Karnataka Board 2nd PUC Mathematics Books

August 19, 202239 Insightful Publications

**Maths Formulas for Class 12:** Students in the CBSE Class 12 typically view mathematics as a difficult subject since there is often a lack of fundamental clarity or a good approach to problem-solving. But did you know that mastering mathematical formulas could help you to get rid of the fear of mathematics? This article shall provide chapter-wise and concept-wise Class 12 maths formulas.

Once students can firmly grasp the various mathematical formulas, it becomes very simple to find the exact solution to any particular problem. Moreover, it allows them to arrive at the answers faster than others. Students must note the formulas and keep them handy for quick reference. Learning all the mathematical formulas and having them at the tip of your fingers is a master trick with multiple benefits. Keep reading this article to learn more.

Whether you are preparing for the CBSE Class 12 board exams, revising for the term-end exams, gearing up for the competitive exams, or solving the questions of the NCERT maths textbook, this article is your one-stop solution. Class 12 math formula will also help understand the chapter in-depth and quickly memorise the formulas. Using these NCERT formulas as a reference, you will complete your assignments on time and learn the formulas on the go.

The main advantage of using the Class 12 Maths formula is that it reduces the need to memorise problems and 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.

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

This article provides a compiled list of all the Class 12 maths formulas. This will help you better understand 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.

*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 ∀ an ∈ 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 that 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(x
_{1}) = f(x_{2}) ⇒ x_{1}= x_{2}∀ x_{1}, x_{2}∈ 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.

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

*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:- A and B are of the same order
- a
_{ij}= b_{ij}for all the certain values of i and j

*Elementary Operations*

- Some basic operations of matrices:
- kA = k[a
_{ij}]_{m × n}= [k(a_{ij})]_{m × n} - – A = (– 1)A
- A – B = A + (– 1)B
- A + B = B + A
- (A + B) + C = A + (B + C); where A, B and C all are of the same order
- k(A + B) = kA + kB; where A and B are of the same order; k is constant
- (k + l)A = kA + lA; where k and l are the constant

- kA = k[a
- If A = [a
_{ij}]_{m × n}and B = [b_{jk}]_{n × p}, then

AB = C = [c_{ik}]_{m × p}; where c_{ik}= \(\sum_{j=1}^{n}a_{ij}b_{jk}\)- A.(BC) = (AB).C
- A(B + C) = AB + AC
- (A + B)C = AC + BC

- If A= [a
_{ij}]_{m × n}, then A’ or AT = [a_{ji}]_{n × m}- (A’)’ = A
- (kA)’ = kA’
- (A + B)’ = A’ + B’
- (AB)’ = B’A’

- Some elementary operations:
- R
_{i}↔ R_{j}or C_{i}↔ C_{j} - R
_{i}→ kR_{i}or C_{i}→ kC_{i} - R
_{i}→ R_{i}+ kR_{j}or C_{i}→ C_{i}+ kC_{j}

- R
- A is said to known as a symmetric matrix if A′ = A
- A is said to be the skew symmetric matrix if A′ = –A

*Definition/Theorems*

- The determinant of a matrix A = [a
_{11}]_{1 × 1}can be given as: |a_{11}| = a_{11}. - For any square matrix A, the |A| will satisfy the following properties:
- |A′| = |A|, where A′ = transpose of A.
- If we interchange any two rows (or columns), then the sign of determinant changes.
- If any two rows or any two columns are identical or proportional, then the value of the determinant is zero.
- 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 (x
_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) 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 a
_{1}x + b_{1}y + c_{1}z = d_{1}a_{2}x + b_{2}y + c_{2}z = d_{2}a_{3}x + b_{3}y + c_{3}z = d_{3}, 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

*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:
- \((f\pm g) (x) = f (x)\pm g(x)\) is continuous.
- \((f.g)(x) = f (x) .g (x)\) is continuous.
- \(\frac{f}{g}(x) = \frac{f(x)}{g(x)}\) (whenever \(g(x)\neq 0\) is continuous.

- Chain Rule:
\(\frac{\mathrm{d} f}{\mathrm{d} x}=\frac{\mathrm{d} v}{\mathrm{d} t}.\frac{\mathrm{d} t}{\mathrm{d} x}\) - 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 |

\(\frac{\mathrm{d} }{\mathrm{d} x}(sin^{-1}x)\) | \(\frac{1}{\sqrt{1-x^2}}\) |

\(\frac{\mathrm{d} }{\mathrm{d} x}(cos^{-1}x)\) | \(-\frac{1}{\sqrt{1-x^2}}\) |

\(\frac{\mathrm{d} }{\mathrm{d} x}(tan^{-1}x)\) | \(\frac{1}{1+x^2}\) |

\(\frac{\mathrm{d} }{\mathrm{d} x}(cot^{-1}x)\) | \(\frac{-1}{1+x^2}\) |

\(\frac{\mathrm{d} }{\mathrm{d} x}(sec^{-1}x)\) | \(\frac{1}{x\sqrt{1-x^2}}\) |

\(\frac{\mathrm{d} }{\mathrm{d} x}(cosec^{-1}x)\) | \(\frac{-1}{x\sqrt{1-x^2}}\) |

\(\frac{\mathrm{d} }{\mathrm{d} x}(e^x)\) | \(e^x\) |

\(\frac{\mathrm{d} }{\mathrm{d} x}(log\:x)\) | \(\frac{1}{x}\) |

*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:
- \(\int [f(x)+g(x)]\:dx=\int f(x)\:dx+\int g(x)\:dx\)
- For any real number k, \(\int k\:f(x)\:dx=k\int f(x)\:dx\)
- \(\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*

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

- 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:
\(Area=\int_{a}^{c}[f(x)-g(x)]\:dx,+\int_{c}^{b}[g(x)-f(x)]\:dx\)

*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:

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

*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 l
^{2}+ m^{2}+ n^{2}= 1.

*Formulas*

- The Direction cosines of a line joining two points P (x
_{1}, y_{1}, z_{1}) and Q (x_{2}, y_{2}, z_{2}) are \(\frac{x_2-x_1}{PQ}\:,\:\frac{y_2-y_1}{PQ}\:,\frac{z_2-z_1}{PQ}\) wherePQ=\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

- Equation of a line through a point (x
_{1}, y_{1}, z_{1}) 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 (x
_{1}, y_{1}, z_{1}) is A (x – x_{1}) + B (y – y_{1}) + C (z – z_{1}) = 0 - The equation of a plane passing through three non-collinear points (x
_{1}, y_{1}, z_{1}); (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) 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 A
_{1}x + B_{1}y + C_{1}z + D_{1}= 0 and A_{2}x + B_{2}y + C_{2}z + D_{2}= 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 (x
_{1}, y_{1}, z_{1}) to the plane Ax + By + Cz + D = 0:\(\left | \frac{Ax_1+By_1+Cz_1+D}{\sqrt{A^2+B^2+C^2}} \right |\)

*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 E_{1}, E_{2}, …. , E_{n}be the partition of a sample space and A be any event; then,P(A) = P(E _{1}) P (A|E_{1}) + P (E_{2}) P (A|E_{2}) + … + P (E_{n}) . P(A|E_{n})

**Bayes Theorem:**If E_{1}, E_{2}, …. , 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 (X
^{2}) – [E(X)]^{2}

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Keeping a separate Maths formula notebook has several benefits for students. Let us check at some of the advantages below:

- The majority of mathematical problems are solved by using formulas; therefore, it is imperative that students know the formulas by heart.
- Solving formulas through the regular method of addition, multiplication, and subtraction would take a longer time.
- Formulas are like short equations that save students the time and energy to arrive at a particular solution.
- Formulas tell students the exact solution to a particular problem.
- They are simpler and faster in arriving at a result.
- By mastering formulas, students can solve the MCQs in competitive exams faster and more effectively.

**Q1: How many formulas are present in the class 12 CBSE Maths?Ans: **It is almost impossible to keep a record of all the formulas given in the Maths book of class 12 CBSE. As for every theory and concept shown in the book, there exist one or more formulas to help find the solutions for the given mathematical problems. The level of formulas increases with each grade making class 12 Mathematics the most difficult at the school level.

**Q2: Why is it critical to memorise Class 12th**

**Q3: 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.

**Q4: 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.

**Q5: Which is the best solution for NCERT Class 12 Mathematics?****Ans:** Students can find 100 per cent 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.

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

**Want help with more formulas? Check out some more formulas given below**.