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

Position Vector	$\overrightarrow{OP}=\overrightarrow{r}=\sqrt{x^2+y^2+z^2}$
Direction Ratios	$l=\frac{a}{r},m=\frac{b}{r},n=\frac{c}{r}$
Vector Addition	$\overrightarrow{PQ}+\overrightarrow{QR}=\overrightarrow{PR}$
Properties of Vector Addition	$CommutativeProperty\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}+\overrightarrow{a}$ $AssociativeProperty(\overrightarrow{a}+\overrightarrow{b})+\overrightarrow{c}=\overrightarrow{a}+(\overrightarrow{b}+\overrightarrow{c})$
Vector Joining Two Points	$\overrightarrow{P_1{P}_2}=\overrightarrow{O{P}_2}-\overrightarrow{O{P}_1}$
Skew Lines	$Cos\theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|$
Equation of a Line	$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$

Algebra Formulas For Class 12

If $\overrightarrow{a}=x\hat{i}+y\hat{j}+z\hat{k}$ then magnitude or length or norm or absolute value of $\overrightarrow{a}$ is $\left|\overrightarrow{a}\right|=a=\sqrt{x^2+y^2+z^2}$

A vector of unit magnitude is unit vector. If $\overrightarrow{a}$ is a vector then unit vector of $\overrightarrow{a}$ is denoted by $\hat{a}$ and $\hat{a}=\frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}$

Important unit vectors are $\hat{i},\hat{j},\hat{k}$ , where $\hat{i}=[1,0,0],\hat{j}=[0,1,0],\hat{k}=[0,0,1]$

If $l=\cos \alpha ,m=\cos \beta ,n=\cos \gamma ,$ then $\alpha ,\beta ,\gamma ,$ are called directional angles of the vectors $\overrightarrow{a}$ and ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$

In Vector Addition

$\overrightarrow{a}+\overrightarrow{b}=\overrightarrow{b}+\overrightarrow{a}$

$\overrightarrow{a}+(\overrightarrow{b}+\overrightarrow{c})=(\overrightarrow{a}+\overrightarrow{b})+\overrightarrow{c}$

$k(\overrightarrow{a}+\overrightarrow{b})=k\overrightarrow{a}+k\overrightarrow{b}$

$\overrightarrow{a}+\overrightarrow{0}=\overrightarrow{0}+\overrightarrow{a}$ , therefore $\overrightarrow{0}$ is the additive identity in vector addition.

$\overrightarrow{a}+(-\overrightarrow{a})=-\overrightarrow{a}+\overrightarrow{a}=\overrightarrow{0}$ , therefore $-\overrightarrow{a}$ is the inverse in vector addition.

Trigonometry Class 12 Formulas

Definition

$\theta ={\sin }^{-1}\left(x\right)isequivalenttox=\sin \theta$

$\theta ={\cos }^{-1}\left(x\right)isequivalenttox=\cos \theta$

$\theta ={\tan }^{-1}\left(x\right)isequivalenttox=\tan \theta$

Inverse Properties

$\sin ({\sin }^{-1}\left(x\right))=x$

$\cos ({\cos }^{-1}\left(x\right))=x$

$\tan ({\tan }^{-1}\left(x\right))=x$

${\sin }^{-1}(\sin \left(\theta \right))=\theta$

${\cos }^{-1}(\cos \left(\theta \right))=\theta$

${\tan }^{-1}(\tan \left(\theta \right))=\theta$

Double Angle and Half Angle Formulas

$\sin \left(2x\right)=2\sin x\cos x$

$\cos \left(2x\right)={\cos }^{2}x–{\sin }^{2}x$

$\tan \left(2x\right)=\frac{2\tan x}{1–{\tan }^{2}x}$

$\sin \frac{x}{2}=\pm \sqrt{\frac{1–\cos x}{2}}$

$\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}$

$\tan \frac{x}{2}=\frac{1-\cos x}{\sin x}=\frac{\sin x}{1–\cos x}$

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:

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

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:

Derivative	Formulas
$\frac{\mathrm{d}}{\mathrm{d}x}(si{n}^{-1}x)$	$\frac{1}{\sqrt{1-x^2}}$
$\frac{\mathrm{d}}{\mathrm{d}x}(co{s}^{-1}x)$	$-\frac{1}{\sqrt{1-x^2}}$
$\frac{\mathrm{d}}{\mathrm{d}x}(ta{n}^{-1}x)$	$\frac{1}{1+x^2}$
$\frac{\mathrm{d}}{\mathrm{d}x}(co{t}^{-1}x)$	$\frac{-1}{1+x^2}$
$\frac{\mathrm{d}}{\mathrm{d}x}(se{c}^{-1}x)$	$\frac{1}{x\sqrt{1-x^2}}$
$\frac{\mathrm{d}}{\mathrm{d}x}(cose{c}^{-1}x)$	$\frac{-1}{x\sqrt{1-x^2}}$
$\frac{\mathrm{d}}{\mathrm{d}x}(e^x)$	$e^x$
$\frac{\mathrm{d}}{\mathrm{d}x}(logx)$	$\frac{1}{x}$

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

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

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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