# Maths Formulas for Class 12

Math Formulas can be quite tricky to understand and memorize. Some may easily remember by using various tricks and some may be stuck at the beginning only. To help students, we have curated Maths Formulas for all classes under one roof. Here, we have gathered all basic maths formulas for class 12 concepts in the form of a list, sheet.

You can utilize this list as a go-to sheet whenever you solve the basic to complex problems using Maths formulas. So, we recommend you to learn and understand the important class 12 maths formulas prevailing here & solve all concepts problems at a faster pace.

## List of Maths Formulas for 12th | Cheat Sheet of Class 12 Maths Formulas

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.

### Vectors and Three Dimensional Geometry Formulas for Class 12

 Position Vector $\stackrel{\to }{OP}=\stackrel{\to }{r}=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$ Direction Ratios $l=\frac{a}{r},m=\frac{b}{r},n=\frac{c}{r}$ Vector Addition $\stackrel{\to }{PQ}+\stackrel{\to }{QR}=\stackrel{\to }{PR}$ Properties of Vector Addition $AssociativeProperty\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)+\stackrel{\to }{c}=\stackrel{\to }{a}+\left(\stackrel{\to }{b}+\stackrel{\to }{c}\right)$ Vector Joining Two Points $\stackrel{\to }{{P}_{1}{P}_{2}}=\stackrel{\to }{O{P}_{2}}-\stackrel{\to }{O{P}_{1}}$ Skew Lines $Cos\theta =|\frac{{a}_{1}{a}_{2}+{b}_{1}{b}_{2}+{c}_{1}{c}_{2}}{\sqrt{{a}_{1}^{2}+{b}_{1}^{2}+{c}_{1}^{2}}\sqrt{{a}_{2}^{2}+{b}_{2}^{2}+{c}_{2}^{2}}}|$ 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 $\stackrel{\to }{a}=x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}$ then magnitude or length or norm or absolute value of $\stackrel{\to }{a}$ is $|\stackrel{\to }{a}|=a=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$ A vector of unit magnitude is unit vector. If $\stackrel{\to }{a}$ is a vector then unit vector of $\stackrel{\to }{a}$ is denoted by $\stackrel{^}{a}$ and $\stackrel{^}{a}=\frac{\stackrel{\to }{a}}{|\stackrel{\to }{a}|}$ Important unit vectors are $\stackrel{^}{i},\stackrel{^}{j},\stackrel{^}{k}$ , where $\stackrel{^}{i}=\left[1,0,0\right],\phantom{\rule{0.222em}{0ex}}\stackrel{^}{j}=\left[0,1,0\right],\phantom{\rule{0.222em}{0ex}}\stackrel{^}{k}=\left[0,0,1\right]$ If $l=\mathrm{cos}\alpha ,m=\mathrm{cos}\beta ,n=\mathrm{cos}\gamma ,$ then $\alpha ,\beta ,\gamma ,$ are called directional angles of the vectors $\stackrel{\to }{a}$ and ${\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\beta +{\mathrm{cos}}^{2}\gamma =1$
 In Vector Addition $\stackrel{\to }{a}+\stackrel{\to }{b}=\stackrel{\to }{b}+\stackrel{\to }{a}$ $\stackrel{\to }{a}+\left(\stackrel{\to }{b}+\stackrel{\to }{c}\right)=\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)+\stackrel{\to }{c}$ $k\left(\stackrel{\to }{a}+\stackrel{\to }{b}\right)=k\stackrel{\to }{a}+k\stackrel{\to }{b}$ $\stackrel{\to }{a}+\stackrel{\to }{0}=\stackrel{\to }{0}+\stackrel{\to }{a}$ , therefore $\stackrel{\to }{0}$ is the additive identity in vector addition. $\stackrel{\to }{a}+\left(-\stackrel{\to }{a}\right)=-\stackrel{\to }{a}+\stackrel{\to }{a}=\stackrel{\to }{0}$ , therefore $-\stackrel{\to }{a}$ is the inverse in vector addition.

### Trigonometry Class 12 Formulas

 Definition $\theta ={\mathrm{sin}}^{-1}\left(x\right)\phantom{\rule{0.167em}{0ex}}is\phantom{\rule{0.167em}{0ex}}equivalent\phantom{\rule{0.167em}{0ex}}to\phantom{\rule{0.167em}{0ex}}x=\mathrm{sin}\theta$ $\theta ={\mathrm{cos}}^{-1}\left(x\right)\phantom{\rule{0.167em}{0ex}}is\phantom{\rule{0.167em}{0ex}}equivalent\phantom{\rule{0.167em}{0ex}}to\phantom{\rule{0.167em}{0ex}}x=\mathrm{cos}\theta$ $\theta ={\mathrm{tan}}^{-1}\left(x\right)\phantom{\rule{0.167em}{0ex}}is\phantom{\rule{0.167em}{0ex}}equivalent\phantom{\rule{0.167em}{0ex}}to\phantom{\rule{0.167em}{0ex}}x=\mathrm{tan}\theta$ Inverse Properties $\mathrm{sin}\left({\mathrm{sin}}^{-1}\left(x\right)\right)=x$ $\mathrm{cos}\left({\mathrm{cos}}^{-1}\left(x\right)\right)=x$ $\mathrm{tan}\left({\mathrm{tan}}^{-1}\left(x\right)\right)=x$ ${\mathrm{sin}}^{-1}\left(\mathrm{sin}\left(\theta \right)\right)=\theta$ ${\mathrm{cos}}^{-1}\left(\mathrm{cos}\left(\theta \right)\right)=\theta$ ${\mathrm{tan}}^{-1}\left(\mathrm{tan}\left(\theta \right)\right)=\theta$ Double Angle and Half Angle Formulas $\mathrm{sin}\left(2x\right)=2\phantom{\rule{0.167em}{0ex}}\mathrm{sin}\phantom{\rule{0.167em}{0ex}}x\phantom{\rule{0.167em}{0ex}}\mathrm{cos}\phantom{\rule{0.167em}{0ex}}x$ $\mathrm{cos}\left(2x\right)={\mathrm{cos}}^{2}x–{\mathrm{sin}}^{2}x$ $\mathrm{tan}\left(2x\right)=\frac{2\phantom{\rule{0.167em}{0ex}}\mathrm{tan}\phantom{\rule{0.167em}{0ex}}x}{1–{\mathrm{tan}}^{2}x}$ $\mathrm{sin}\frac{x}{2}=±\sqrt{\frac{1–\mathrm{cos}x}{2}}$ $\mathrm{cos}\frac{x}{2}=±\sqrt{\frac{1+\mathrm{cos}x}{2}}$ $\mathrm{tan}\frac{x}{2}=\frac{1-\mathrm{cos}\phantom{\rule{0.167em}{0ex}}x}{\mathrm{sin}\phantom{\rule{0.167em}{0ex}}x}=\frac{\mathrm{sin}\phantom{\rule{0.167em}{0ex}}x}{1–\mathrm{cos}\phantom{\rule{0.167em}{0ex}}x}$

Important Maths Formulas for Entrance Exams:

### 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 = IX and fog = IY. 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] $\left[\frac{-\pi }{2},\frac{\pi }{2}\right]$ 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

1. $y=si{n}^{-1}x⇒x=sin\phantom{\rule{0.222em}{0ex}}y$

2. $x=sin\phantom{\rule{0.222em}{0ex}}y⇒y=si{n}^{-1}x$

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. $co{s}^{-1}\left(-x\right)=\pi -co{s}^{-1}x$

7. $co{t}^{-1}\left(-x\right)=\pi -co{t}^{-1}x$

8. $se{c}^{-1}\left(-x\right)=\pi -se{c}^{-1}x$

9. $si{n}^{-1}\left(-x\right)=-si{n}^{-1}x$

10. $ta{n}^{-1}\left(-x\right)=-ta{n}^{-1}x$

11. $cose{c}^{-1}\left(-x\right)=-cose{c}^{-1}x$

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. $2\phantom{\rule{0.222em}{0ex}}ta{n}^{-1}x=si{n}^{-1}\frac{2x}{1+{x}^{2}}=co{s}^{-1}\frac{1-{x}^{2}}{1+{x}^{2}}$

17. $2\phantom{\rule{0.222em}{0ex}}ta{n}^{-1}x=ta{n}^{-1}\frac{2x}{1-{x}^{2}}$

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 = [aij]m × m will be known as diagonal matrix if aij = 0, when i ≠ j.
4. A = [aij]n × n is a scalar matrix if aij = 0, when i ≠ j, aij = k, (where k is some constant); and i = j.
5. A = [aij]n × n is an identity matrix, if aij = 1, when i = j and aij = 0, when i ≠ j.
6. A zero matrix will contain all its element as zero.
7. A = [aij] = [bij] = B if and only if:
• (i) A and B are of the same order
• (ii) aij = bij for all the certain values of i and j

Elementary Operations

1. Some basic operations of matrices:
• (i) kA = k[aij]m × n = [k(aij)]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 = [aij]m × n and B = [bjk]n × p, then
AB = C = [cik]m × p ; where cik =
$\sum _{j=1}^{n}{a}_{ij}{b}_{jk}$

• (i) A.(BC) = (AB).C
• (ii) A(B + C) = AB + AC
• (iii) (A + B)C = AC + BC
3. If A= [aij]m × n, then A’ or AT = [aji]n × m
• (i) (A’)’ = A
• (ii) (kA)’ = kA’
• (iii) (A + B)’ = A’ + B’
• (iv) (AB)’ = B’A’
4. Some elementary operations:
• (i) Ri ↔ Rj or Ci ↔ Cj
• (ii) Ri → kRi or Ci → kCi
• (iii) Ri → Ri + kRj or Ci → Ci + kCj
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| =

$|\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}\\ {a}_{3}& {b}_{3}& {c}_{3}\end{array}|={a}_{1}|\begin{array}{cc}{b}_{2}& {c}_{2}\\ {b}_{3}& {c}_{3}\end{array}|-{b}_{1}|\begin{array}{cc}{a}_{2}& {c}_{2}\\ {a}_{3}& {c}_{3}\end{array}|+{c}_{1}|\begin{array}{cc}{a}_{2}& {b}_{2}\\ {a}_{3}& {b}_{3}\end{array}|$

2. Area of triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is:
∆ =
$\frac{1}{2}$

$|\begin{array}{ccc}{x}_{1}& {y}_{1}& 1\\ {x}_{2}& {y}_{2}& 1\\ {x}_{3}& {y}_{3}& 1\end{array}|$

3. Cofactor of aij of given by Aij = (– 1)i+ j Mij
4. If A =
$\left[\begin{array}{ccc}{a}_{11}& {a}_{12}& {a}_{13}\\ {a}_{21}& {a}_{22}& {a}_{23}\\ {a}_{31}& {a}_{32}& {a}_{33}\end{array}\right]$

$\left[\begin{array}{ccc}{A}_{11}& {A}_{21}& {A}_{31}\\ {A}_{12}& {A}_{22}& {A}_{32}\\ {A}_{13}& {A}_{23}& {A}_{33}\end{array}\right]$

; where Aij is the cofactor of aij.

5. ${A}^{-1}=\frac{1}{|A|}\left(adj\phantom{\rule{0.222em}{0ex}}A\right)$

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)
$\left(f±g\right)\left(x\right)=f\left(x\right)±g\left(x\right)$

is continuous.

• (ii)
$\left(f.g\right)\left(x\right)=f\left(x\right).g\left(x\right)$

is continuous.

• (iii)
$\frac{f}{g}\left(x\right)=\frac{f\left(x\right)}{g\left(x\right)}$

(whenever

$g\left(x\right)\ne 0$

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

${f}^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Formulas

Given below are the standard derivatives:

 Derivative Formulas $\frac{\mathrm{d}}{\mathrm{d}x}\left(si{n}^{-1}x\right)$ $\frac{1}{\sqrt{1-{x}^{2}}}$ $\frac{\mathrm{d}}{\mathrm{d}x}\left(co{s}^{-1}x\right)$ $-\frac{1}{\sqrt{1-{x}^{2}}}$ $\frac{\mathrm{d}}{\mathrm{d}x}\left(ta{n}^{-1}x\right)$ $\frac{1}{1+{x}^{2}}$ $\frac{\mathrm{d}}{\mathrm{d}x}\left(co{t}^{-1}x\right)$ $\frac{-1}{1+{x}^{2}}$ $\frac{\mathrm{d}}{\mathrm{d}x}\left(se{c}^{-1}x\right)$ $\frac{1}{x\sqrt{1-{x}^{2}}}$ $\frac{\mathrm{d}}{\mathrm{d}x}\left(cose{c}^{-1}x\right)$ $\frac{-1}{x\sqrt{1-{x}^{2}}}$ $\frac{\mathrm{d}}{\mathrm{d}x}\left({e}^{x}\right)$ ${e}^{x}$ $\frac{\mathrm{d}}{\mathrm{d}x}\left(log\phantom{\rule{0.222em}{0ex}}x\right)$ $\frac{1}{x}$

### Class 12 Maths Formulas: Integrals

Definition/Properties

1. Integration is the inverse process of differentiation. Suppose,
$\frac{\mathrm{d}}{\mathrm{d}x}F\left(x\right)=f\left(x\right)$

; then we can write

$\int f\left(x\right)\phantom{\rule{0.222em}{0ex}}dx=F\left(x\right)+C$

2. Properties of indefinite integrals:
• (i)
$\int \left[f\left(x\right)+g\left(x\right)\right]\phantom{\rule{0.222em}{0ex}}dx=\int f\left(x\right)\phantom{\rule{0.222em}{0ex}}dx+\int g\left(x\right)\phantom{\rule{0.222em}{0ex}}dx$

• (ii) For any real number k,
$\int k\phantom{\rule{0.222em}{0ex}}f\left(x\right)\phantom{\rule{0.222em}{0ex}}dx=k\int f\left(x\right)\phantom{\rule{0.222em}{0ex}}dx$

• (iii)
$\int \left[{k}_{1}\phantom{\rule{0.222em}{0ex}}{f}_{1}\left(x\right)+{k}_{2}\phantom{\rule{0.222em}{0ex}}{f}_{2}\left(x\right)+\dots +{k}_{n}\phantom{\rule{0.222em}{0ex}}{f}_{n}\left(x\right)\right]\phantom{\rule{0.222em}{0ex}}dx=\phantom{\rule{0ex}{0ex}}{k}_{1}\int {f}_{1}\left(x\right)\phantom{\rule{0.222em}{0ex}}dx+{k}_{2}\int {f}_{2}\left(x\right)\phantom{\rule{0.222em}{0ex}}dx+\dots +{k}_{n}\int {f}_{n}\left(x\right)\phantom{\rule{0.222em}{0ex}}dx$

3. First fundamental theorem of integral calculus: Let the area function be defined as:
$A\left(x\right)={\int }_{a}^{x}f\left(x\right)\phantom{\rule{0.222em}{0ex}}dx$

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:
$\frac{\mathrm{d}}{\mathrm{d}x}F\left(x\right)=f\left(x\right)$

for every x falling in the domain of f; then,

${\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.222em}{0ex}}dx=\left[F\left(x\right)+C{\right]}_{a}^{b}=F\left(b\right)-F\left(a\right)$

Formulas – Standard Integrals

1. $\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C,n\ne -1$

. Particularly,

$\int dx=x+C\right)$

2. $\int cos\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}dx=sin\phantom{\rule{0.222em}{0ex}}x+C$

3. $\int sin\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}dx=-cos\phantom{\rule{0.222em}{0ex}}x+C$

4. $\int se{c}^{2}x\phantom{\rule{0.222em}{0ex}}dx=tan\phantom{\rule{0.222em}{0ex}}x+C$

5. $\int cose{c}^{2}x\phantom{\rule{0.222em}{0ex}}dx=-cot\phantom{\rule{0.222em}{0ex}}x+C$

6. $\int sec\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}tan\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}dx=sec\phantom{\rule{0.222em}{0ex}}x+C$

7. $\int cosec\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}cot\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}dx=-cosec\phantom{\rule{0.222em}{0ex}}x+C$

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. $\int {a}^{x}dx=\frac{{a}^{x}}{log\phantom{\rule{0.222em}{0ex}}a}+C$

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. $\int \frac{1}{x}\phantom{\rule{0.222em}{0ex}}dx=log\phantom{\rule{0.222em}{0ex}}|x|+C$

Formulas – Partial Fractions

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

Formulas – Integration by Substitution

1. $\int tan\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}dx=log\phantom{\rule{0.222em}{0ex}}|sec\phantom{\rule{0.222em}{0ex}}x|+C$

2. $\int cot\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}dx=log\phantom{\rule{0.222em}{0ex}}|sin\phantom{\rule{0.222em}{0ex}}x|+C$

3. $\int sec\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}dx=log\phantom{\rule{0.222em}{0ex}}|sec\phantom{\rule{0.222em}{0ex}}x+tan\phantom{\rule{0.222em}{0ex}}x|+C$

4. $\int cosec\phantom{\rule{0.222em}{0ex}}x\phantom{\rule{0.222em}{0ex}}dx=log\phantom{\rule{0.222em}{0ex}}|cosec\phantom{\rule{0.222em}{0ex}}x-cot\phantom{\rule{0.222em}{0ex}}x|+C$

Formulas – Integrals (Special Functions)

1. $\int \frac{dx}{{x}^{2}-{a}^{2}}=\frac{1}{2a}\phantom{\rule{0.222em}{0ex}}log\phantom{\rule{0.222em}{0ex}}|\frac{x-a}{x+a}|+C$

2. $\int \frac{dx}{{a}^{2}-{x}^{2}}=\frac{1}{2a}\phantom{\rule{0.222em}{0ex}}log\phantom{\rule{0.222em}{0ex}}|\frac{a+x}{a-x}|+C$

3. $\int \frac{dx}{{x}^{2}+{a}^{2}}=\frac{1}{a}\phantom{\rule{0.222em}{0ex}}ta{n}^{-1}\frac{x}{a}+C$

4. $\int \frac{dx}{\sqrt{{x}^{2}-{a}^{2}}}=log\phantom{\rule{0.222em}{0ex}}|x+\sqrt{{x}^{2}-{a}^{2}}|+C$

5. $\int \frac{dx}{\sqrt{{x}^{2}+{a}^{2}}}=log\phantom{\rule{0.222em}{0ex}}|x+\sqrt{{x}^{2}+{a}^{2}}|+C$

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}

$\int {f}_{1}\left(x\right).{f}_{2}\left(x\right)={f}_{1}\left(x\right)\int {f}_{2}\left(x\right)\phantom{\rule{0.222em}{0ex}}dx-\int \left[\frac{\mathrm{d}}{\mathrm{d}x}{f}_{1}\left(x\right).\int {f}_{2}\left(x\right)\phantom{\rule{0.222em}{0ex}}dx\right]dx$

2. $\int {e}^{x}\left[f\left(x\right)+{f}^{\prime }\left(x\right)\right]\phantom{\rule{0.222em}{0ex}}dx=\int {e}^{x}\phantom{\rule{0.222em}{0ex}}f\left(x\right)\phantom{\rule{0.222em}{0ex}}dx+C$

Formulas – Special Integrals

1. $\int \sqrt{{x}^{2}-{a}^{2}}\phantom{\rule{0.222em}{0ex}}dx=\frac{x}{2}\sqrt{{x}^{2}-{a}^{2}}-\frac{{a}^{2}}{2}\phantom{\rule{0.222em}{0ex}}log|x+\sqrt{{x}^{2}-{a}^{2}}|+C$

2. $\int \sqrt{{x}^{2}+{a}^{2}}\phantom{\rule{0.222em}{0ex}}dx=\frac{x}{2}\sqrt{{x}^{2}+{a}^{2}}+\frac{{a}^{2}}{2}\phantom{\rule{0.222em}{0ex}}log|x+\sqrt{{x}^{2}+{a}^{2}}|+C$

3. $\int \sqrt{{a}^{2}-{x}^{2}}\phantom{\rule{0.222em}{0ex}}dx=\frac{x}{2}\sqrt{{a}^{2}-{x}^{2}}+\frac{a}{2}\phantom{\rule{0.222em}{0ex}}si{n}^{-1}\frac{x}{a}+C$

4. $a{x}^{2}+bx+c=a\left[{x}^{2}+\frac{b}{a}x+\frac{c}{a}\right]=a\left[{\left(x+\frac{b}{2a}\right)}^{2}+\left(\frac{c}{a}-\frac{{b}^{2}}{4{a}^{2}}\right)\right]$

### 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:

• $Area={\int }_{a}^{b}y\phantom{\rule{0.222em}{0ex}}dx={\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.222em}{0ex}}dx$

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:

• $Area={\int }_{c}^{d}x\phantom{\rule{0.222em}{0ex}}dy={\int }_{c}^{d}\varphi \left(y\right)\phantom{\rule{0.222em}{0ex}}dy$

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:

• $Area={\int }_{a}^{b}\left[f\left(x\right)-g\left(x\right)\right]\phantom{\rule{0.222em}{0ex}}dx,\phantom{\rule{0.222em}{0ex}}where,f\left(x\right)\ge g\left(x\right)\phantom{\rule{0.222em}{0ex}}in\phantom{\rule{0.222em}{0ex}}\left[a,b\right]$

4. If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in [c, b], a < c < b, then:

• $Area={\int }_{a}^{c}\left[f\left(x\right)-g\left(x\right)\right]\phantom{\rule{0.222em}{0ex}}dx,+{\int }_{c}^{b}\left[g\left(x\right)-f\left(x\right)\right]\phantom{\rule{0.222em}{0ex}}dx$

### 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:

$\stackrel{\to }{OP}\left(=\stackrel{\to }{r}\right)=x\stackrel{^}{i}+y\stackrel{^}{j}+z\stackrel{^}{k}$

2. The scalar product of two given vectors
$\stackrel{\to }{a}$

and

$\stackrel{\to }{b}$

having angle θ between them is defined as:

• $\stackrel{\to }{a}\phantom{\rule{0.222em}{0ex}}.\phantom{\rule{0.222em}{0ex}}\stackrel{\to }{b}=|\stackrel{\to }{a}||\stackrel{\to }{b}|\phantom{\rule{0.222em}{0ex}}cos\phantom{\rule{0.222em}{0ex}}\theta$

3. The position vector of a point R dividing a line segment joining the points P and Q whose position vectors
$\stackrel{\to }{a}$

and

$\stackrel{\to }{b}$

are respectively, in the ratio m : n is given by:

• (i) internally:
$\frac{n\stackrel{\to }{a}+m\stackrel{\to }{b}}{m+n}$

• (ii) externally:
$\frac{n\stackrel{\to }{a}-m\stackrel{\to }{b}}{m-n}$

Formulas

If two vectors

$\stackrel{\to }{a}$

and

$\stackrel{\to }{b}$

are given in its component forms as

$\stackrel{^}{a}={a}_{1}\stackrel{^}{i}+{a}_{2}\stackrel{^}{j}+{a}_{3}\stackrel{^}{k}$

and

$\stackrel{^}{b}={b}_{1}\stackrel{^}{i}+{b}_{2}\stackrel{^}{j}+{b}_{3}\stackrel{^}{k}$

and λ as the scalar part; then:

• (i)
$\stackrel{\to }{a}+\stackrel{\to }{b}=\left({a}_{1}+{b}_{1}\right)\stackrel{^}{i}+\left({a}_{2}+{b}_{2}\right)\stackrel{^}{j}+\left({a}_{3}+{b}_{3}\right)\stackrel{^}{k}$

;

• (ii)
$\lambda \stackrel{\to }{a}=\left(\lambda {a}_{1}\right)\stackrel{^}{i}+\left(\lambda {a}_{2}\right)\stackrel{^}{j}+\left(\lambda {a}_{3}\right)\stackrel{^}{k}$

;

• (iii)
$\stackrel{\to }{a}\phantom{\rule{0.222em}{0ex}}.\phantom{\rule{0.222em}{0ex}}\stackrel{\to }{b}=\left({a}_{1}{b}_{1}\right)+\left({a}_{2}{b}_{2}\right)+\left({a}_{3}{b}_{3}\right)$

• (iv) and
$\stackrel{\to }{a}×\stackrel{\to }{b}=\left[\begin{array}{ccc}\stackrel{^}{i}& \stackrel{^}{j}& \stackrel{^}{k}\\ {a}_{1}& {b}_{1}& {c}_{1}\\ {a}_{2}& {b}_{2}& {c}_{2}\end{array}\right]$

.

### 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
$\frac{{x}_{2}-{x}_{1}}{PQ}\phantom{\rule{0.222em}{0ex}},\phantom{\rule{0.222em}{0ex}}\frac{{y}_{2}-{y}_{1}}{PQ}\phantom{\rule{0.222em}{0ex}},\frac{{z}_{2}-{z}_{1}}{PQ}$

where

• PQ=
$\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}+\left({z}_{2}-{z}_{1}{\right)}^{2}}$

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
$\stackrel{\to }{a}$

and

$\stackrel{\to }{b}$

is

$\stackrel{\to }{r}=\stackrel{\to }{a}+\lambda \left(\stackrel{\to }{b}-\stackrel{\to }{a}\right)$

4. The shortest distance between
$\stackrel{\to }{r}=\stackrel{\to }{{a}_{1}}+\lambda \phantom{\rule{0.222em}{0ex}}\stackrel{\to }{{b}_{1}}$

and

$\stackrel{\to }{r}=\stackrel{\to }{{a}_{2}}+\mu \phantom{\rule{0.222em}{0ex}}\stackrel{\to }{{b}_{2}}$

is:

• $|\frac{\left(\stackrel{\to }{{b}_{1}}×\stackrel{\to }{{b}_{2}}\right).\left(\stackrel{\to }{{a}_{2}}-\stackrel{\to }{{a}_{1}}\right)}{|\stackrel{\to }{{b}_{1}}×\stackrel{\to }{{b}_{2}}|}|$

5. The distance between parallel lines
$\stackrel{\to }{r}=\stackrel{\to }{{a}_{1}}+\lambda \phantom{\rule{0.222em}{0ex}}\stackrel{\to }{b}$

and

$\stackrel{\to }{r}=\stackrel{\to }{{a}_{2}}+\mu \phantom{\rule{0.222em}{0ex}}\stackrel{\to }{b}$

is

• $|\frac{\stackrel{\to }{b}×\left(\stackrel{\to }{{a}_{2}}-\stackrel{\to }{{a}_{1}}\right)}{|\stackrel{\to }{b}|}|$

6. The equation of a plane through a point whose position vector is
$\stackrel{\to }{a}$

and perpendicular to the vector

$\stackrel{\to }{N}$

is

$\left(\stackrel{\to }{r}-\stackrel{\to }{a}\right)\phantom{\rule{0.222em}{0ex}}.\phantom{\rule{0.222em}{0ex}}\stackrel{\to }{N}=0$

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:

• $|\begin{array}{ccc}x-{x}_{1}& y-{y}_{1}& z-{z}_{1}\\ {x}_{2}-{x}_{1}& {y}_{2}-{y}_{1}& {z}_{2}-{z}_{1}\\ {x}_{3}-{x}_{1}& {y}_{3}-{y}_{1}& {z}_{3}-{z}_{1}\end{array}|=0$

9. The two lines
$\stackrel{\to }{r}=\stackrel{\to }{{a}_{1}}+\lambda \phantom{\rule{0.222em}{0ex}}\stackrel{\to }{{b}_{1}}$

and

$\stackrel{\to }{r}=\stackrel{\to }{{a}_{2}}+\mu \phantom{\rule{0.222em}{0ex}}\stackrel{\to }{{b}_{2}}$

are coplanar if:

• $\left(\stackrel{\to }{{a}_{2}}-\stackrel{\to }{{a}_{1}}\right)\phantom{\rule{0.222em}{0ex}}.\phantom{\rule{0.222em}{0ex}}\left(\stackrel{\to }{{b}_{1}}×\stackrel{\to }{{b}_{2}}\right)=0$

10. The angle φ between the line
$\stackrel{\to }{r}=\stackrel{\to }{a}+\lambda \phantom{\rule{0.222em}{0ex}}\stackrel{\to }{b}$

and the plane

$\stackrel{\to }{r}\phantom{\rule{0.222em}{0ex}}.\phantom{\rule{0.222em}{0ex}}\stackrel{^}{n}=d$

is given by:

• $sin\phantom{\rule{0.222em}{0ex}}\varphi =|\frac{\stackrel{\to }{b}\phantom{\rule{0.222em}{0ex}}.\phantom{\rule{0.222em}{0ex}}\stackrel{^}{n}}{|\stackrel{\to }{b}||\stackrel{^}{n}|}|$

11. The angle θ between the planes A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 is given by:

• $cos\phantom{\rule{0.222em}{0ex}}\theta =|\frac{{A}_{1}\phantom{\rule{0.222em}{0ex}}{A}_{2}+{B}_{1}\phantom{\rule{0.222em}{0ex}}{B}_{2}+{C}_{1}\phantom{\rule{0.222em}{0ex}}{C}_{2}}{\sqrt{{A}_{1}^{2}+{B}_{1}^{2}+{C}_{1}^{2}}\phantom{\rule{0.222em}{0ex}}\sqrt{{A}_{2}^{2}+{B}_{2}^{2}+{C}_{2}^{2}}}|$

12. The distance of a point whose position vector is
$\stackrel{\to }{a}$

from the plane

$\stackrel{\to }{r}\phantom{\rule{0.222em}{0ex}}.\phantom{\rule{0.222em}{0ex}}\stackrel{^}{n}=d$

is given by:

$|d-\stackrel{\to }{a}\phantom{\rule{0.222em}{0ex}}.\phantom{\rule{0.222em}{0ex}}\stackrel{^}{n}|$

13. The distance from a point (x1 , y1 , z1) to the plane Ax + By + Cz + D = 0:

• $|\frac{A{x}_{1}+B{y}_{1}+C{z}_{1}+D}{\sqrt{{A}^{2}+{B}^{2}+{C}^{2}}}|$

### 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:

• $P\left(E\phantom{\rule{0.222em}{0ex}}|\phantom{\rule{0.222em}{0ex}}F\right)=\frac{E\cap F}{P\left(F\right)}\phantom{\rule{0.222em}{0ex}},\phantom{\rule{0.222em}{0ex}}P\left(F\right)\ne 0$

2. Total Probability: Let E1 , E2 , …. , En 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 (En) . P(A|En)
3. Bayes Theorem: If E1 , E2 , …. , En are events contituting in a sample space S; then,

• $P\left({E}_{i}\phantom{\rule{0.222em}{0ex}}|\phantom{\rule{0.222em}{0ex}}A\right)=\frac{P\left({E}_{i}\right)\phantom{\rule{0.222em}{0ex}}P\left(A|{E}_{i}\right)}{\sum _{j=1}^{n}P\left({E}_{j}\right)\phantom{\rule{0.222em}{0ex}}P\left(A|{E}_{j}\right)}$

4. Var (X) = E (X2) – [E(X)]2

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