All Maths Formulas for Class 12

All Formulas for Maths Class 12

If you are searching for all the formulas of Maths Class 12 at one place, then you have come to the right place. Maths-formula brings you all the important concepts, theories and Maths formulas for class 12 to help you in your preparation for mathematics class 12 board examination. These concepts, theories and formulas are extremely important from the examinations’ point of view.

Maths is a subject where reasoning and logic are very important. Students should have a clear understanding of the underlying theories, concepts and formulas. Only then, you will be able to Solve the NCERT Mathematics Questions asked in the board examination. Here, we are providing all the concepts and theories of those chapters also which do not have the formulas and have only theories and definitions.

Chapter-wise All Maths Formulas for Class 12

Maths Formulas for Class 12: Chapter 1 Relations and Functions

Definition/Theorems

Relations:

1. An empty relation holds a specific relation R in X as: R = φ ⊂ X × X.

2. A symmetric relation R in X satisfies a certain relation as: (a, b) ∈ R implies (b, a) ∈ R.

3. A reflexive relation R in X can be given as: (a, a) ∈ R; ∀ a ∈ X.

4. A transitive relation R in X can be given as: (a, b) ∈ R and (b, c) ∈ R, thereby, implying (a, c) ∈ R.

5. A universal relation is the relation R in X can be given by R = X × X.

6. An equivalence relation R in X is a relation that shows all the reflexive, symmetric and transitive relations.

Functions:

1. A function f: X → Y is one-one or injective, if f(x₁) = f(x₂) ⇒ x₁ = x₂ ∀ x₁, x₂ ∈ X.

2. A function f: X → Y is onto or surjective, if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.

3. A function f: X → Y is one-one and onto or bijective, if f follows both the one-one and onto properties.

4. A function f: X → Y is invertible if ∃ g: Y → X such that gof = I_X and fog = I_Y.

5. A function f: X → Y is invertible if and only if f is one-one and onto.

6. A binary operation ∗ on a set A is a function ∗ from A × A to A.

7. An element e ∈ X is the identity element for binary operation ∗ : X × X → X, if a ∗ e = a = e ∗ a; ∀ a ∈ X.

8. An element a ∈ X is invertible for binary operation ∗ : X × X → X, if there exists b ∈ X such that a ∗ b = e = b ∗ a, where e is the identity for the binary operation ∗. The element b is called the inverse of a and is denoted by a^–1.

9. An operation ∗ on X is commutative if a ∗ b = b ∗ a; ∀ a, b in X.

10. An operation ∗ on X is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c); ∀ a, b, c in X.

Maths Formulas for Class 12: Chapter 2: Inverse Trigonometric Functions

Important Formulas:

Maths Formulas for Class 12: Chapter 3 Matrices

Definition/Theorems

1. A matrix is said to have an ordered rectangular array of numbers or functions. A matrix of order m × n consists of m rows and n columns.

2. An m × n matrix is a square matrix, if m = n.

3. A = [a_ij]_m_{× m} is a diagonal matrix if a_ij = 0, when i ≠ j.

4. 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.

5. A = [a_ij]_n_{× n} is an identity matrix, if a_ij = 1, when i = j and a_ij = 0, when i ≠ j.

6. A zero matrix has all its elements as zero.

7. A = [a_ij] = [b_ij] = B if and only if:

(i) A and B are of the same order

(ii) a_ij = b_ij for all the possible values of i and j.

Properties of Matrices

1. Some basic operations of matrices:

(i) kA = k[a_ij]_{m × n} = [k(a_ij)]_{m × n}

(ii) – A = (– 1)A

(iii) A – B = A + (– 1)B

(iv) A + B = B + A

(v) (A + B) + C = A + (B + C); where A, B and C all are of the same order

(vi) k(A + B) = kA + kB; where A and B are of the same order; k is constant

(vii) (k + l)A = kA + lA; where k and l are the constant

2. If A = [a_ij]_{m × n} and B = [b_jk]_{n × p}, then AB = C = [c_ik]_{m × p} ; where c_ik = ∑a_ijb_jk

(i) A.(BC) = (AB).C

(ii) A(B + C) = AB + AC

(iii) (A + B)C = AC + BC

3. If A= [a_ij]_{m × n}, then A’ or A^T = [a_ji]_{n × m}

(i) (A’)’ = A

(ii) (kA)’ = kA’

(iii) (A + B)’ = A’ + B’

(iv) (AB)’ = B’A’

4. Some elementary operations:

(i) R_i ↔ R_j or C_i ↔ C_j

(ii) R_i → kR_i or C_i → kC_i

(iii) R_i → R_i + kR_j or C_i → C_i + kC_j

5. The matrix A is a symmetric matrix, if A′ = A.

6. The matrix A is a skew-symmetric matrix, if A′ = –A.

7. If A and B are two square matrices such that AB = BA = I, then B is the inverse matrix of A and is denoted by A^-1 and A is the inverse of B. The inverse of a square matrix, if exists, is unique.

Maths Formulas for Class 12: Chapter 4 Determinants

Definition/Theorems

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

· A determinant of order 2 × 2 matrix A = is given by

· A determinant of order 3 × 3 matrix A =

is given by (expanding along R₁)

· We can find the value of a determinant by expanding along any one of the three rows (or columns) and the value remains the same.

· Generally, we find the value of a determinant by expanding along a row or column which has maximum number of zeroes.

For any square matrix A, the |A| satisfy following properties.

1. |A′| = |A|, where A′ = transpose of A.

2. If we interchange any two rows (or columns), then sign of determinant changes.

3. If any two rows or any two columns are identical or proportional, then value of determinant is zero.

4. If we multiply each element of a row or a column of a determinant by constant k, then value of determinant is multiplied by k.

5. Multiplying a determinant by k means multiply elements of only one row (or one column) by k.

6. If A = [a_ij]_{3 × 3}, then |k.A| = k³|A|

7. If elements of a row or a column in a determinant can be expressed as sum of two or more elements, then the given determinant can be expressed as sum of two or more determinants.

8. If to each element of a row or a column of a determinant the equimultiples of corresponding elements of other rows or columns are added, then value of determinant remains the same.

9. If A is a skew symmetric matrix of odd order, then |A| = 0.

10. Area of a triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is given by

11. Minors: Minor of an element a_ij of the determinant of matrix A is the determinant obtained by deleting i^th row and j^th column and denoted by M_ij.

12. Cofactors: Cofactor of a_ij of given by A_ij = (– 1)^{i+ j} M_ij.

13. Value of determinant of a matrix A is obtained by sum of product of elements of a row (or a column) with corresponding cofactors. For example, |A| = a₁₁ A₁₁ + a₁₂ A₁₂ + a₁₃ A₁₃.

14. If elements of one row (or column) are multiplied with cofactors of elements of any other row (or column), then their sum is zero. For example, a₁₁ A₂₁ + a₁₂ A₂₂ + a₁₃ A₂₃ = 0.

15. Adjoint of a matrix: A (adj A) = (adj A) A = |A| I, where A is a square matrix of order n.

16. Singular matrix: A square matrix A is said to be singular or non-singular according as |A| = 0 or |A| ≠ 0.

17. Inverse of a square matrix: If AB = BA = I, where B is a square matrix, then B is called inverse of A. Also A^-1 = B or B^-1 = A and hence (A^-1)^-1 = A.

18. A square matrix A has inverse if and only if A is non-singular.

19.

20. If a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂

a₃x + b₃y + c₃z = d₃

then these equations can be written as AX = B, where

21. Unique solution of equation AX = B is given by X = A^-1B, where |A| ≠ 0.

22. A system of equation is consistent or inconsistent according as its solution exists or not.

23. For a square matrix A in matrix equation AX = B

• |A| ≠ 0, there exists unique solution

• |A| = 0 and (adj A) B ≠ 0, then there exists no solution

• |A| = 0 and (adj A) B = 0, then system may or may not be consistent and has infinite solutions.

Maths Formulas for Class 12: Chapter 5 Continuity and Differentiability

Definition/Properties

1. A function is said to be continuous at a given point if the limit of that function at the point is equal to the value of the function at the same point.

Formulas

Given below are the standard derivatives:

Maths Formulas for Class 12: Chapter 6 Application of Derivatives

If a quantity y varies with another quantity x, satisfying some rule y = f(x),

(i) If f′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maxima.

(ii) If f′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at every point sufficiently close to and to the left of c, and f ′(x) > 0 at every point sufficiently close to and to the right of c, then c is a point of local minima.

(iii) If f′(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. In fact, such a point is called point of inflexion.

· Second Derivative Test: Let f be a function defined on an interval I and c ∈ I. Let f be twice differentiable at c. Then,

(i) x = c is a point of local maxima if f′(c) = 0 and f″(c) < 0
The values f(c) is local maximum value of f.

(ii) x = c is a point of local minima if f′(c) = 0 and f″(c) > 0
In this case, f(c) is local minimum value of f.

(iii) The test fails if f′(c) = 0 and f″(c) = 0.
In this case, we go back to the first derivative test and find whether c is a point of maxima, minima or a point of inflexion.

· Working rule for finding absolute maxima and/or absolute minima

Step 1: Find all critical points of f in the interval, i.e., find points x where either f′(x) = 0 or f is not differentiable.

Step 2: Take the end points of the interval.

Step 3: At all these points (listed in Step 1 and 2), calculate the values of f.

Step 4: Identify the maximum and minimum values of f out of the values calculated in Step 3. This maximum value will be the absolute maximum value of f and the minimum value will be the absolute minimum value of f.

Maths Formulas for Class 12: Chapter 7 Integrals

Definition/Properties

Formulas – Standard Integrals

Formulas – Partial Fractions

Formulas – Integration by Substitution

Formulas – Integrals (Special Functions)

Formulas – Integration by Parts

1. The integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}

Formulas – Special Integrals

Maths Formulas for Class 12: Chapter 8 Application of Integrals

Definition/Formulas

· The area of the region bounded by the curve y = f (x), x-axis and the lines x = a and x = b (b > a) is given by the formula:

· The area of the region bounded by the curve x = φ(y), y-axis and the lines y = c, y = d is given by the formula:

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

· If f(x) ≥ g(x) in [a, c] and f(x) ≤ g(x) in [c, b], a < c < b, then:

Maths Formulas for Class 12: Chapter 9 Differential Equations

Definition/Properties

Differential Equation: An equation involving derivatives of the dependent variable with respect to independent variable (variables) is known as a differential equation.
Linear and non-linear differential equation: A differential equation is said to be linear if unknown function (dependent variable) as its derivative which occurs in the equation, occur only in the first degree and are not multiplied together. Otherwise the differential equation is said to be non-linear.
Order: Order of a differential equation is the order of the highest order derivative occurring in the differential equation.
Degree: Degree of a differential equation is defined if it is a polynomial equation in its derivatives.
Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it.
Solution: A function which satisfies the given differential equation is called its solution.
General Solution: The solution which contains as many arbitrary constants as the order of the differential equation is called a general solution.
Particular Solution: The solution free from arbitrary constants is called particular solution.
To form a differential equation from a given function, we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants.
Variable Separable Method: Variable separable method is used to solve such an equation in which variables can be separated completely, i.e., terms containing y should remain with dy and terms containing x should remain with dx.
A differential equation which can be expressed in the formwhere f(x, y) and g(x, y) are homogenous functions of degree zero is called a homogeneous differential equation.
A differential equation of the formwhere P and Q are constants or functions of x only is called a first order linear differential equation.

Maths Formulas for Class 12: Chapter 10 Vector Algebra

Definition/Properties

1. Vector is a certain quantity that has both the magnitude and the direction. The position vector of a point P(x, y, z) is given by:

Maths Formulas for Class 12: Chapter 11 Three Dimensional Geometry

Definition/Properties

1. Direction cosines of a line are the cosines of the angle made by a particular line with the positive directions on coordinate axes.

2. Skew lines are lines in space which are neither parallel nor intersecting. These lines lie in separate planes.

3. If l, m and n are the direction cosines of a line, then l² + m² + n² = 1.

Formulas

Maths Formulas for Class 12: Chapter 12 Linear Programming

Definitions/Theorems

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.
A few important linear programming problems are:

(i) Diet problems

(ii) Manufacturing problems

(iii) Transportation problems

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

Theorem 1: 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.

Theorem 2: Let R be the feasible region for a linear programming problem, and let Z = ax + by 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, then 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 of the following steps:

(i) Find the feasible region of the linear programming problem and determine its corner points (vertices).

(ii) 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.

(iii) 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,

(i) 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.

(ii) 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.