Математические формулы CBSE Class 11

Опубликовано: 8 Октября, 2022

GeeksforGeeks предлагает лист формул для учащихся 11-го класса по математике, который строго основан на учебной программе NCERT, чтобы упростить процесс подготовки и повторения учащихся к школьным экзаменам, а также к различным конкурсным экзаменам, таким как JEE и NEET. Эта статья не только включает список формул, но и предлагает учащимся краткое изложение глав, важные моменты, которые следует помнить, краткое объяснение важных понятий и вывод формул для лучшего понимания и запоминания глав. Следовательно, эти математические формулы CBSE Class 11 по главам подготовлены для обеспечения максимальной подготовки и хороших оценок на любом экзамене.

Глава 1: Наборы

В главе объясняется концепция множеств вместе с их представлением. Обсуждаемые темы: пустые множества, равные множества, подмножества, конечные и бесконечные множества, множества мощности и универсальные множества. Набор — это хорошо согласованная коллекция объектов. Множество, состоящее из определенных элементов, является конечным множеством. В противном случае это бесконечное множество. Ниже перечислены важные термины и свойства, используемые в наборах:

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

Глава 2: Отношения и функции

В главе «Отношения и функции» объясняются передовые концепции теории множеств с использованием концепции упорядоченных и неупорядоченных пар элементов. Упорядоченная пара — это пара элементов, сгруппированных вместе в определенном порядке. Отношение R к множеству A и множеству B может быть описано как подмножество декартова произведения A × B, которое получается путем описания отношения между первым из его элементов x и вторым из его элементов y, заданных в упорядоченных парах из А × В.

The below-mentioned properties will surely assist students to solve various maths problems:
Relations: 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.
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, Domainof R = Range of R^-1 and Rangeof R = Domain of R^-1.
Functions: 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.
Acartesian 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.
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

Глава 3: Тригонометрические функции

В математике тригонометрические функции — это действительные функции, которые относятся к углу прямоугольного треугольника, образуя некоторые конечные отношения длин двух сторон. Найдите ниже важные математические формулы для 11-го класса, связанные с тригонометрическими функциями.

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
sin² θ + cos² θ = 1 ⇒ sin² θ = 1 – cos² θ ⇒ cos² θ = 1 – sin² θ
cosec² θ – cot² θ = 1 ⇒ cosec² θ = 1 + cot² θ ⇒ cot² θ = cosec² θ – 1
sec² θ – tan² θ = 1 ⇒ sec² θ = 1 + tan² θ ⇒ tan² θ = sec² θ – 1
Product 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]

Глава 4: Принцип математической индукции

Как следует из названия, в главе объясняется концепция Принципа математической индукции. Обсуждаемые темы - это процесс доказательства индукции и мотивация приложения, использующего натуральные числа как наименее индуктивное подмножество действительных чисел. Одной из ключевых основ математического мышления является дедуктивное рассуждение. В отличие от дедукции, индуктивное рассуждение зависит от работы с различными случаями и выработки гипотезы путем наблюдения за случаями до тех пор, пока мы не наблюдаем каждый случай. Таким образом, простым языком мы можем сказать, что слово «индукция» означает обобщение частных случаев или фактов.

Below mentioned is the list of some important terms and steps used in the chapter mentioned above:
Statement: A sentence is called a statement if it is either true or false.
Motivation: Motivation is tending to initiate an action. Here Basis step motivate us for mathematical induction.
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. 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.
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.

Глава 5: Комплексные числа и квадратные уравнения

Поэтому, как следует из названия главы, в этой главе объясняется концепция комплексных чисел и квадратных уравнений, а также их свойства. Обсуждаемые темы: квадратный корень, алгебраические свойства, плоское и полярное представление комплексных чисел, решения квадратных уравнений в комплексной системе счисления. Вот несколько важных моментов, связанных с комплексными числами и квадратными уравнениями:

Complex Numbers: A 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 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, i² = -1, i³ = -i, i⁴ = 1
Equality of Complex Number: Two complex numbers z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ are equal, iff x₁ = x₂ and y₁ = y₂ i.e. Re(z₁) = Re(z₂) and Im(z₁) = Im(z₂)
Algebra of Complex Numbers
Addition: Consider z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ are any two complex numbers, then their sum is defined as
z₁ + z₂ = (x₁ + iy₁) + (x₂ + iy₂) = (x₁ + x₂) + i (y₁ + y₂)
Subtraction: Consider z₁ = (x₁ + iy₁) and z₂ = (x₂ + iy₂) are any two complex numbers, then their difference is defined as
z₁ – z₂ = (x₁ + iy₁) – (x₂ + iy₂) = (x₁ – x₂) + i(y₁ – y₂)
Multiplication: Consider z₁ = (x₁ + iy₁) and z₂ = (x₂ + iy₂) be any two complex numbers, then their multiplication is defined as
z₁z₂ = (x₁ + iy₁) (x₂ + iy₂) = (x₁x₂ – y₁y₂) + i (x₁y₂ + x₂y₁)
Division: Consider z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂ 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 + 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| = √x₂+y₂
Argand Plane
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 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.

Глава 6: Линейные неравенства

В главе объясняется концепция линейных неравенств. Обсуждаемые темы - алгебраические решения и графическое представление линейных неравенств с одной и двумя переменными соответственно. В математике неравенство — это отношение, которое имеет место между двумя значениями, когда они различны. Решение линейных неравенств очень похоже на решение линейных уравнений, за исключением одной небольшой, но важной детали: вы переворачиваете знак неравенства всякий раз, когда умножаете или делите неравенство на отрицательный.

Inequation: An inequation or inequality is a statement involving variables and the sign of inequality like >, <, ≥ or ≤.
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 Variable and its Graphical Representation
Using 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:
Rule 1: Don’t change the sign of an inequality by adding or subtracting the same integer on both sides of an equation.
Rule 2: Add or subtract the same positive integer from both sides of an inequality equation.

Глава 7: Перестановки и комбинации

В настоящей главе разъясняются понятия перестановки (расположение ряда объектов в определенном порядке) и комбинации (совокупность объектов независимо от порядка). Обсуждаемые темы - это фундаментальный принцип подсчета, факториала, перестановок, комбинаций и их приложений, а также концепция ограниченной перестановки. Если определенное событие происходит «m» разными способами, за которым следует событие, которое происходит «n» разными способами, то общее количество событий может быть задано в порядке m × n. Найдите важные математические формулы для класса 11 перестановок и комбинаций, как показано ниже:

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 × 1 and 0! = 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 ⁿP_r is given by
ⁿP_r = n! / (n−r)!
The number of permutation of n objects of which p₁ are of one kind, p₂ are of second kind,… pk are of kth kind such that p₁ + p₂ + p₃ + … + p_k = n is
n! / p₁! p₂! p₃! ….. 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 ⁿC_r, and it is given by
ⁿC_r = n! / r!(n−r)!
Relation Between Permutation and combination: The relationship between the two concepts is given by two theorems as,
ⁿP_r = ⁿC_rr! when 0 < r ≤ n.
ⁿC_r + ⁿC_r-1= ⁿ⁺¹C_r

Глава 8: Биномиальная теорема

В этой главе обсуждается биномиальная теорема для положительных целых чисел, используемая для решения сложных вычислений. Обсуждаемые темы - история, формулировка и доказательство биномиальной теоремы и ее расширения вместе с треугольником Паскаля. Биномиальная теорема помогает расширить бином, заданный для любого положительного целого числа n.

Binomial Theorem: The expansion of a binomial for any positive integer n is given by Binomial Theorem, which is
(a + b)ⁿ = ⁿC₀ a_n + ⁿC₁ a_n-1 b + ⁿC2 a_n-2 b₂ + … + ⁿC_n-1 a b_n-1 + ⁿC_n b_n
Some special cases from the binomial theorem can be written as:
(x – y)ⁿ = ⁿC₀ x_n – ⁿC₁ x_n-1 y + ⁿC₂ x_n-2 y₂ + … + (-1)ⁿ ⁿC_n x_n
(1 – x)ⁿ = ⁿC₀ – ⁿC₁ x + ⁿC₂ x² – …. (-1)ⁿ ⁿC_n x_n
ⁿC₀ = ⁿC_n = 1
Pascal’s triangle: The coefficients of the expansions are arranged in an array called Pascal’s triangle.
General Term of following expansions are:
(a + b)ⁿ is T_r+1= ⁿC_r a^n−r.b^r
(a – b)ⁿ is (-1)^r ⁿC_r a^n−r.b^r
(1 + x)_n = ⁿC_r . x_r
(1 – x)_n = (-1)^r ⁿC_n x^r
Middle Terms: In the expansion (a + b)ⁿ, if n is even, then the middle term is the (n/2 + 1)^th term. If n is odd, then the middle terms are (n/2 + 1)^th and ((n+1)/2+1)^th terms.

Глава 9: Последовательности и серии

В главе «Последовательности и серии» обсуждаются концепции последовательности (упорядоченный список чисел) и серии (сумма всех членов последовательности). Обсуждаемые темы: последовательность и ряды, арифметическая и геометрическая прогрессия, арифметика и среднее геометрическое. Вот список некоторых важных терминов, используемых в Sequence и Series, как указано ниже:

Progression: A sequence whose terms follow certain patterns is known as 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: a₁ = a,
The second term: a₂ = a + d,
The third term: a₃ = 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 (a₁+ a_n)
A sequence is an AP If the sum of n terms is of the form An² + 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 a₁, a₂, a₃,……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 b² = 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,G₁, G₂, G₃,….. G_n, b are in GP then G₁, G₂, G₃,……G_n are in GM’s between a and b, then
common ratio is:
If a₁, a₂, a₃,…, a_n are n numbers are non-zero and non-negative, then their GM is given by
GM = (a₁ . a₂ . a₃ …a_n)^1/n
Product of n GM is G₁ × G₂ × G₃ ×… × 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: Σn² = 1² + 2² + 3² + … + n² = n(n+1)(2n+1)/6
Sum of cubes of first n natural numbers is: Σn³ = 1³ + 2³ + 3³ + .. + n³ = (n(n+1)(2n+1)/6)²

Глава 10: Прямые линии

Прямые линии определяли понятие линии, ее угол, наклон и общее уравнение. Обсуждаемые темы: наклон линии, угол между двумя линиями, различные формы уравнений линии, общее уравнение линии и семейство линий соответственно. Вот несколько важных формул, используемых в главе «Прямые линии»:

Distance Formula: The distance between two points A(x₁, y₁) and B (x₂, y₂) is given by,
The distance of a point A(x, y) from the origin 0 (0, 0) is given by OA = √(x² + y²).
Section Formula: The coordinates of the point which divides the joint of (x₁, y₁) and (x₂, y₂) in the ratio m:n internally, is
And externally is:
Mid-Point of the joint of (x₁, y₁) and (x₂, y₂) is: .
X-axis divides the line segment joining (x₁, y₁) and (x₂, y₂) in the ratio -y₁ : y₂.
Y-axis divides the line segment joining (x₁, y₁) and (x₂, y₂) in the ratio -x₁ : x₂.
Coordinates of Centroid of a Triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is
Area of Triangle: The area of the triangle, the coordinates of whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃) is,
If the points (x₁, y₁), (x₂, y₂) and (x₃, y₃) are collinear, then x₁ (y₂ – y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂) = 0.
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
The slope of a line passing through points P(x₁, y₁) and Q(x₂, y₂) is given by,
Angle between Two Lines: The angle θ between two lines having slope m₁ and m₂ is,
If two lines are parallel, their slopes are equal i.e. m₁ = m₂.
If two lines are perpendicular to each other, then their product of slopes is -1 i.e. m₁m₂ = -1.
Point of intersection of two lines: Let equation of lines be ax₁ + by₁ + c₁ = 0 and a₂x + b₂y + c₂ = 0, then their point of intersection is
Distance of a Point from a Line: The perpendicular distanced of a point P(x₁, y₁)from the line Ax + By + C = 0 is given by,
Distance Between Two Parallel Lines: The distance d between two parallel lines y = mx + c₁ and y = mx + c₂ is given by,
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.
General Equation of a Line: Any equation of the form Ax + By + C = 0, where A and B are simultaneously not zero is called the general equation of a line
Normal 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 (x₁, y₁) and has the slope of m is given by y – y₁ = m (x – x₁).
Two points form: The equation of a line passing through the points (x₁, y₁) and (x₂, y₂) is given by

Глава 11: Конические сечения

Темы, обсуждаемые в главе «Конические сечения», — это сечения конуса, вырожденный случай аконического сечения по уравнениям и свойства конических сечений. Окружность – это геометрическая фигура, все точки которой расположены на равном расстоянии от фиксированной точки данной плоскости. Ниже приведен список некоторых важных формул, обсуждаемых в главе «Конические сечения».

Equation of a circle with radius r having a centre (h, k) is given by (x – h)² + (y – k)² = r².
The general equation of the circle is given by x² + y² + 2gx + 2fy + c = 0 , where, g, f and c are constants.
The centre of the circle is (-g, -f).
The radius of the circle is r = √(g²+ f²− c)
The parametric equation of the circle x² + y² = r² are given by x = r cos θ, y = r sin θ, where θ is the parameter.
And the parametric equation of the circle (x – h)² + (y – k)² = r² are given by x = 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
y²= 4ax
y² = -4ax
x² = 4ay
x² = -4ay
Axis of parabola
y = 0
y = 0
x = 0
x = 0
Directrix of parabola
x = -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 rectum
4a
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
x²/a²+ y²/b²= 1, a > b
x²/b²+ y²/a²= 1, a > b
Equation of Major Axis
y = 0
x = 0
Length of Major Axis
2a
2a
Equation of Minor Axis
x = 0
y = 0
Length of Minor Axis
2b
2b
Equation of Directrix
x = ±a/e
y = ±a/e
Vertex
(±a, 0)
(0, ±a)
Focus
(±ae, 0)
(0, ±ae)
Length of latus rectum
2b²/a
2b²/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
x²/a²– y²/b²= 1
x²/a²– y²/b²= 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 axis
2b
2b
Length of Transverse axis
2a
2a
Equation of Conjugate axis
x = 0
y = 0
Equation of Transverse axis
y = 0
x = 0
Equation of Directrix
x = ±a/e
y = ±a/e
Eccentricity (e)
√(a²+b²)/a²
√(a²+b²)/a²
Length of latus rectum
2b²/a
2b²/a

Глава 12: Введение в трехмерную геометрию

Как следует из названия, в главе объясняются концепции геометрии в трехмерном пространстве. Обсуждаемые темы - координатные оси и плоскости соответственно, координаты точек, расстояние и сечение для точек. Три плоскости, определяемые парой осей, известны как координатные плоскости с плоскостями XY, YZ и ZX. Найдите ниже важные математические формулы для 11 класса «Введение в трехмерную геометрию»:

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)
Distance Formula: The distance between two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is given by,
While, the distance between two points A(x, y, z) from the origin O(0, 0, 0) is given by,
Section Formula: The coordinates of the point R which divides the line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally or externally in the ratio m : n are given by,
Mid-Point of the joint of (x₁, y₁) and (x₂, y₂) is:
Coordinates of Centroid of a Triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is

Глава 13: Пределы и производные

В главе объясняется концепция исчисления, которая занимается изучением изменения значения функции, когда изменение происходит в точках домена. Обсуждаемые темы - определение и алгебраические операции пределов и производных соответственно.

Предел функции в некоторой точке имеет общее значение как левого, так и правого пределов, если они совпадают друг с другом. Вот список некоторых важных формул, используемых для решения задач на пределы и производные, как,

Limit: Let 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 .
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 o a}f(x) and lim_{x o a}g(x) exists, then:
Some Standard Limits are 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:
Some Standard Derivatives are given as:

Глава 14: Математические рассуждения

Как следует из названия, в главе объясняются концепции математического мышления (важный навык для анализа любой данной гипотезы в контексте математики). Обсуждаемые темы - утверждения, индуктивное рассуждение и дедуктивное рассуждение. Ниже приведен список важных терминов, обсуждаемых как:

Statements: A 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: Negation of a statement p: If p denote a statement, then the negation of p is denoted by ∼p.
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
Contrapositive: The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒∼p.
Contradiction: If to check whether p is true we assume negation p is true.
Validating statements: 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:
direct method
contrapositive method
method of contradiction
using a counterexample.

Глава 15: Статистика

В этой главе объясняются концепции статистики (данные, собранные для определенных целей), дисперсии и методов расчета для негруппированных и сгруппированных данных. Обсуждаемые темы: диапазон, среднее отклонение, дисперсия и стандартное отклонение, а также анализ частотных распределений. Здесь можно найти основные математические формулы для класса 11 статистики, приведенные ниже:

Measure of Dispersion: The 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.
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 x₁, x₂, x₃,…, 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 x₁, x₂, ……x_n be n observations with x¯ as the mean, then the variance denoted by σ², is given by
Standard deviation: If σ² is the variance, then σ is called the standard deviation is given by
Standard 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 as
Coefficient of variation = (Standard deviation / Mean) × 100

Глава 16: Вероятность

Вероятность является фундаментальной частью учебной программы по математике в 11 классе и важна для экзаменов в 11 классе и различных конкурсных экзаменов, таких как JEE и NEET. На предыдущих занятиях учащиеся, возможно, усвоили основное понятие вероятности как меры неопределенности различных явлений. Здесь обсуждается список формул, важных свойств, приложений и резюме главы, что поможет учащимся быстро изучить связанные понятия и получить хорошие оценки на экзамене.

В главе рассматривается понятие вероятности (мера неопределенности различных явлений или шанс наступления события). Обсуждаемые темы: случайные эксперименты, результаты, выборочные пространства, события и их типы. Ниже приведен список важных формул и краткое изложение глав для класса 11 вероятности:

Probability= Number of Favourable outcomes in an Event / Total number of Outcomes
Event: An 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: P(E) ≥ 0 and P(S) = 1
Mutually 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 P₁, P₂, . . . . . , P_n are exhaustive and mutually exclusive if P₁ ∪ P₂ ∪ . . . . . ∪ P_n = S and E_i ∩ E_j = φ for all i ≠ j.

Different forms of parabola	y²= 4ax	y² = -4ax	x² = 4ay	x² = -4ay
Axis of parabola	y = 0	y = 0	x = 0	x = 0
Directrix of parabola	x = -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 rectum	4a	4a	4a	4a
Focal length	\|x + a\|	\|x – a\|	\|y + a\|	\|y – a\|

Different forms of Ellipse	x²/a²+ y²/b²= 1, a > b	x²/b²+ y²/a²= 1, a > b
Equation of Major Axis	y = 0	x = 0
Length of Major Axis	2a	2a
Equation of Minor Axis	x = 0	y = 0
Length of Minor Axis	2b	2b
Equation of Directrix	x = ±a/e	y = ±a/e
Vertex	(±a, 0)	(0, ±a)
Focus	(±ae, 0)	(0, ±ae)
Length of latus rectum	2b²/a	2b²/a

Different forms of Hyperbola	x²/a²– y²/b²= 1	x²/a²– y²/b²= 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 axis	2b	2b
Length of Transverse axis	2a	2a
Equation of Conjugate axis	x = 0	y = 0
Equation of Transverse axis	y = 0	x = 0
Equation of Directrix	x = ±a/e	y = ±a/e
Eccentricity (e)	√(a²+b²)/a²	√(a²+b²)/a²
Length of latus rectum	2b²/a	2b²/a