**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

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

### 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(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.

### 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

- (i) 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}= \(\sum_{j=1}^{n}a_{ij}b_{jk}\)- (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}

- (i) 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

### Class 12 Maths Formulas: Determinants

*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:
- (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 (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
- (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 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:\(\left | \frac{Ax_1+By_1+Cz_1+D}{\sqrt{A^2+B^2+C^2}} \right |\)

### 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 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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**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 free **Class 12 Maths questions** of Embibe which will help you a lot. Make the best use of these resources and master the subject.

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

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Trigonometric Table | Differentiation Formulas |

Maths Formulas For Class 11 | Class 11 & 12 Maths Practice Questions |

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