Cbse Class 12 Mathematics Formulas Pdf

Maths Formulas For Class 12: You might have heard many students say that Class 12 Maths is the most difficult of all subjects that one has to study in their entire school life. Such negativity is enough to bring failures in even small class tests, let alone the Class 12 Board Exams. Experts at Embibe do not want you to believe in such myths because Class 12th Maths encourages you to develop your logic and implementation. We advise you to learn and understand the important Maths Formulas for Class 12 provided here which will let you evolve faster in the subject as well as help you build concrete knowledge.

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Maths Formulas For Class 12

Class 12 is one of the most challenging years of a student's academic career. Furthermore, the CBSE class 12 maths syllabus is broad. As a result, the most important step students should do when studying for their exam is to properly memorise the Class 12 Maths formulas. Class 12 acts as a critical link between higher education and regular education. However, when students get hold of inappropriate study material, this might be difficult. In class 12, math is a core subject. Students can enhance their problem-solving skills by learning mathematical formulas efficiently. This is why the free PDF of CBSE maths formulas for class 12 is such an useful learning tool.

Once you start solving the NCERT exercise questions, you can use the Maths formulas for Class 12 provided here as a reference. Maths formulas for Class 12 PDF will also help in understanding the chapter in-depth and easily memorizing the formulas. Using these NCERT formulas as a reference, you will be able to complete your assignments on time and learn the formulas on go. This article provides a compiled list of all the Class 12 Maths all formulas. This will help you have a better understanding of the concepts which will eventually result in a higher score in the exam. So, go through the detailed Class 12 Maths all formulas provided below.

The Class 12 Maths Formulas provided here will assist you in conquering your Board exams as well as the entrance examinations. Let's take a look at the important chapters of Class 12 Maths for which we need formulas:

Relations and Functions
Inverse Trigonometric Functions
Matrices
Determinants
Continuity and Differentiability
Integrals
Application of Integrals
Vector Algebra
Three Dimensional Geometry
Probability

The main advantage of using the class 12 mathematics formula PDF is that it reduces the need to memorise problems and instead teaches you how to solve them. Instead of reading through textbooks, students may use these formulas to save time and study for their examinations. It can be time-consuming to write out each of these formulas. Referring to them online, on the other hand, may assist you have a fun-loving and exciting learning experience.

Maths Formulas For Class 12: Relations And Functions

Definition/Theorems

Empty relation holds a specific relation R in X as: R = φ ⊂ X × X.
A Symmetric relation R in X satisfies a certain relation as: (a, b) ∈ R implies (b, a) ∈ R.
A Reflexive relation R in X can be given as: (a, a) ∈ R; for all ∀ a ∈ X.
A Transitive relation R in X can be given as: (a, b) ∈ R and (b, c) ∈ R, thereby, implying (a, c) ∈ R.
A Universal relation is the relation R in X can be given by R = X × X.
Equivalence relation R in X is a relation that shows all the reflexive, symmetric and transitive relations.

Properties

A function f: X → Y is one-one/injective; if f(x₁) = f(x₂) ⇒ x₁ = x₂ ∀ x₁ , x₂ ∈ X.
A function f: X → Y is onto/surjective; if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.
A function f: X → Y is one-one and onto or bijective; if f follows both the one-one and onto properties.
A function f: X → Y is invertible if ∃ g: Y → X such that gof = I_X and fog = I_Y. This can happen only if f is one-one and onto.
A binary operation \(\ast\) performed on a set A is a function \(\ast\) from A × A to A.
An element e ∈ X possess the identity element for binary operation \(\ast\) : X × X → X, if a \(\ast\) e = a = e \(\ast\) a; ∀ a ∈ X.
An element a ∈ X shows the invertible property for binary operation \(\ast\) : X × X → X, if there exists b ∈ X such that a \(\ast\) b = e = b \(\ast\) a where e is said to be the identity for the binary operation \(\ast\). The element b is called the inverse of a and is denoted by a^–1.
An operation \(\ast\) on X is said to be commutative if a \(\ast\) b = b \(\ast\) a; ∀ a, b in X.
An operation \(\ast\) on X is said to associative if (a \(\ast\) b) \(\ast\) c = a \(\ast\) (b \(\ast\) c); ∀ a, b, c in X.

Class 12 Maths Formulas: Inverse Trigonometric Functions

Inverse Trigonometric Functions are quite useful in Calculus to define different integrals. You can also check the Trigonometric Formulas here.

Properties/Theorems

The domain and range of inverse trigonometric functions are given below:

Functions	Domain	Range
y = sin^-1 x	[–1, 1]	\(\left [ \frac{-\pi }{2},\frac{\pi }{2} \right ]\)
y = cos^-1 x	[–1, 1]	\(\left [0,\pi \right ]\)
y = cosec^-1 x	R – (–1, 1)	\(\left [ \frac{-\pi }{2},\frac{\pi }{2} \right ]\) – {0}
y = sec^-1 x	R – (–1, 1)	\(\left [0,\pi \right ]\) – {\(\frac{\pi }{2}\)}
y = tan^-1 x	R	\(\left ( \frac{-\pi}{2},\frac{\pi}{2} \right )\)
y = cot^-1 x	R	\(\left (0,\pi \right )\)

Formulas

\(y=sin^{-1}x\Rightarrow x=sin\:y\)
\(x=sin\:y\Rightarrow y=sin^{-1}x\)
\(sin^{-1}\frac{1}{x}=cosec^{-1}x\)
\(cos^{-1}\frac{1}{x}=sec^{-1}x\)
\(tan^{-1}\frac{1}{x}=cot^{-1}x\)
\(cos^{-1}(-x)=\pi-cos^{-1}x\)
\(cot^{-1}(-x)=\pi-cot^{-1}x\)
\(sec^{-1}(-x)=\pi-sec^{-1}x\)
\(sin^{-1}(-x)=-sin^{-1}x\)
\(tan^{-1}(-x)=-tan^{-1}x\)
\(cosec^{-1}(-x)=-cosec^{-1}x\)
\(tan^{-1}x+cot^{-1}x=\frac{\pi}{2}\)
\(sin^{-1}x+cos^{-1}x=\frac{\pi}{2}\)
\(cosec^{-1}x+sec^{-1}x=\frac{\pi}{2}\)
\(tan^{-1}x+tan^{-1}y=tan^{-1}\frac{x+y}{1-xy}\)
\(2\:tan^{-1}x=sin^{-1}\frac{2x}{1+x^2}=cos^{-1}\frac{1-x^2}{1+x^2}\)
\(2\:tan^{-1}x=tan^{-1}\frac{2x}{1-x^2}\)
\(tan^{-1}x+tan^{-1}y=\pi+tan^{-1}\left (\frac{x+y}{1-xy} \right )\); xy > 1; x, y > 0

Maths Formulas For Class 12: Matrices

Definition/Theorems

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.
An m × n matrix will be known as a square matrix when m = n.
A = [a_ij]_{m × m} will be known as diagonal matrix if a_ij = 0, when i ≠ j.
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.
A = [a_ij]_{n × n} is an identity matrix, if a_ij = 1, when i = j and a_ij = 0, when i ≠ j.
A zero matrix will contain all its element as zero.
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 certain values of i and j

Elementary Operations

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
If A = [a_ij]_{m × n} and B = [b_jk]_{n × p}, then
AB = C = [c_ik]_{m × p} ; where c_ik = \(\sum_{j=1}^{n}a_{ij}b_{jk}\)
- (i) A.(BC) = (AB).C
- (ii) A(B + C) = AB + AC
- (iii) (A + B)C = AC + BC
If A= [a_ij]_{m × n}, then A' or AT = [a_ji]_{n × m}
- (i) (A')' = A
- (ii) (kA)' = kA'
- (iii) (A + B)' = A' + B'
- (iv) (AB)' = B'A'
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
A is said to known as a symmetric matrix if A′ = A
A is said to be the skew symmetric matrix if A′ = –A

Class 12 Maths Formulas: Determinants

Definition/Theorems

The determinant of a matrix A = [a₁₁]_{1 × 1} can be given as: |a₁₁| = a₁₁.
For any square matrix A, the |A| will satisfy the following properties:
- (i) |A′| = |A|, where A′ = transpose of A.
- (ii) If we interchange any two rows (or columns), then sign of determinant changes.
- (iii) If any two rows or any two columns are identical or proportional, then the value of the determinant is zero.
- (iv) If we multiply each element of a row or a column of a determinant by constant k, then the value of the determinant is multiplied by k.

Formulas

Determinant of a matrix \(A=\begin{bmatrix} a_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3 \end{bmatrix}\) can be expanded as:
|A| = \(\begin{vmatrix} a_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3 \end{vmatrix}=a_1 \begin{vmatrix} b_2& c_2\\ b_3& c_3 \end{vmatrix}-b_1 \begin{vmatrix} a_2& c_2\\ a_3& c_3 \end{vmatrix}+c_1 \begin{vmatrix} a_2& b_2\\ a_3& b_3 \end{vmatrix}\)
Area of triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is:
∆ = \(\frac{1}{2}\)\(\begin{vmatrix} x_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1 \end{vmatrix}\)
Cofactor of aij of given by A_ij = (– 1)^{i+ j} M_ij
If A = \(\begin{bmatrix} a_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33} \end{bmatrix}\), then adj A = \(\begin{bmatrix} A_{11}& A_{21}& A_{31}\\ A_{12}& A_{22}& A_{32}\\ A_{13}& A_{23}& A_{33} \end{bmatrix}\) ; where A_ij is the cofactor of a_ij.
\(A^{-1}=\frac{1}{|A|}(adj\:A)\)
If a₁x + b₁y + c₁z = d₁ a₂x + b₂y + c₂z = d₂ a₃x + b₃ y + c₃z = d₃ , then these equations can be written as A X = B, where:
A=\(\begin{bmatrix} a_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3 \end{bmatrix}\), X = \(\begin{bmatrix} x\\ y\\ z \end{bmatrix}\) and B = \(\begin{bmatrix} d_1\\ d_2\\ d_3 \end{bmatrix}\)
For a square matrix A in matrix equation AX = B
- (i) | A| ≠ 0, there exists unique solution
- (ii) | A| = 0 and (adj A) B ≠ 0, then there exists no solution
- (iii) | A| = 0 and (adj A) B = 0, then the system may or may not be consistent.

Maths Formulas For Class 12: Continuity And Differentiability

Definition/Properties

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.
Properties related to the functions:
- (i) \((f\pm g) (x) = f (x)\pm g(x)\) is continuous.
- (ii) \((f.g)(x) = f (x) .g (x)\) is continuous.
- (iii) \(\frac{f}{g}(x) = \frac{f(x)}{g(x)}\) (whenever \(g(x)\neq 0\) is continuous.
Chain Rule: If f = v o u, t = u (x) and if both \(\frac{\mathrm{d} t}{\mathrm{d} x}\) and \(\frac{\mathrm{d} v}{\mathrm{d} x}\) exists, then:
\(\frac{\mathrm{d} f}{\mathrm{d} x}=\frac{\mathrm{d} v}{\mathrm{d} t}.\frac{\mathrm{d} t}{\mathrm{d} x}\)
Rolle's Theorem: If f: [a, b] → R is continuous on [a, b] and differentiable on (a, b) where as f(a) = f(b), then there exists some c in (a, b) such that f ′(c) = 0.
Mean Value Theorem: 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
\(f'(c)=\frac{f(b)-f(a)}{b-a}\)

Formulas

Given below are the standard derivatives:

Derivative	Formulas
\(\frac{\mathrm{d} }{\mathrm{d} x}(sin^{-1}x)\)	\(\frac{1}{\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(cos^{-1}x)\)	\(-\frac{1}{\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(tan^{-1}x)\)	\(\frac{1}{1+x^2}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(cot^{-1}x)\)	\(\frac{-1}{1+x^2}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(sec^{-1}x)\)	\(\frac{1}{x\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(cosec^{-1}x)\)	\(\frac{-1}{x\sqrt{1-x^2}}\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(e^x)\)	\(e^x\)
\(\frac{\mathrm{d} }{\mathrm{d} x}(log\:x)\)	\(\frac{1}{x}\)

Class 12 Maths Formulas: Integrals

Definition/Properties

Integration is the inverse process of differentiation. Suppose, \(\frac{\mathrm{d} }{\mathrm{d} x}F(x)=f(x)\); then we can write \(\int f(x)\:dx=F(x)+C\)
Properties of indefinite integrals:
- (i) \(\int [f(x)+g(x)]\:dx=\int f(x)\:dx+\int g(x)\:dx\)
- (ii) For any real number k, \(\int k\:f(x)\:dx=k\int f(x)\:dx\)
- (iii) \(\int [k_1\:f_1(x)+k_2\:f_2(x)+…+k_n\:f_n(x)]\:dx=\\
  k_1\int f_1(x)\:dx+k_2\int f_2(x)\:dx+…+k_n\int f_n(x)\:dx\)
First fundamental theorem of integral calculus: Let the area function be defined as: \(A(x)=\int_{a}^{x}f(x)\:dx\) for all \(x\geq a\), 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]; Furthermore, let's assume F another function as: \(\frac{\mathrm{d} }{\mathrm{d} x}F(x)=f(x)\) for every x falling in the domain of f; then,
\(\int_{a}^{b}f(x)\:dx=[F(x)+C]_{a}^{b}=F(b)-F(a)\)

Formulas – Standard Integrals

\(\int x^ndx=\frac{x^{n+1}}{n+1}+C,n\neq -1\). Particularly, \(\int dx=x+C)\)
\(\int cos\:x\:dx=sin\:x+C\)
\(\int sin\:x\:dx=-cos\:x+C\)
\(\int sec^2x\:dx=tan\:x+C\)
\(\int cosec^2x\:dx=-cot\:x+C\)
\(\int sec\:x\:tan\:x\:dx=sec\:x+C\)
\(\int cosec\:x\:cot\:x\:dx=-cosec\:x+C\)
\(\int \frac{dx}{\sqrt{1-x^2}}=sin^{-1}x+C\)
\(\int \frac{dx}{\sqrt{1-x^2}}=-cos^{-1}x+C\)
\(\int \frac{dx}{1+x^2}=tan^{-1}x+C\)
\(\int \frac{dx}{1+x^2}=-cot^{-1}x+C\)
\(\int e^xdx=e^x+C\)
\(\int a^xdx=\frac{a^x}{log\:a}+C\)
\(\int \frac{dx}{x\sqrt{x^2-1}}=sec^{-1}x+C\)
\(\int \frac{dx}{x\sqrt{x^2-1}}=-cosec^{-1}x+C\)
\(\int \frac{1}{x}\:dx=log\:|x|+C\)

Formulas – Partial Fractions

Partial Fraction	Formulas
\(\frac{px+q}{(x-a)(x-b)}\)	\(\frac{A}{x-a}+\frac{B}{x-b},a\neq b\)
\(\frac{px+q}{(x-a)^2}\)	\(\frac{A}{x-a}+\frac{B}{(x-b)^2}\)
\(\frac{px^2+qx+r}{(x-a)(x-b)(x-c)}\)	\(\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{x-c}\)
\(\frac{px^2+qx+r}{(x-a)^2(x-b)}\)	\(\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{x-b}\)
\(\frac{px^2+qx+r}{(x-a)(x^2+bx+c)}\)	\(\frac{A}{x-a}+\frac{Bx+C}{x^2+bx+c}\)

Formulas – Integration by Substitution

\(\int tan\:x\:dx=log\:|sec\:x|+C\)
\(\int cot\:x\:dx=log\:|sin\:x|+C\)
\(\int sec\:x\:dx=log\:|sec\:x+tan\:x|+C\)
\(\int cosec\:x\:dx=log\:|cosec\:x-cot\:x|+C\)

Formulas – Integrals (Special Functions)

\(\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\:log\:\left |\frac{x-a}{x+a} \right |+C\)
\(\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\:log\:\left |\frac{a+x}{a-x} \right |+C\)
\(\int \frac{dx}{x^2+a^2}=\frac{1}{a}\:tan^{-1}\frac{x}{a}+C\)
\(\int \frac{dx}{\sqrt{x^2-a^2}}=log\:\left |x+\sqrt{x^2-a^2} \right |+C\)
\(\int \frac{dx}{\sqrt{x^2+a^2}}=log\:\left |x+\sqrt{x^2+a^2} \right |+C\)
\(\int \frac{dx}{\sqrt{x^2-a^2}}=sin^{-1}\frac{x}{a}+C\)

Formulas – Integration by Parts

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}
\(\int f_1(x).f_2(x)=f_1(x)\int f_2(x)\:dx-\int \left [ \frac{\mathrm{d} }{\mathrm{d} x}f_1(x).\int f_2(x)\:dx \right ]dx\)
\(\int e^x\left [ f(x)+f'(x) \right ]\:dx=\int e^x\:f(x)\:dx+C\)

Formulas – Special Integrals

\(\int \sqrt{x^2-a^2}\:dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\:log\left | x+\sqrt{x^2-a^2} \right |+C\)
\(\int \sqrt{x^2+a^2}\:dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\:log\left | x+\sqrt{x^2+a^2} \right |+C\)
\(\int \sqrt{a^2-x^2}\:dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a}{2}\:sin^{-1}\frac{x}{a}+C\)
\(ax^2+bx+c=a\left [ x^2+\frac{b}{a}x+\frac{c}{a} \right ]=a\left [ \left ( x+\frac{b}{2a} \right )^2+\left ( \frac{c}{a}-\frac{b^2}{4a^2} \right ) \right ]\)

Maths Formulas For Class 12: Application Of Integrals

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:
- \(Area=\int_{a}^{b}y\:dx=\int_{a}^{b}f(x)\:dx\)
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:
- \(Area=\int_{c}^{d}x\:dy=\int_{c}^{d}\phi (y)\:dy\)
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:
- \(Area=\int_{a}^{b}[f(x)-g(x)]\:dx,\: where, f(x)\geq g(x)\:in\:[a,b]\)
If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in [c, b], a < c < b, then:
- \(Area=\int_{a}^{c}[f(x)-g(x)]\:dx,+\int_{c}^{b}[g(x)-f(x)]\:dx\)

Class 12 Maths Formulas: Vector Algebra

Definition/Properties

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:\(\overrightarrow{OP}(=\vec{r})=x\hat{i}+y\hat{j}+z\hat{k}\)
The scalar product of two given vectors \(\vec{a}\) and \(\vec{b}\) having angle θ between them is defined as:
- \(\vec{a}\:.\:\vec{b}=|\vec{a}||\vec{b}|\:cos\:\theta\)
The position vector of a point R dividing a line segment joining the points P and Q whose position vectors \(\vec{a}\) and \(\vec{b}\) are respectively, in the ratio m : n is given by:
- (i) internally: \(\frac{n\vec{a}+m\vec{b}}{m+n}\)
- (ii) externally: \(\frac{n\vec{a}-m\vec{b}}{m-n}\)

Formulas

If two vectors \(\vec{a}\) and \(\vec{b}\) are given in its component forms as \(\hat{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) and \(\hat{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\) and λ as the scalar part; then:

(i) \(\vec{a}+\vec{b}=(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}\) ;
(ii) \(\lambda \vec{a}=(\lambda a_1)\hat{i}+(\lambda a_2)\hat{j}+(\lambda a_3)\hat{k}\) ;
(iii) \(\vec{a}\:.\:\vec{b}=(a_1b_1)+(a_2b_2)+(a_3b_3)\)
(iv) and \(\vec{a}\times \vec{b}= \begin{bmatrix} \hat{i}& \hat{j}& \hat{k}\\ a_{1}& b_{1}& c_{1}\\ a_{2}& b_{2}& c_{2} \end{bmatrix}\).

Maths Formulas For Class 12: Three Dimensional Geometry

Definition/Properties

Direction cosines of a line are the cosines of the angle made by a particular line with the positive directions on coordinate axes.
Skew lines are lines in space which are neither parallel nor intersecting. These lines lie in separate planes.
If l, m and n are the direction cosines of a line, then l² + m² + n² = 1.

Formulas

The Direction cosines of a line joining two points P (x₁ , y₁ , z₁) and Q (x₂ , y₂ , z₂) are \(\frac{x_2-x_1}{PQ}\:,\:\frac{y_2-y_1}{PQ}\:,\frac{z_2-z_1}{PQ}\) where
- PQ=\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)
Equation of a line through a point (x₁ , y₁ , z₁ ) and having direction cosines l, m, n is: \(\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}\)
The vector equation of a line which passes through two points whose position vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{r}=\vec{a}+\lambda (\vec{b}-\vec{a})\)
The shortest distance between \(\vec{r}=\vec{a_1}+\lambda\: \vec{b_1}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b_2}\) is:
- \(\left | \frac{(\vec{b_1}\times \vec{b_2}).(\vec{a_2}-\vec{a_1})}{|\vec{b_1}\times \vec{b_2}|} \right |\)
The distance between parallel lines \(\vec{r}=\vec{a_1}+\lambda\: \vec{b}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b}\) is
- \(\left | \frac{\vec{b}\times (\vec{a_2}-\vec{a_1})}{|\vec{b}|} \right |\)
The equation of a plane through a point whose position vector is \(\vec{a}\) and perpendicular to the vector \(\vec{N}\) is \((\vec{r}-\vec{a})\:.\:\vec{N}=0\)
Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point (x₁ , y₁ , z₁) is A (x – x₁) + B (y – y₁) + C (z – z₁) = 0
The equation of a plane passing through three non-collinear points (x₁ , y₁ , z₁); (x₂ , y₂ , z₂) and (x₃ , y₃ , z₃) is:
- \(\begin{vmatrix} x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1 \end{vmatrix}=0\)
The two lines \(\vec{r}=\vec{a_1}+\lambda\: \vec{b_1}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b_2}\) are coplanar if:
- \((\vec{a_2}-\vec{a_1})\:.\:(\vec{b_1}\times \vec{b_2})=0\)
The angle φ between the line \(\vec{r}=\vec{a}+\lambda\: \vec{b}\) and the plane \(\vec{r}\:.\:\hat{n}=d\) is given by:
- \(sin\:\phi =\left |\frac{\vec{b}\:.\:\hat{n}}{|\vec{b}||\hat{n}|} \right |\)
The angle θ between the planes A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0 is given by:
- \(cos\:\theta =\left | \frac{A_1\:A_2+B_1\:B_2+C_1\:C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\:\sqrt{A_2^2+B_2^2+C_2^2}} \right |\)
The distance of a point whose position vector is \(\vec{a}\) from the plane \(\vec{r}\:.\:\hat{n}=d\) is given by: \(\left | d-\vec{a}\:.\:\hat{n} \right |\)
The distance from a point (x₁ , y₁ , z₁) to the plane Ax + By + Cz + D = 0:
- \(\left | \frac{Ax_1+By_1+Cz_1+D}{\sqrt{A^2+B^2+C^2}} \right |\)

Class 12 Maths Formulas: Probability

Definition/Properties

The conditional probability of an event E holds the value of the occurrence of the event F as:
- \(P(E\:|\:F)=\frac{E\cap F}{P(F)}\:,\:P(F)\neq 0\)
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)
Bayes Theorem: If E₁ , E₂ , …. , E_n are events contituting in a sample space S; then,
- \(P(E_i\:|\:A)=\frac{P(E_i)\:P(A|E_i)}{\sum_{j=1}^{n}P(E_j)\:P(A|E_j)}\)
Var (X) = E (X²) – [E(X)]²

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Frequently Asked Questions: FAQs

Here we have provided the most frequently asked questions (FAQs) related to NCERT Class 12 Maths formulas:

Ques: How many formulas are present in the class 12 CBSE Maths?

Ans: It is almost next to impossible to keep a record of all the formulas given in the Maths book of class 12 CBSE. As for each and every theory and concept, given in the book, there exist one or more formulas to help find the solutions for the given mathematical problems. As well the level of formulas increases with each grade making class 12 Mathematics the most difficult on the school level.

Ques: Give a list of basic maths formulas used in CBSE class 12th?

Ans: A list containing the basic maths formulas that have been introduced for the students of class 12th have been listed in this above article. The basic list of topics include:
1. Algebra
2. Matrices
3. Geometry
4. Linear Programming
5. Calculus
6. Probability

Ques: What is the formula used for the trigonometric ratio integration?

Ans: ∫sin (x) dx = -Cos x + C

∫cos(x) dx = Sin x + C

∫sec^2x dx = tan x + C, etc.

Ques: Why are the mathematical formulas important?

Ans: The mathematical formulas are important because it helps in solving the mathematical problems with utmost ease. Hence it is important to learn these mathematical formulas for solving the problems in a given time span and in an efficient manner. Mathematical formulas are in the generalized form and at the time of solving the mathematical problems, all we need to do is put the value of entities in the formula given and make the whole process easier and swifter.

Ques: Where can I find the complete class 12 Mathematics formula for the NCERT book?

Ans: Students can find the compiled list of formulas in this article on the embibe platform for free. Students can also find direct links to Class 12 Mathematics Notes, Solutions, practice papers, mock tests, important questions, and much more. Students can go through this article to find the complete NCERT Solutions for Class 12 Mathematics along with the links of complete NCERT Book Class 12 Mathematics & important study material.

Ques: Which is the best solution for NCERT Class 12 Mathematics?

Ans: Students can find 100 percent accurate Solutions for the NCERT Class 12 Mathematics on the Embibe platform. This article contains the complete solution which has been solved by expert mathematics teachers associated with embibe. We, at Embibe, provide solutions for all the questions given in the Class 12 Mathematics textbook after taking the CBSE Board guidelines under strict consideration from the latest NCERT book for Class 12 Mathematics.

Ques: What is the formula used for the exponential function integration?

Ans: If an exponential function is integrated, the function will remain unchanged with a constant being added to it.

CHECK OUT DETAILED CBSE SYLLABUS FOR CLASS 12 MATHS

So, now you have all the Maths formulas for Class 12. We hope this article has helped you. Understand these Class 12 Maths formulas while implementing them to solve questions. You can solve the freeClass 12 Maths questions of Embibe which will help you a lot. Make the best use of these resources and master the subject.

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