Class 12 Maths Formulas

Last Updated : 04 Apr, 2024

Class 12 maths formulas page is designed for the convenience of the learners so that one can understand all the important concepts of Class 12 Mathematics directly and easily. Math formulae for Class 12 are for the students who find mathematics to be a nightmare and difficult to grasp. They may become hesitant and lose interest in studies as a result of this. We have included all of the key formulae for the 12th standard Maths topic, which students may simply recall, to assist them in understanding Maths in a straightforward manner. For all courses such as Integration, Differentiation, Trignometry, Relation and Functions, and so on, the formulae are provided here according to the NCERT curriculum.

Table of Content

Chapter 1: Relations and Functions
Chapter 2: Inverse Trigonometric Functions
Chapter 3: Matrices
Chapter 4: Determinants
Chapter 5: Continuity and Differentiability
Chapter 6: Applications of Derivatives
Chapter 7: Integrals
Chapter 8: Applications of Integrals
Chapter 9: Differential Equations
Chapter 10: Vector Algebra
Chapter 11: Three-Dimensional Geometry
Chapter 12: Linear Programming
Chapter 13: Probability

Chapter 1: Relations and Functions

The Chapter Relation and Functions discusses the introduction of relations and functions, types of relation, types of functions, the composition of functions and invertible function, binary operations, inverse function, and miscellaneous examples.

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, Domain of R = Range of R^-1 and Range of R = Domain of R^-1.

Functions: A relation f from a set A to set B is said to be a function if every element of set A has one and only image in set B.
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 × ϕ = ϕ
- 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

Learn More, Domain and Range of a Function

Chapter 2: Inverse Trigonometric Functions

Inverse Trigonometric Functions of NCERT Class 12 Maths, gives an account of various topics such as the remarks based on basic concepts of inverse trigonometric functions, properties of inverse trigonometric functions, and miscellaneous examples. Inverse Trigonometric Functions are quite useful in Calculus to define different integrals. You can also check the Trigonometric Formulas here.

Inverse trigonometric functions: Inverse trigonometric functions map real numbers back to angles. e.g. Inverse of sine function denoted by sin^-1 or arc sin(x) is defined on [-1,1].

Some of the important useful properties of Inverse Trigonometric Functions are:

Functions	Domain	Range
y = sin^-1 x	[–1, 1]	[−π/2,π/2]
y = cos^-1 x	[–1, 1]	[0,π]
y = cosec^-1 x	R – (–1, 1)	[−π/2, π/2] – {0}
y = sec^-1 x	R – (–1, 1)	[0,π] – {π/2, π/2}
y = tan^-1 x	R	(−π/2, π/2)
y = cot^-1 x	R	(0,π)

Self-Adjusting Trigonometric Property

Self Adjusting Inverse Trigonometric Properties are,

sin(sin^-1 x)=x
sin^-1 (sin x) = x
cos(cos^-1 x) = x
cos^-1 (cos x) = x
tan(tan^-1 x) = x
tan^-1 (tan x) = x
sec(sec^-1x) = x
sec^-1(sec x) = x
cosec^-1(cosec x) = x
cosec(cosec^-1 x) = x
cot^-1(cot x) = x
cot(cot^-1x) = x

Reciprocal Relations

Reciprocal Relations of the Inverse Trigonometric Relations are,

sin^-1 (1/x) = cosec^-1 x, x ≥ 1 or x ≤ -1
cos^-1 (1/x) = sec^-1 x, x ≥ 1 or x ≤ -1
tan^-1 (1/x) = cot^-1 x, x > 0

Even and Odd Functions

Even and Odd Functions of the Inverse Trigonometry Functions are,

sin^-1 (-x) = -sin^-1 (x), x ∈ [-1, 1]
tan^-1 (-x) = -tan^-1 (x), x ∈ R
cosec^-1 (-x) = -cosec^-1 (x), |x| ≥1
cos^-1 (-x) = π – cos^-1 (x), x ∈ [-1, 1]
sec^-1 (-x) = π – sec^-1 (x), |x| ≥1
cot^-1 (-x) = π – cot^-1 (x), x ∈ R

Complementary Relations

The complementary relation of the Inverse trigonometry functions is,

sin^-1 x + cos^-1 x = π/2
tan^-1 x + cot^-1 x = π/2
cosec^-1 x + sec^-1 x = π/2

Sum and Difference Formulae

The sum and difference formulas are the important formulas used in inverse trigonometric functions, some of the important inverse trigonometric sum and difference formulas are,

tan^-1 x + tan^-1 y = tan^-1 {(x+y)/(1−xy)}
tan^-1 x – tan^-1 y = tan^-1 {(x-y)/(1+xy)}
sin^-1 x + sin^-1 y = sin^-1 [x√(1-y²)+y√(1-x²)]
sin^-1 x – sin^-1 y = sin^-1 [x√(1-y²)-y√(1-x²)]
cos^-1 x + cos^-1 y = cos^-1 [xy-√(1-x²)√(1-y²)]
cos^-1 x – cos^-1 y = cos^-1 [xy+√(1-x²)√(1-y²)]
cot^-1 x + cot^-1 y = cot^-1 [(xy-1)/(x+y)]
cot^-1 x + cot^-1 y = cot^-1 [(xy+1)/(y-x)]

Double Angle Formula

The double angle formula for the inverse trigonometric functions is,

2tan^-1 x = sin^-1 (2x/1+x²)
2tan^-1 x = cos^-1 (1-x²/1+x²)
2tan^-1 x = tan^-1 (2x/1-x²)
2sin^-1 x = sin^-1 (2x√(1+x²))
2cos^-1 x = sin^-1 (2x√(1-x²))

Conversion Properties

The conversion properties for the inverse trigonometric function are,

sin^-1 x = cos^-1 √(1-x²) = tan^-1 {x/x√(1-x²)} = cot^-1 {√(1-x²)/x}
cos^-1 x = sin^-1 √(1-x²) = tan^-1 {√(1-x²)/x} = cot^-1 {x/√(1-x²)}
tan^-1 x = sin^-1 {x/√(1-x²)} = cos^-1 {x/√(1+x²)} = sec^-1 √(1+x²) = cosec^-1 {√(1+x²)/x}

Chapter 3: Matrices

In Chapter 3 of NCERT textbook, we shall see the definition of a matrix, types of matrices, equality of matrices, operations on matrices such as the addition of matrices and multiplication of a matrix by a scalar, properties of matrix addition, properties of scalar multiplication, multiplication of matrices, properties of multiplication of matrices, transpose of a matrix, properties of the transpose of the matrix, symmetric and skew-symmetric matrices, elementary operation or transformation of a matrix, the inverse of a matrix by elementary operations and miscellaneous examples.

Matrix: A matrix is said to have an ordered rectangular array of functions or numbers. A matrix of order m × n consists of m rows and n columns.
Order of Matrix: If a matrix has m rows and n columns, then its order is written as m × n. If a matrix has order m × n, then it has mn elements
- Square Matrix: An m × n matrix will be known as a square matrix when m = n.
- Diagonal Matrix: A = [a_ij]_{m × m} will be known as a diagonal matrix if a_ij = 0, when i ≠ j.
- Scalar Matrix: A = [a_ij]_{n × n} is a scalar matrix if a_ij = 0, when i ≠ j, a_ij = k, (where k is some constant); and i = j.
- Identity Matrix: A = [a_ij]_{n × n} is an identity matrix, if a_ij = 1, when i = j and a_ij = 0, when i ≠ j.
- Zero Matrix: A zero matrix will contain all its elements as zero.
- Column Matrix: A matrix which is of the form [A]_n×1 is called the column matrix.
- Row Matrix: A matrix that is of the form [A]_1×n is called the row matrix.
A = [a_ij] = [b_ij] = B if and only if:
- A and B are of the same order
- a_ij = b_ij for all the certain values of i and j
Operations on Matrices: Between two or more two matrices, the following operations are defined below:
- Addition of Matrix: If A = [a_ij]_m×n and B = [y_ij]_m×n, then A + B = [a_ij +b_ij]_m×n, 1 ≤ i ≤ m, 1 ≤ j ≤ n
- Subtraction of Matrix: If A = [a_ij]_m×n and B = [b_ij]_m×n, then A – B = [a_ij – b_ij]_m×n, 1 ≤ i ≤ m, 1 ≤ j ≤ n
- Multiplication of a matrix by scalar number: Let A = [a_ij]_m×n be a matrix and k is scalar, then kA is another matrix obtained by multiplying each element of A by the scalar k, i.e. if A = [a_ij]_m×n, then kA = [ka_ij]_m×n
- Multiplication of Matrices: Let A and B be two matrices. Then, their product AB is defined, if the number of columns in matrix A is equal to the number of rows in matrix B.

Chapter 4: Determinants

Chapter 4 of 12 Class NCERT Maths Solution discusses the topic of determinants. Students will get to learn about the definition and meaning of determinants, remarks based on the order of determinants, properties of determinants, finding the area of a triangle using determinants, minors and cofactors of determinants, adjoint of a matrix, the inverse of a matrix, applications of determinants, and matrices and miscellaneous examples. Below we have links provided to each exercise solution covered in this chapter. The determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, co-factors, and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency, and the number of solutions of a system of linear equations by examples, solving a system of linear equations in two or three variables (having unique solution) using the inverse of a matrix.

Determinant of a Matrix

The determinant is the numerical value of the square matrix. So, to every square matrix A = [a_ij] of order n, we can associate a number (real or complex) called the determinant of the square matrix A. It is denoted by det A or |A|.

e.g. The determinant of a matrix A = [a₁₁]_{1 × 1} can be given as: |a₁₁| = a₁₁.

Before finding the Determinant of a Matrix we must learn the Minor and Cofactor of a Matrix.

Minor

Minor of an element ay of a determinant is a determinant obtained by deleting the ith row and j^thcolumn in which element ay lies. Minor of an element a_ij is denoted by M_ij.

Note: Minor of an element of a determinant of order n(n ≥ 2) is a determinant of order (n – 1).

Cofactor

Cofactor of an element a_ij of a determinant, denoted by Aij or Cij is defined as A_ij = (-1)^i+jM_ij, where M_ijis a minor of an element a_ij.

Value of a Determinant

Value of determinant of a matrix of order 2,

A = [Tex]\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix} [/Tex]

[Tex]|A|=\begin{vmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{vmatrix} [/Tex]

[Tex]\Rightarrow |A|=a_{11}\cdot a_{22}-a_{21}\cdot a_{12} [/Tex]

Value of determinant of a matrix of order 3,

[Tex]A = \begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix} [/Tex]

[Tex]|A|=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix}=a_{11}\cdot \begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix}-a_{12}\cdot \begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix}+ a_{13}\cdot \begin{vmatrix}a_{22}&a_{23}\\a_{31}&a_{32}\end{vmatrix} [/Tex]

Singular and Non-Singular Matrix

If the value of the determinant corresponding to a square matrix is zero, then the matrix is said to be a singular matrix, otherwise it is a non-singular matrix, i.e. for a square matrix A, if |A| ≠ 0, then it is said to be a non-singular matrix and of |A| = 0, then it is said to be a singular matrix.

Determinant Theorems

If A and B are non-singular matrices of the same order, then AB and BA are also non-singular matrices of the same order.
The determinant of the product of matrices is equal to the product of their respective determinants, i.e. |AB| = |A||B|, where A and B are a square matrix of the same order.

Adjoint of a Matrix

The adjoint of a square matrix ‘A’ is the transpose of the matrix which is obtained by cofactors of each element of a determinant corresponding to that given matrix. It is denoted by adj(A).

In general, the adjoint of a matrix A = [a_ij]_n×n is a matrix [A_ji]_n×n, where A_ji is a cofactor of element a_ji.

Properties of Adjoint of a Matrix

If A is a square matrix of order n × n, then

A(adj A) = (adj A)A = |A| I_n
|adj A| = |A|^n-1
adj (A^T) = (adj A)^T

The area of a triangle whose vertices are (x₁, y₁), (x₂, y₂) and (x₃, y₃) is given by:

[Tex]\Delta=\dfrac{1}{2}\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix} [/Tex]

Inverse of a Square Matrix

Let A be a non-singular matrix such that |A| ≠ 0 then the inverse of the matrix is defined as

[Tex]A^{-1}=\dfrac{1}{|A|}adj(A) [/Tex]

Properties of an Inverse Matrix

(A^-1)^-1 = A
(A^T)^-1=(A^-1)^T
(AB)^-1 = B^-1A^-1
(ABC)^-1 =C^-1B^-1A^-1
adj (A^-1) = (adj A)^-1

Solution of System of Linear Equations using Inverse of a Matrix

Let the given system of equations be a₁x + b₁y + c₁z = d₁; a₂x + b₂y + c₂z = d₂ and a₃x + b₃y + c₃z = d₃.

Write the following system of linear equations in matrix form as

AX = B

where,

[Tex]A=\begin{bmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{bmatrix} [/Tex]
[Tex]X=\begin{bmatrix}x\\y\\z\end{bmatrix} [/Tex]
[Tex]B=\begin{bmatrix}d_1\\d_2\\d_3\end{bmatrix} [/Tex].

Case I: If |A| ≠ 0, then the system is consistent and has a unique solution which is given by X = A^-1B.

Case II: If |A| = 0 and (adj A) B ≠ 0, then the system is inconsistent and has no solution.

Case III: If |A| = 0 and (adj A) B = 0, then the system may be either consistent or inconsistent according to as the system has either infinitely many solutions or no solutions

Chapter 5: Continuity and Differentiability

Chapter 5 of the NCERT textbook starts with the definition of continuity. Students go on to learn about continuity, the algebra of continuous functions, the definition and meaning of differentiability, derivatives of composite functions, the derivative of implicit functions, the derivative of inverse trigonometric functions, derivative of exponential and logarithmic functions, logarithmic differentiation, derivatives of functions in parametric forms, second-order derivatives, mean value theorem via miscellaneous examples. Here, students can find the exercises explaining these concepts properly with solutions. Continuity and differentiability, the derivative of composite functions, chain rule, the derivative of inverse trigonometric functions, and the derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation is the derivative of functions expressed in parametric forms. Second-order derivatives. Rolle’s and Lagrange’s Mean Value Theorems (without proof) and their geometric interpretation.

Continuity: Continuity of function at a point: Geometrically we say that a function y=f(x) is continuous at x=a if the graph of the function y=f(x) is continuous (without any break) at x = a.

A function f(x) is said to be continuous at a point x = a when:

f(a) exists i.e. f(a) is finite, definite and real.

[Tex]\lim_{x\to a}f(x) [/Tex] exists.

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

A function f(x) is continuous at x = a if

[Tex]\lim_{h\to 0}f(a+h)=\lim_{h\to 0}f(a-h) = f(a) [/Tex]

where h→0 through positive values.

Continuity of a Function in a Closed Interval: A function f(x) is said to be continuous in the closed interval if it is continuous for every value of x lying between a and b continuous to the right of a and to the left of x = b i.e. [Tex]\lim_{x\to a-0}f(x)=f(a)\text{ and }\lim_{x\to b-0}f(x) = f(b) [/Tex]

Continuity of a Function in an Open Interval: A function f(x) is said to be continuous in an open interval (a,b) if it is continuous at every point in (a,b).

Discontinuity (Discontinuous function): A function f(x) is said to be discontinuous in an interval if it is discontinuous even at a single point of the interval.

The sum, difference, product, and quotient of continuous functions are continuous. i.e. if f and g are continuous functions, then

(f ± g) (x) = f(x) ± g(x) is continuous
(f . g) (x) = f(x) . g(x) is continuous
{f/g)(x)=f(x)/g(x) (provided g (x) ≠ 0) is continuous)

Chain Rule: If f = v o u, t = u (x), and if both dt/dx and dv/dx exist, then: df/dx = dv/dt. dt/dx

Addition Rule: (u±v)′ = u′ ± v’

Product Rule: (uv)′ = u′v + uv’

Mean Value Theorem

Mean Value Theorem states that,

If f : [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that,

f′(c) = (f(b)−f(a))/(b−a)

Rolle’s Theorem

Rolle’s Theorem states that,

If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) whereas f(a) = f(b), then there exists some c in (a, b) such that f ′(c) = 0.

Lagrange’s Mean Value Theorem

Lagrange’s Mean Value states that,

If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b). Then there exists some c in (a, b) such that [Tex]f'(c)=\dfrac{f(b)-f(a)}{b-a} [/Tex]

Learn more about, Rolle’s and Lagrange’s Mean Value Theorem

Derivatives of Some Standard Functions

The derivative of some standard functions are,

[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 6: Applications of Derivatives

Chapter 6 of the NCERT textbook, provides a definition of derivatives, rate of change of quantities, increasing and decreasing functions, tangents and normals, approximations, maxima and minima, first derivative test, maximum and minimum values of a function in a closed interval, and miscellaneous examples. Here, students can find the exercises explaining these concepts properly. Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima, and minima (the first derivative test is motivated geometrically, and the second derivative test is given as a provable tool).

Derivative for Rate of Change of a Quantity

Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. By using the application of derivatives we can find the approximate change in one quantity with respect to the change in the other quantity. Assume we have a function y = f(x), which is defined in the interval [a, a+h], then the average rate of change in the function in the given interval is

(f(a + h)-f(a))/h

Now using the definition of the derivative, we can write

f'(a)=limh→0f(a+h)−f(a)hf′(a)=limh→0f(a+h)−f(a)h

which is also the instantaneous rate of change of the function f(x) at a.

Now, for a very small value of h, we can write

f'(a) ≈ {(f(a+h) − f(a)}/h

f(a+h) ≈ {f(a) + f'(a)}/h

Approximation Value

Derivative of a function can be used to find the linear approximation of a function at a given value. The linear approximation method was given by Newton and he suggested finding the value of the function at the given point and then finding the equation of the tangent line to find the approximately close value to the function. The equation of the function of the tangent is

L(x) = f(a) + f'(a)(x−a)

Learn more about, Approximation

Tangent and Normal To a Curve

A tangent is a line to a curve that will only touch the curve at a single point and its slope is equal to the derivative of the curve at that point. The slope(m) of the tangent to a curve of a function y = f(x) at a point (x₁,y₁)(x₁,y₁) is obtained by taking the derivative of the function (m = f'(x) ).

By finding the slope of the tangent line to the curve and using the equation

m = (y₂−y₁)/(x₂−x₁)

we can find the equation of the tangent line to the curve. Similarly, we can find the equation of the normal line to the curve of a function at a point. This normal line will be normal(perpendicular) to the tangent line. Hence the slope of the normal line to a curve of a function y = f(x) at a point (x₁,y₁)(x₂,y₂) is given as follows.

n = -1/m = – 1/ f'(x)

By using the equation

−1/m = (y₂−y₁)/(x₂−x₁)

We can find the equation of the normal line to the curve.

Learn more about, Tangents and Normals

Maxima, Minima, and Point of Inflection

Maxima and minima are the peaks and valleys of a curve, whereas the point of inflection is the part of the curve where the curve changes its nature(from convex to concave or vice versa). We can find the maxima, minima, and point of inflection by using the first-order derivative test. According to this test, we first find the derivative of the function at a given point and equate it to 0, i.e., f'(c) = 0, (here we have found the slope of the curve equal to 0, which means it is a line parallel to the x-axis). Now if the function is defined in the given interval, then we check the value of f'(x) at the points lying to the left of the curve and to the right of the curve and check the nature of the f'(x), then we can say, that the given point is maxima or minima based on the below conditions.

Maxima when the slope or f’(x) changes its sign from +ve to -ve as we move via point c. And f(c) is the maximum value.
Minima when the slope or f’(x) changes its sign from -ve to +ve as we move via point c. And f(c) is the minimum value.
Point C is called the Point of inflection when the sign of slope or sign of the f’(x) doesn’t change as we move via c.

Increasing and Decreasing Functions

The increasing function is a function that seems to reach the top of the x-y plane whereas the decreasing function seems to reach the downside corner of the x-y plane. Let us say we have a function f(x) differentiable within the limits (a, b). Then we check any two points on the curve of the function.

If at any two points x1x1 and x2x2 such that x1x1 < x2x2, there exists a relation f(x1)f(x1) ≤ f(x2)f(x2), then the given function is increasing function in the given interval, and if f(x1)f(x1) < f(x2)f(x2), then the given function is strictly increasing function in the given interval.
And, if at any two points x1x1 and x2x2 such that x1x1 < x2x2, there exists a relation f(x1)f(x1) ≥ f(x2)f(x2), then the given function is decreasing function in the given interval and if f(x1)f(x1) > f(x2)f(x2), then the given function is strictly decreasing function in the given interval

Learn more about, Increasing and Decreasing Function

Chapter 7: Integrals

In this chapter, the methods to determine the function when its derivative is given and the area under a graph of a function are discussed. The basic properties of integrals and the fundamental theorem of calculus are also included in this chapter.

Integration is the inverse process of differentiation. In differential calculus, we are given a function and we have to find the derivative or differential of this function, but in integral calculus, we are to find a function whose differential is given. Thus, integration is a process that is the inverse of differentiation.

Then, ∫f(x) dx = F(x) + C, these integrals are called indefinite integrals or general integrals. C is an arbitrary constant by varying which one gets different anti-derivatives of the given function.

Derivative of a function is unique but a function can have infinite anti-derivatives or integrals.

Properties of Indefinite Integral

Various properties of the integral are,

∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

For any real number k, ∫k f(x) dx = k∫f(x)dx.

In general, if f₁, f₂,………, f_n are functions and k₁, k₂,…, k_n are real numbers, then

∫[k₁f₁(x) + k₂ f₂(x)+…+ k_nf_n(x)] dx = k₁ ∫f₁(x) dx + k₂ ∫ f₂(x) dx+…+ k_n ∫f_n(x) dx

First Fundamental Theorem of Integral Calculus

Let the area function be defined as

A(x) = ∫_a^xf(x)dx for all x ≥ a

where the function f is assumed to be continuous on [a, b]

Then A’ (x) = f (x) for every x ∈ [a, b].

Second Fundamental Theorem of Integral Calculus

Let f be the certain continuous function of x defined on the closed interval [a, b] then,

∫^b_af(x)dx = [F(x) + C]^b_a= F(b)−F(a)

Learn more about, Fundamental Theorem of Calculus

Standard Integrals Formulas

Standard Integral Formulas are,

∫xⁿdx = xⁿ⁺¹/(n+1) + C (where, n ≠ −1)
∫cos x dx = sin x + C
∫sin x dx = −cos x + C
∫sec²x dx = tan x + C
∫cosec²x dx = −cot x + C
∫sec x.tan x dx = sec x + C
∫cosec x.cot x dx = −cosec x + C
∫e^xdx = e^x+ C
∫a^xdx = a^xlog_ea + C
∫1/x dx = log|x| + C

Other Integral Formulas

Other integral formulas that are widely used are,

∫tan x dx = log|sec x| + C
∫cot x dx = log|sin x| + C
∫sec x dx =log|sec x + tan x| + C
∫cosec x dx = log|cosec x − cot x| + C

Chapter 8: Applications of Integrals

This chapter included topics like how to find the area of different geometrical figures such as circles, parabolas, and ellipses. The area enclosed by the curve y = f (x); x-axis and the lines x = a and x = b (b > a) is given by the formula,

Area = ∫^b_ay.dx =∫^b_af(x).dx

Area of the region bounded by the curve x = φ (y) as its y-axis and the lines y = c, y = d is given by the formula:

Area = ∫^d_cx.dy = ∫^d_cϕ(y).dy

The area enclosed in between the two given curves y = f (x), y = g (x), and the lines x = a, x = b is given by the following formula:

Area = ∫^b_a[f(x)−g(x)].dx, {where, f(x) ≥ g(x) in [a,b]}

If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in [c, b], a < c < b, then the resultant area between the curve is given as,

Area=∫^c_a[f(x)−g(x)].dx, + ∫^b_c[g(x)−f(x)].dx

Chapter 9: Differential Equations

In this chapter, the concepts of differential equations and how to find solutions to differential equations are discussed. This topic holds various applications in Physics, Economics, Chemistry, and Biology.

Chapter 10: Vector Algebra

In this chapter, the concepts of vector quantities, how to find the position vector of a point, geometrical interpretation of vectors, and dot and cross product of vectors are discussed. These concepts have great importance in higher education (engineering and technology). The important formulas used in vector algebras are,

For vector a as,

[Tex]\vec a [/Tex] = xi + yj + zk, then the magnitude of the vector is,

I[Tex]\vec{a} [/Tex]I= √(x² + y² + z²)

The vector law used is,

A + B = B + A (Commutative Law)
A + (B + C) = (A + B) + C (Associative Law)
(A • B )= |P| |Q| cos θ ( Dot Product )
(A × B )= |P| |Q| sin θ (Cross Product)
k (A + B )= kA + kB
A + 0 = 0 + A (Additive Identity)

Learn more about, Vector Algebra

Chapter 11: Three-Dimensional Geometry

Based on the vector algebra discussed in the previous chapter, here are the concepts like, how it can be applied to three-dimensional geometry. Also, the introduction to topics like direction cosines and direction ratios, cartesian and vector equations of a line, and how to find the shortest distance between two lines using these concepts are discussed in this part. Various formulas used in 3-D geometry are,

Distance Formula: The distance between two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂) 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 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,

[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]

Mid-Point of the joint of (x₁, y₁) and (x₂, y₂) is,

[Tex]\left(\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2}\right) [/Tex]

Coordinates of Centroid of a Triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is,

[Tex]\left(\dfrac{x_1+x_2+x_3}{3},\,\dfrac{y_1+y_2+y_3}{3}\right) [/Tex]

Incentre of triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is,

[Tex]\left(\dfrac{ax_1+bx_2+cx_3}{a+b+c}, \dfrac{ay_1+by_2+cy_3}{a+b+c}, \dfrac{az_1+bz_2+cz_3}{a+b+c}\right) [/Tex]

Centroid of a Tetrahedron with vertices (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃), and (x₁, y₄, z₄) is,

[Tex]\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}, \dfrac{z_1+z_2+z_3+z_4}{4}\right) [/Tex]

Direction Cosines of a Line: If the directed line OP makes angles α, β, and γ with positive X-axis, Y-axis, and Z-axis respectively, then cos α, cos β, and cos γ, are called direction cosines of a line. They are denoted by l, m, and n. Therefore, l = cos α, m = cos β, and n = cos γ. Also, the sum of squares of direction cosines of a line is always 1, i.e.

l² + m² + n² = 1

cos² α + cos² β + cos² γ = 1

Direction Ratios of a Line: Numbers proportional to the direction cosines of a line, are called direction ratios of a line.

If a, b, and c are direction ratios of a line, then

[Tex]\dfrac{l}{a}~=~\dfrac{m}{b}~=~\dfrac{n}{c} [/Tex]

If a, b, and care direction ratios of a line, then its direction cosines are

[Tex]l~=~\pm \dfrac{a}{\sqrt{a^2+b^2+c^2}}\\m~=~\pm \dfrac{b}{\sqrt{a^2+b^2+c^2}}\\n~=~\pm \dfrac{c}{\sqrt{a^2+b^2+c^2}} [/Tex]

Angle between two line segments: When a₁, b₁, c₁ and a₂, b₂, c₂ are the direction ratios of two lines and θ is the acute angle between them, then:

[Tex]\cos\theta~=~\left|\dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a^2_1+b^2_1+c^2_1}\sqrt{a^2_2+b^2_2+c^2_2}}\right| [/Tex]

Two lines will be perpendicular if their direction ratios have a relation,

a₁a₂+ b₁b₂+ c₁c₂ = 0

Two lines will be parallel if their direction ratios have a relation,

a₁/a₂ = b₁/b₂ = c₁/c₂

Projection of a line segment on a line: When P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) then the projection of PQ on a line having direction cosines l, m, n is

|l(x₂-x₁)+m(y₂-y₁)+n(z₂-z₁)|

Equation of a Plane

General form: ax+by+cz+d = 0, where a, b, c are not all zero, a, b, c, d ∈ R.
Normal form: lx+my+nz = p
Plane through the point (x₁, y₁, z₁): a(x-x₁)+b(y-y₁)+c(z-z₁) = 0
Intercept form: [Tex]\dfrac{x}{a}~+~\dfrac{y}{b}~+~\dfrac{z}{c}~=~1 [/Tex]
Vector form: [Tex] (\vec{r}~-~\vec{a} ).\vec{n}= 0 \text{ or }\vec{r}.\: \vec{n}= \vec{a}.\: \vec{n} [/Tex]

Planes Parallel to Axes

Plane Parallel to X-axis is by + cz + d = 0
Plane Parallel to Y-axis is ax + cz + d = 0
Plane Parallel to Z-axis is ax + by + d = 0

Chapter 12: Linear Programming

This chapter is a continuation of the concepts of linear inequalities and the system of linear equations in two variables studied in the previous class. This chapter helps to learn how these concepts can be applied to solve real-world problems and how to optimize the problems of linear programming so that one can maximize resource utilization, minimize profits, etc.

Linear Programming Problem: A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called objective function) subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are non-negative.

Feasible Region: The common region determined by all the constraints including the non-negative constraints, x ≥ 0, y ≥ 0 of a linear programming problem is called the feasible region (or solution region) for the problem.

Infeasible Solution: Points within and on the boundary of the feasible region represent feasible solutions to the constraints. Any point outside the feasible region is an infeasible solution.

Optimal Solution: Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.

The following Theorems are fundamental in solving linear programming problems:

Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by, be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
Let R be the feasible region for a linear programming problem, and let be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R, and each of these occurs at a corner point (vertex) of R. If the feasible region is unbounded. A maximum or a minimum may not exist. However, if it exists, it must occur at a corner point of R.
Corner point method: For solving a linear programming problem. The method comprises the following steps:
- Find the feasible region of the linear programming problem and determine its corner points (vertices).
- Evaluate the objective function: Z = ax + by, at each corner point. Let M and m respectively be the largest and smallest values at these points.
- If the feasible region is bounded, M and m respectively are the maximum and minimum values of the objective function.

If the feasible region is unbounded, then,

M is the maximum value of the objective function, if the open half-plane determined by ax + by > M, has no point in common with the feasible region. Otherwise, the objective function has no maximum value.
m is the minimum value of the objective function if the open half-plane determined by ax + by < M, has no point in common with the feasible region. Otherwise, the objective function has no minimum value.

If two corner points of the feasible region are both optimal solutions of the same type, i.e., both produce the same maximum or minimum, then any point on the line segment joining these two points is also an optimal solution of the same type.

Learn more about, Graphical Solutions to Linear Programming Problems

Chapter 13: Probability

This chapter deals with probability, the concept of probability is also studied in earlier classes. This chapter in the present class helps to learn about conditional probability. Further, topics like Bayes’ theorem, independence of events, the probability distribution of random variables, mean and variance of a probability distribution, and Binomial distribution are discussed in this chapter.

Conditional Probability: The possibility of an event or outcome occurring dependent on the occurrence of a preceding event or outcome is known as conditional probability. It simply depends on any previous occurrence that has already occurred. Consider two events A and B with the same sample space of a random experiment, then the conditional probability of the event A gives that B has occurred, i.e. P(A|B) is,

P(A|B) = P(A ∩ B)/P(B)

(for P(B) ≠ 0).

When E and F are the events of a sample space S of an experiment: P(S|F) = P(F|F) = 1
When A and B are any two events in a sample space S and F an event of S, such that P(F)≠0: P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)
P(E′|F) = 1 − P(E|F)

Multiplication Rule: Consider two events such as, E and F from a sample space S. Here, the set E ∩ F denotes the event that both E and F have occurred. Or we can say, E ∩ F represents the simultaneous occurrence of the events E and F. The event E ∩ F is also written as EF. According to this rule, if E and F are the events in a sample space, then;

P(E ∩ F) = P(E) P(F|E) = P(F) P(E|F)

where P(E) ≠ 0 and P(F) ≠ 0.

Similarly, for three events E, F, and G from a sample S:

P(E ∩ F ∩ G) = P(E) P(F|E) P(G|(E ∩ F)) = P(E) P(F|E) P(G|EF)

Independent Events: Two experiments are said to be independent if the probability of the events E and F occurring simultaneously when the two experiments are performed is the product of P(E) and P(F) calculated separately on the basis of two experiments, i.e., for every pair of events E and F, where E is associated with the first experiment and F with the second experiment.

P (E ∩ F) = P (E).P(F)

Baye’s Theorem: A set of events E₁, E₂, …, E_n is said to denote a partition of the sample space S when,

E_i ∩ E_j = φ, i ≠ j, i, j = 1, 2, 3, …, n
E₁ ∪ Ε₂ ∪ … ∪ E_n = S and
P(E_i )> 0 for all i = 1, 2, …, n.

Also, the events E₁, E₂, …, E_n denotes a partition of the sample space S if they are pairwise disjoint, exhaustive, and have nonzero probabilities, and A be any event with non-zero probability, so:

[Tex]P(E_i|A)=\dfrac{P(E_i)P(A|E_i)}{\sum_{j=1}^{n}P(E_j)P(A|E_j)} [/Tex]

Theorem of Total Probability: Let E₁, E₂, …., E_n be the partition of a sample space and A be any event; then,

P(A) = P(E₁) P (A|E₁) + P (E₂) P (A|E₂) + … + P (E_n) . P(A|E_n)

Random Variables and their Probability Distributions: A random variable is a real-valued function whose domain is a random experiment’s sample space. The probability distribution of a random variable X is the system of numbers:

X: x₁ x₂ … x_n and P(X): p₁ p₂ … p_nwhere p_i > 0,

[Tex]\sum_{i=1}^{n}p_i=1 [/Tex]

where i = 1, 2, 3, … , n.

The real numbers x₁, x₂, …, x_n are the possible values of the random variable X, and p_i (i = 1, 2, …, n) is the probability of the random variable X taking the value xi i.e.

P(X = x_i ) = p_i

The mean, variance and standard deviation of a random variable X can be written as:

Mean: [Tex]E(x)=\mu=\sum^{n}_{i+1}x_ip_i [/Tex]
Variance: [Tex]\sigma^2_x=Var(X)=\sum^{n}_{i=1}(x_i-\mu)^2p(x_i)=E(X-\mu)^2 [/Tex]
Standard Deviation: [Tex]\sigma_x=\sqrt{Var(X)}=\sqrt{\sum^{n}_{i=1}(x_i-\mu)^2p(x_i)} [/Tex]

Bernoulli Trials and Binomial Distribution: Trials of a random experiment are called Bernoulli trials if they satisfy the following conditions:

There should be a finite number of trials.
The trials should be independent.
Each trial has exactly two outcomes: success or failure.
The probability of success remains the same in each trial.

P (X = x) = P(x) = ⁿC_x q^n-x p^x

where,

x = 0, 1, …, n,
n is the Total number of trials
p is the probability of success
q = 1 – p