-0)=\lim_{x\to a^-}f(x)=\lim_{h\to 0}f(a-h)[/Tex]

- Similarly, right hand limit is given as,
[Tex]f(a+0)=\lim_{x\to a^+}f(x)=\lim_{h\to 0}f(a+h)[/Tex]

- A limit [Tex]\lim_{x\to a}f(x) [/Tex] exists when:
[Tex]\lim_{x\to a^-}f(x) [/Tex] and [Tex]\lim_{x\to a^+}f(x) [/Tex] both exists or,

[Tex]\lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x)[/Tex]

Some Important Properties of LimitsConsider f and g be two functions such that both \lim_{x\to a}f(x) and \lim_{x\to a}g(x) exists, then::[Tex]\begin{aligned}\lim_{x\to a}[f(x)\pm g(x)]&=\lim_{x\to a}f(x)\pm \lim_{x\to a} g(x)\\\lim_{x\to a}kf(x)&=k\lim_{x\to a}f(x)\\\lim_{x\to a}f(x)\cdot g(x)&=\lim_{x\to a}f(x)\times\lim_{x\to a}g(x)\\\lim_{x\to a}\dfrac{f(x)}{g(x)}&=\dfrac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}\end{aligned}[/Tex]

SomeStandard Limitsare given as:[Tex]\begin{aligned}\lim_{x\to a}\dfrac{x^n-a^n}{x-a}&=na^{n-1}\\\lim_{x\to 0}\dfrac{\sin x}{x}&=1\\\lim_{x\to 0}\dfrac{\tan x}{x}&=1\\\lim_{x\to 0}\dfrac{a^x-1}{x}&=\log_e a\\\lim_{x\to 0}\dfrac{e^x-1}{x}&=1\\\lim_{x\to 0}\dfrac{\log(1+x)}{x}&=1\end{aligned} [/Tex]

DerivativesConsider a real-valued function f, such that::[Tex]f'(x)=\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h} [/Tex]

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

[Tex]\lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h} [/Tex] exists finitely.

Consider f and g be two functions such that their derivatives can be defined in a common domain as:Some Important Properties of Derivatives:[Tex]\begin{aligned}\dfrac{\mathrm{d}}{\mathrm{d}x}[f(x)+g(x)]&=\dfrac{\mathrm{d}}{\mathrm{d}x}[f(x)]+\dfrac{\mathrm{d}}{\mathrm{d}x}[g(x)]\\\dfrac{\mathrm{d}}{\mathrm{d}x}[f(x)-g(x)]&=\dfrac{\mathrm{d}}{\mathrm{d}x}[f(x)]-\dfrac{\mathrm{d}}{\mathrm{d}x}[g(x)]\\\dfrac{\mathrm{d}}{\mathrm{d}x}[f(x)\cdot g(x)]&=\left[\dfrac{\mathrm{d}}{\mathrm{d}x}f(x)\right]\cdot g(x)+f(x)\cdot\left[\dfrac{\mathrm{d}}{\mathrm{d}x}g(x)\right]\\\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\dfrac{f(x)}{g(x)}\right]&=\dfrac{\left[\dfrac{\mathrm{d}}{\mathrm{d}x}f(x)\right]\cdot g(x)-f(x)\cdot \left[\dfrac{\mathrm{d}}{\mathrm{d}x}g(x)\right]}{[g(x)]^2}\end{aligned}[/Tex]

SomeStandard Derivativesare given as:[Tex]\begin{aligned}\dfrac{\mathrm{d}}{\mathrm{d}x}(x^n)&=nx^{n-1}\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin x)&=\cos x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\cos x)&=-\sin x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\tan x)&=\sec^2 x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\cot x)&=-\cosec^2 x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\sec x)&=\sec x \tan x\\\dfrac{\mathrm{d}}{\mathrm{d}x}\cosec x&=-\cosec x \cot x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(a^x)&=a^x\log_e a\\\dfrac{\mathrm{d}}{\mathrm{d}x}(e^x)&=e^x\\\dfrac{\mathrm{d}}{\mathrm{d}x}(\log_e x)&=\dfrac{1}{x}\end{aligned}[/Tex]

### Chapter 14: Mathematical Reasoning

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 discussed are statements, inductive reasoning, and deductive reasoning. Following are the list of important terms discussed as:

StatementsA statement is a sentence which either true or false, but not both simultaneously. For example: “A triangle has four sides.”, “New Delhi is the capital of India.” are the statements.:: Negation of a statement p: If p denote a statement, then the negation of p is denoted by ∼p.Negation of a 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:Compound statement

: “AND”, “OR”Connectives: “there exists”, “For every”QuantifiersThe meaning of implications “If ”, “only if ”, “ if and only if ”.Implications:“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
: The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒∼p.ContrapositiveIf to check whether p is true we assume negation p is true.Contradiction:: Checking of a statement whether it is true or false. The validity of a statement depends upon which of the special. The following methods are used to check the validity of statements:Validating statements

- direct method
- contrapositive method
- method of contradiction
- using a counterexample.

### Chapter 15: Statistics

This chapter 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. Here one will find the essential maths formulas for Class 11 of Statistics given below:

Measure of DispersionThe dispersion is the measure of variations in the values of the variable. It measures the degree of scatteredness of the observation in a distribution around the central value.:RangeThe 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 for ungrouped data- For n observations x

_{1}, x_{2}, x_{3},…, x_{n}, the mean deviation about their mean x¯ is given by:[Tex]MD(\bar x)=\dfrac{\sum |x_i – \bar x|}{n}[/Tex]

And, the Mean deviation about its median M is given by,

[Tex]MD(M)=\dfrac{\sum |x_i – M|}{n}[/Tex]

Mean deviation for discrete frequency distribution-

[Tex]MD(\bar x)=\dfrac{\sum f_i|x_i – \bar x|}{\sum f_i}=\dfrac{\sum f_i|x_i – \bar x|}{N}[/Tex]

VarianceVariance 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[Tex]\sigma^2=\dfrac{\sum(x_i-\bar x)^2}{n}[/Tex]

Standard deviationIf σ:^{2}is the variance, then σ is called the standard deviation is given by[Tex]\sigma=\sqrt{\dfrac{\sum(x_i-\bar x)^2}{n}}[/Tex]

Standard deviation of a discrete frequency distribution is given by

[Tex]\sigma=\sqrt{\dfrac{\sum f_i(x_i-\bar x)^2}{N}}[/Tex]

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

[Tex]CV=\dfrac{\sigma}{\bar x}\times 100[/Tex]

### Chapter 16: Probability

Probability is a fundamental part of the Class 11 Maths curriculum and is important for Class 11 exams and different competitive exams like JEE and NEET. In previous classes, students may have learned the basic concept of probability as a measure of uncertainty of various phenomena. Here, a list of formulas, important properties, applications and a summary of the chapter is discussed, which will help students learn the related concepts quickly and score good marks in the exam.

The chapter 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. Following is the list of important formulas and chapter summary for Class 11 Probability as:

ProbabilityNumber of Favourable outcomes in an Event / Total number of Outcomes=EventAn event is a subset of the S (sample space). An empty set is also known as the Impossible event.:

- 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:
andP(E) ≥ 0P(S) = 1Mutually exclusive eventsIf E and F are mutually exclusive events, then::P(E ∪ F) = P(E) + P(F)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:Equally likely outcomes: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 SP(A) = n(A) / n(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.