Maths Formulas for Class 12

Memorizing and Practicing with the Maths Formulas for every class 12 concept builds strong basics and confidence on the subject. Class 12 Maths is the most important pillar for higher studies. So, utilize the Class 12 Mathematics Formula Sheet and Tables provided here for a quick revision of all the concepts during exams. Here is the list of Maths formulas for Class 12 cover all related concepts formulas as per Latest Syllabus.

Class 12 Relations and Functions
Class 12 Inverse Trigonometric Functions
Class 12 Matrices
Class 12 Determinants
Class 12 Continuity and Differentiability
Class 12 Integrals
Class 12 Application of Integrals
Class 12 Vector Algebra
Class 12 Three Dimensional Geometry
Class 12 Probability

Vectors and Three Dimensional Geometry Formulas for Class 12

Algebra Formulas For Class 12

Trigonometry Class 12 Formulas

Important Maths Formulas for Entrance Exams:

3-Dimensional Coordinate Geometry Formulas
Addition of Vectors Formulas
Area Under The Curve Formulas
Binomial Theorem Formulas
Circle Formulas
Complex Number Formulas
Continuity Formulas
Correlation and Regression Formulas
Definite Integration Formulas
De-Moivre's Theorem and Euler Formulas
Determinants Formulas
Differential Equation Formulas
Differentiation Formulas
Ellipse Formulas
Exponential And Logarithmic Series Formulas
Function Formulas
Hyperbola Formulas
Hyperbolic Functions Formulas
Indefinite Integral Formulas
Limits Formulas
Linear Programming Problems Formulas
Logarithm and Modulus Function Formulas
Matrices Formulas
Maxima and Minima Formulas
Measure of Central Tendency and Dispersion Formulas
Monotonicity Formulas
Numerical Methods Formulas
Parabola Formulas
Permutations and Combinations Formulas
Points Formulas
Probability Formulas
Product of Vectors Formulas
Progression and Series Formulas
Quadratic Equation and Expressions Formulas
Separation of Real and Imaginary Parts Formulas
Statics and dynamics Formulas
Straight Line Formulas
Tangent and Normal Formulas

Maths Formulas For Class 12: Relations And Functions

Definition/Theorems

1. 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; for all ∀ 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. Equivalence relation R in X is a relation that shows all the reflexive, symmetric and transitive relations.

Properties

1. A function f: X → Y is one-one/injective; if f(x₁) = f(x₂) ⇒ x₁ = x₂ ∀ x₁ , x₂ ∈ X.
2. A function f: X → Y is onto/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. This can happen only if f is one-one and onto.
5. A binary operation $\ast$ performed on a set A is a function $\ast$ from A × A to A.
6. 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.
7. 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.
8. An operation $\ast$ on X is said to be commutative if a $\ast$ b = b $\ast$ a; ∀ a, b in X.
9. 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:

Formulas

1. $y=si{n}^{-1}x\Rightarrow x=siny$
2. $x=siny\Rightarrow y=si{n}^{-1}x$
3. $si{n}^{-1}\frac{1}{x}=cose{c}^{-1}x$
4. $co{s}^{-1}\frac{1}{x}=se{c}^{-1}x$
5. $ta{n}^{-1}\frac{1}{x}=co{t}^{-1}x$
6. $co{s}^{-1}(-x)=\pi -co{s}^{-1}x$
7. $co{t}^{-1}(-x)=\pi -co{t}^{-1}x$
8. $se{c}^{-1}(-x)=\pi -se{c}^{-1}x$
9. $si{n}^{-1}(-x)=-si{n}^{-1}x$
10. $ta{n}^{-1}(-x)=-ta{n}^{-1}x$
11. $cose{c}^{-1}(-x)=-cose{c}^{-1}x$
12. $ta{n}^{-1}x+co{t}^{-1}x=\frac{\pi }{2}$
13. $si{n}^{-1}x+co{s}^{-1}x=\frac{\pi }{2}$
14. $cose{c}^{-1}x+se{c}^{-1}x=\frac{\pi }{2}$
15. $ta{n}^{-1}x+ta{n}^{-1}y=ta{n}^{-1}\frac{x+y}{1-xy}$
16. $2ta{n}^{-1}x=si{n}^{-1}\frac{2x}{1+x^2}=co{s}^{-1}\frac{1-x^2}{1+x^2}$
17. $2ta{n}^{-1}x=ta{n}^{-1}\frac{2x}{1-x^2}$
18. $ta{n}^{-1}x+ta{n}^{-1}y=\pi +ta{n}^{-1}\left(\frac{x+y}{1-xy}\right)$; xy > 1; x, y > 0

Maths Formulas For Class 12: Matrices

Definition/Theorems

1. 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.
2. An m × n matrix will be known as a square matrix when m = n.
3. A = [a_ij]_{m × m} will be known as 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 will contain all its element 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 certain values of i and j

Elementary Operations

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 = $\sum _{j=1}^n{a}_{ij}{b}_{jk}$
   • (i) A.(BC) = (AB).C
   • (ii) A(B + C) = AB + AC
   • (iii) (A + B)C = AC + BC
3. 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’
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. A is said to known as a symmetric matrix if A′ = A
6. A is said to be the skew symmetric matrix if A′ = –A

Class 12 Maths Formulas: Determinants

Definition/Theorems

1. The determinant of a matrix A = [a₁₁]_{1 × 1} can be given as: |a₁₁| = a₁₁.
2. 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

1. Determinant of a matrix $A=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$ can be expanded as:
   |A| = $\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=a_1\left|\begin{array}{cc}{b}_2& c_2\\ b_3& c_3\end{array}\right|-b_1\left|\begin{array}{cc}{a}_2& c_2\\ a_3& c_3\end{array}\right|+c_1\left|\begin{array}{cc}{a}_2& b_2\\ a_3& b_3\end{array}\right|$
2. Area of triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is:
   ∆ = $\frac{1}{2}$$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$
3. Cofactor of aij of given by A_ij = (– 1)^{i+ j} M_ij
4. If A = $\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, then adj A = $\left[\begin{array}{ccc}{A}_{11}& A_{21}& A_{31}\\ A_{12}& A_{22}& A_{32}\\ A_{13}& A_{23}& A_{33}\end{array}\right]$ ; where A_ij is the cofactor of a_ij.
5. $A^{-1}=\frac{1}{|A|}(adjA)$
6. 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=$\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$, X = $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and B = $\left[\begin{array}{c}{d}_1\\ d_2\\ d_3\end{array}\right]$
7. 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

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.
2. 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 $g(x)\ne 0$ is continuous.
3. 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:
   $\frac{\mathrm{d}f}{\mathrm{d}x}=\frac{\mathrm{d}v}{\mathrm{d}t}.\frac{\mathrm{d}t}{\mathrm{d}x}$
4. 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.
5. 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
   $f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$

Formulas

Given below are the standard derivatives:

Class 12 Maths Formulas: Integrals

Definition/Properties

1. Integration is the inverse process of differentiation. Suppose, $\frac{\mathrm{d}}{\mathrm{d}x}F(x)=f(x)$; then we can write $\int f(x)dx=F(x)+C$
2. 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, $\int kf(x)dx=k\int f(x)dx$
   • (iii) $$\begin{gathered} \int [k_1{f}_1(x)+k_2{f}_2(x)+\dots +k_n{f}_n(x)]dx= \\ {k}_1\int f_1(x)dx+k_2\int f_2(x)dx+\dots +k_n\int f_n(x)dx \end{gathered}$$
3. First fundamental theorem of integral calculus: Let the area function be defined as: $A(x)={\int }_a^{x}f(x)dx$ for all $x\ge a$, where the function f is assumed to be continuous on [a, b]. Then A’ (x) = f (x) for every x ∈ [a, b].
4. 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

1. $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1$. Particularly, $\int dx=x+C)$
2. $\int cosxdx=sinx+C$
3. $\int sinxdx=-cosx+C$
4. $\int se{c}^{2}xdx=tanx+C$
5. $\int cose{c}^{2}xdx=-cotx+C$
6. $\int secxtanxdx=secx+C$
7. $\int cosecxcotxdx=-cosecx+C$
8. $\int \frac{dx}{\sqrt{1-x^2}}=si{n}^{-1}x+C$
9. $\int \frac{dx}{\sqrt{1-x^2}}=-co{s}^{-1}x+C$
10. $\int \frac{dx}{1+x^2}=ta{n}^{-1}x+C$
11. $\int \frac{dx}{1+x^2}=-co{t}^{-1}x+C$
12. $\int e^{x}dx=e^x+C$
13. $\int a^{x}dx=\frac{a^x}{loga}+C$
14. $\int \frac{dx}{x\sqrt{x^2-1}}=se{c}^{-1}x+C$
15. $\int \frac{dx}{x\sqrt{x^2-1}}=-cose{c}^{-1}x+C$
16. $\int \frac{1}{x}dx=log|x|+C$

Formulas – Partial Fractions

Formulas – Integration by Substitution

1. $\int tanxdx=log|secx|+C$
2. $\int cotxdx=log|sinx|+C$
3. $\int secxdx=log|secx+tanx|+C$
4. $\int cosecxdx=log|cosecx-cotx|+C$

Formulas – Integrals (Special Functions)

1. $\int \frac{dx}{x^2-a^2}=\frac{1}{2a}log\left|\frac{x-a}{x+a}\right|+C$
2. $\int \frac{dx}{a^2-x^2}=\frac{1}{2a}log\left|\frac{a+x}{a-x}\right|+C$
3. $\int \frac{dx}{x^2+a^2}=\frac{1}{a}ta{n}^{-1}\frac{x}{a}+C$
4. $\int \frac{dx}{\sqrt{x^2-a^2}}=log|x+\sqrt{x^2-a^2}|+C$
5. $\int \frac{dx}{\sqrt{x^2+a^2}}=log|x+\sqrt{x^2+a^2}|+C$
6. $\int \frac{dx}{\sqrt{x^2-a^2}}=si{n}^{-1}\frac{x}{a}+C$

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}
   $\int f_1(x).f_2(x)=f_1(x)\int f_2(x)dx-\int [\frac{\mathrm{d}}{\mathrm{d}x}{f}_1(x).\int f_2(x)dx]dx$
2. $\int e^x[f(x)+f^{\prime }(x)]dx=\int e^{x}f(x)dx+C$

Formulas – Special Integrals

1. $\int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}log|x+\sqrt{x^2-a^2}|+C$
2. $\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}log|x+\sqrt{x^2+a^2}|+C$
3. $\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a}{2}si{n}^{-1}\frac{x}{a}+C$
4. $a{x}^2+bx+c=a[x^2+\frac{b}{a}x+\frac{c}{a}]=a[{(x+\frac{b}{2a})}^2+(\frac{c}{a}-\frac{b^2}{4a^2})]$

Maths Formulas For Class 12: Application Of Integrals

1. 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:
   • $Area={\int }_a^{b}ydx={\int }_a^{b}f(x)dx$
2. 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:
   • $Area={\int }_c^{d}xdy={\int }_c^d\varphi (y)dy$
3. 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:
   • $Area={\int }_a^b[f(x)-g(x)]dx,where,f(x)\ge g(x)in[a,b]$
4. 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

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

Formulas

If two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ 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) $\overrightarrow{a}+\overrightarrow{b}=(a_1+b_1)\hat{i}+(a_2+b_2)\hat{j}+(a_3+b_3)\hat{k}$ ;
• (ii) $\lambda \overrightarrow{a}=(\lambda a_1)\hat{i}+(\lambda a_2)\hat{j}+(\lambda a_3)\hat{k}$ ;
• (iii) $\overrightarrow{a}.\overrightarrow{b}=(a_1{b}_1)+(a_2{b}_2)+(a_3{b}_3)$
• (iv) and $\overrightarrow{a}\times \overrightarrow{b}=\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ a_1& b_1& c_1\\ a_2& b_2& c_2\end{array}\right]$.

Maths Formulas For Class 12: 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

1. 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=$\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$
2. Equation of a line through a point (x₁ , y₁ , z₁ ) and having direction cosines l, m, n is: $\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$
3. The vector equation of a line which passes through two points whose position vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\overrightarrow{r}=\overrightarrow{a}+\lambda (\overrightarrow{b}-\overrightarrow{a})$
4. The shortest distance between $\overrightarrow{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}$ and $\overrightarrow{r}=\overrightarrow{a_2}+\mu \overrightarrow{b_2}$ is:
   • $\left|\frac{(\overrightarrow{b_1}\times \overrightarrow{b_2}).(\overrightarrow{a_2}-\overrightarrow{a_1})}{|\overrightarrow{b_1}\times \overrightarrow{b_2}|}\right|$
5. The distance between parallel lines $\overrightarrow{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b}$ and $\overrightarrow{r}=\overrightarrow{a_2}+\mu \overrightarrow{b}$ is
   • $\left|\frac{\overrightarrow{b}\times (\overrightarrow{a_2}-\overrightarrow{a_1})}{|\overrightarrow{b}|}\right|$
6. The equation of a plane through a point whose position vector is $\overrightarrow{a}$ and perpendicular to the vector $\overrightarrow{N}$ is $(\overrightarrow{r}-\overrightarrow{a}).\overrightarrow{N}=0$
7. 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
8. The equation of a plane passing through three non-collinear points (x₁ , y₁ , z₁); (x₂ , y₂ , z₂) and (x₃ , y₃ , z₃) is:
   • $\left|\begin{array}{ccc}x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1\end{array}\right|=0$
9. The two lines $\overrightarrow{r}=\overrightarrow{a_1}+\lambda \overrightarrow{b_1}$ and $\overrightarrow{r}=\overrightarrow{a_2}+\mu \overrightarrow{b_2}$ are coplanar if:
   • $(\overrightarrow{a_2}-\overrightarrow{a_1}).(\overrightarrow{b_1}\times \overrightarrow{b_2})=0$
10. The angle φ between the line $\overrightarrow{r}=\overrightarrow{a}+\lambda \overrightarrow{b}$ and the plane $\overrightarrow{r}.\hat{n}=d$ is given by:
    • $sin\varphi =\left|\frac{\overrightarrow{b}.\hat{n}}{|\overrightarrow{b}||\hat{n}|}\right|$
11. The angle θ between the planes A₁x + B₁y + C₁z + D₁ = 0 and A₂x + B₂y + C₂z + D₂ = 0 is given by:
    • $cos\theta =\left|\frac{A_1{A}_2+B_1{B}_2+C_1{C}_2}{\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}\right|$
12. The distance of a point whose position vector is $\overrightarrow{a}$ from the plane $\overrightarrow{r}.\hat{n}=d$ is given by: $|d-\overrightarrow{a}.\hat{n}|$
13. The distance from a point (x₁ , y₁ , z₁) to the plane Ax + By + Cz + D = 0:
    • $\left|\frac{A{x}_1+B{y}_1+C{z}_1+D}{\sqrt{A^2+B^2+C^2}}\right|$

Class 12 Maths Formulas: Probability

Definition/Properties

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

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