## 12th Maths Formulas List PDF

PDF Name | 12th Maths Formulas List |
---|---|

No. of Pages | 39 |

PDF Size | 1.22 MB |

Language | English |

PDF Category | Education & Jobs |

Published/Updated | July 4, 2020 |

Source / Credits | vthometutor.com |

Comments ✎ | |

Uploaded By | pk |

**Download 12th Maths Formulas List PDF for free using the direct download link given at the bottom of this article.**

## 12th Maths Formulas List

**Areas**

Square | A=l2 | l : length of side | |

Rectangle | A=w×h | w : width h : height | |

Triangle | A=b×h2 | b : base h : height | |

Rhombus | A=D×d2 | D : large diagonal d : small diagonal | |

Trapezoid | A=B+b2×h | B : large side b : small side h: height | |

Regular polygon | A=P2×a | P : perimeter a : apothem | |

Circle | A=πr2 P=2πr | r : radius P : perimeter | |

Cone (lateral surface) | A=πr×s | r : radius s : slant height | |

Sphere (surface area) | A=4πr2 | r: radius |

#### Volumes

Cube | V=s3V=s3 | ss: side | |

Parallelepiped | V=l×w×hV=l×w×h | ll: length ww: width hh: height | |

Regular prism | V=b×hV=b×h | bb: base hh: height | |

Cylinder | V=πr2×hV=πr2×h | rr: radius hh: height | |

Cone (or pyramid) | V=13b×hV=13b×h | bb: base hh: height | |

Sphere | V=43πr3V=43πr3 | rr: radius |

#### Functions and Equations

Directly Proportional | y=kxy=kx k=yxk=yx | kk: Constant of Proportionality |

Inversely Proportional | y=kxy=kx k=yxk=yx | |

ax2+bx+c=0ax2+bx+c=0 | Quadratic formula | x=−b±b2−4ac−−−−−−−√2ax=-b±b2-4ac2a |

Concavity | Concave up: a>0a>0 | |

Concave down: a<0a<0 | ||

Discriminant | Δ=b2−4acΔ=b2-4ac | |

Vertex of the parabola | V(−b2a,−Δ4a)V(-b2a,-Δ4a) | |

y=a(x−h)2+ky=a(x-h)2+k | Concavity | Concave up: a>0a>0 |

Concave down: a<0a<0 | ||

Vertex of the parabola | V(h,k)V(h,k) | |

Zero-product property | A×B=0⇔A=0∨B=0A×B=0⇔A=0∨B=0 | ex : (x+2)×(x−1)=0⇔(x+2)×(x-1)=0⇔ x+2=0∨x−1=0⇔x=−2∨x=1x+2=0∨x-1=0⇔x=-2∨x=1 |

Difference of two squares | (a−b)(a+b)=a2−b2(a-b)(a+b)=a2-b2 | ex : (x−2)(x+2)=x2−22=x2−4(x-2)(x+2)=x2-22=x2-4 |

Perfect square trinomial | (a+b)2=a2+2ab+b2(a+b)2=a2+2ab+b2 | ex : (2x+3)2=(2x)2+2⋅2x⋅3+32=(2x+3)2=(2x)2+2⋅2x⋅3+32= 4×2+12x+94×2+12x+9 |

Binomial theorem | (x+y)n=∑k=0nnCkxn−kyk |

#### Probability and Sets

Commutative | A∪B=B∪AA∪B=B∪A | A∩B=B∩AA∩B=B∩A |

Associative | A∪(B∪C)=A∪(B∪C)A∪(B∪C)=A∪(B∪C) | A∩(B∩C)=A∩(B∩C)A∩(B∩C)=A∩(B∩C) |

Neutral element | A∪∅=AA∪∅=A | A∩E=AA∩E=A |

Absorbing element | A∪E=EA∪E=E | A∩∅=∅A∩∅=∅ |

Distributive | A∪(B∩C)=(A∪B)∩(A∪C)A∪(B∩C)=(A∪B)∩(A∪C) | A∩(B∪C)=(A∩B)∪(A∩C)A∩(B∪C)=(A∩B)∪(A∩C) |

De Morgan’s laws | A∩B¯¯¯¯¯¯¯¯¯=A¯¯¯∪B¯¯¯A∩B¯=A¯∪B¯ | A∪B¯¯¯¯¯¯¯¯¯=A¯¯¯∩B¯¯¯A∪B¯=A¯∩B¯ |

Laplace laws | P(A)=Number of ways it can happenTotal number of outcomesP(A)=Number of ways it can happenTotal number of outcomes | |

Complement of an Event | P(A¯¯¯)=1−P(A)P(A¯)=1-P(A) | |

Union of Events | P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)-P(A∩B) | |

Conditional Probability | P(A∣B)=P(A∩B)P(B)P(A∣B)=P(A∩B)P(B) | |

Independent Events | P(A∣B)=P(A)P(A∣B)=P(A) | P(A∩B)=P(A)×P(B)P(A∩B)=P(A)×P(B) |

Permutation | Pn=n!=n×(n−1)×…×2×1Pn=n!=n×(n-1)×…×2×1 | ex : P4=4!=4×3×2×1=24P4=4!=4×3×2×1=24 |

Permutations without repetition | nAp=n!(n−p)!nAp=n!(n-p)! | ex : 6A2=6!(6−2)!=306A2=6!(6-2)!=30 |

Permutations with repetition | nA′p=npnAp′=np | ex : 5A′3=53=1255A3′=53=125 |

Combination | nCp=nApp!=n!(n−p)!×p!nCp=nApp!=n!(n-p)!×p! | ex : 5C4=5A44!=55C4=5A44!=5 |

Probability Distribution | Average value | μ=x1p1+x2p2+…+xkpkμ=x1p1+x2p2+…+xkpk |

Standard deviation | σ=∑i=1kpi(xi−μ)2−−−−−−−−−−−−⎷σ=∑i=1kpi(xi-μ)2 | |

Binomial distribution | P(X=k)=nCk.pk.(1−p)n−kP(X=k)=nCk.pk.(1-p)n-k | ex : B(10;0,6)B(10;0,6) P(X=3)=10C3×0,63×0,47P(X=3)=10C3×0,63×0,47 |

# Maths Formulas for Class 12: Check Class 12 Maths All Formulas

**Maths Formulas For Class 12:**Many students have said that the hardest subject in school is 12th grade maths. This negativity can cause failures even in small classes, and even on the Board Exams. Embibe experts don’t want you to believe such myths. Class 12th Mathematics encourages you develop logic and implement. You should learn the Maths Formulas for Class 12 which will help you grow faster and provide concrete knowledge.

**Please note: If you have difficulty accessing the formulas on your mobile device, open the Desktop site from your mobile’s browser settings.**

**SOLVE CLASS 12 MATHHS PRACTICE QUESTIONS. NOW**

After you have completed the NCERT exercises, you can refer to the Maths formulas of Class 12, which are available here. The Maths formulas PDF for Class 12 will help you understand the chapter and memorize the formulas. These NCERT formulas will help you to finish your assignments quickly and easily learn the formulas. This article contains a list of all Class 12 Maths all formulas. This will allow you to better understand the concepts and will ultimately result in a higher exam score. You can now review the Class 12 Maths formulas below.

These Class 12 Maths Formulas will help you conquer your Board exams and the entrance examinations. Let’s look at the most important chapters in Class 12 Maths where we will need formulas.

- Relationships and Functions
- Inverse Trigonometric Functions
- Matrices
- Determinants
- Continuity, Differentiability
- Integrals
- Integration
- Vector Algebra
- Three Dimensional Geometry
- Probability

*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.

Inverse Trigonometric Functions are quite useful in Calculus to define different integrals. You can also check the Trigonometric Formulas here.

*Properties/Theorems*

The domain and range of inverse trigonometric functions are given below:

Functions | Domain | Range |

y = sin^{-1} x | [–1, 1] | \(\left [ \frac{-\pi }{2},\frac{\pi }{2} \right ]\) |

y = cos^{-1} x | [–1, 1] | \(\left [0,\pi \right ]\) |

y = cosec^{-1} x | R – (–1, 1) | \(\left [ \frac{-\pi }{2},\frac{\pi }{2} \right ]\) – {0} |

y = sec^{-1} x | R – (–1, 1) | \(\left [0,\pi \right ]\) – {\(\frac{\pi }{2}\)} |

y = tan^{-1} x | R | \(\left ( \frac{-\pi}{2},\frac{\pi}{2} \right )\) |

y = cot^{-1} x | R | \(\left (0,\pi \right )\) |

*Formulas*

- \(y=sin^{-1}x\Rightarrow x=sin\:y\)
- \(x=sin\:y\Rightarrow y=sin^{-1}x\)
- \(sin^{-1}\frac{1}{x}=cosec^{-1}x\)
- \(cos^{-1}\frac{1}{x}=sec^{-1}x\)
- \(tan^{-1}\frac{1}{x}=cot^{-1}x\)
- \(cos^{-1}(-x)=\pi-cos^{-1}x\)
- \(cot^{-1}(-x)=\pi-cot^{-1}x\)
- \(sec^{-1}(-x)=\pi-sec^{-1}x\)
- \(sin^{-1}(-x)=-sin^{-1}x\)
- \(tan^{-1}(-x)=-tan^{-1}x\)
- \(cosec^{-1}(-x)=-cosec^{-1}x\)
- \(tan^{-1}x+cot^{-1}x=\frac{\pi}{2}\)
- \(sin^{-1}x+cos^{-1}x=\frac{\pi}{2}\)
- \(cosec^{-1}x+sec^{-1}x=\frac{\pi}{2}\)
- \(tan^{-1}x+tan^{-1}y=tan^{-1}\frac{x+y}{1-xy}\)
- \(2\:tan^{-1}x=sin^{-1}\frac{2x}{1+x^2}=cos^{-1}\frac{1-x^2}{1+x^2}\)
- \(2\:tan^{-1}x=tan^{-1}\frac{2x}{1-x^2}\)
- \(tan^{-1}x+tan^{-1}y=\pi+tan^{-1}\left (\frac{x+y}{1-xy} \right )\); xy > 1; x, y > 0

*Definition/Theorems*

- A matrix is said to have an ordered rectangular array of functions or numbers. A matrix of order m × n consists of m rows and n columns.
- An m × n matrix will be known as a square matrix when m = n.
- A = [a
_{ij}]_{m × m}will be known as diagonal matrix if a_{ij}= 0, when i ≠ j. - A = [a
_{ij}]_{n × n}is a scalar matrix if a_{ij}= 0, when i ≠ j, a_{ij}= k, (where k is some constant); and i = j. - A = [a
_{ij}]_{n × n}is an identity matrix, if a_{ij}= 1, when i = j and a_{ij}= 0, when i ≠ j. - A zero matrix will contain all its element as zero.
- A = [a
_{ij}] = [b_{ij}] = B if and only if:- (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

*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.

*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 |

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

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

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

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

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

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

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

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

*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*

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

- The area enclosed by the curve y = f (x) ; x-axis and the lines x = a and x = b (b > a) is given by the formula:
- \(Area=\int_{a}^{b}y\:dx=\int_{a}^{b}f(x)\:dx\)

- Area of the region bounded by the curve x = φ (y) as its y-axis and the lines y = c, y = d is given by the formula:
- \(Area=\int_{c}^{d}x\:dy=\int_{c}^{d}\phi (y)\:dy\)

- The area enclosed in between the two given curves y = f (x), y = g (x) and the lines x = a, x = b is given by the following formula:
- \(Area=\int_{a}^{b}[f(x)-g(x)]\:dx,\: where, f(x)\geq g(x)\:in\:[a,b]\)

- If f (x) ≥ g (x) in [a, c] and f (x) ≤ g (x) in [c, b], a < c < b, then:
- \(Area=\int_{a}^{c}[f(x)-g(x)]\:dx,+\int_{c}^{b}[g(x)-f(x)]\:dx\)

*Definition/Properties*

- Vector is a certain quantity that has both the magnitude and the direction. The position vector of a point P (x, y, z) is given by:\(\overrightarrow{OP}(=\vec{r})=x\hat{i}+y\hat{j}+z\hat{k}\)
- The scalar product of two given vectors \(\vec{a}\) and \(\vec{b}\) having angle θ between them is defined as:
- \(\vec{a}\:.\:\vec{b}=|\vec{a}||\vec{b}|\:cos\:\theta\)

- The position vector of a point R dividing a line segment joining the points P and Q whose position vectors \(\vec{a}\) and \(\vec{b}\) are respectively, in the ratio m : n is given by:
- (i) internally: \(\frac{n\vec{a}+m\vec{b}}{m+n}\)
- (ii) externally: \(\frac{n\vec{a}-m\vec{b}}{m-n}\)

*Formulas*

If two vectors \(\vec{a}\) and \(\vec{b}\) are given in its component forms as \(\hat{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) and \(\hat{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\) and λ as the scalar part; then:

- (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 \(\vec{a}\times \vec{b}= \begin{bmatrix} \hat{i}& \hat{j}& \hat{k}\\ a_{1}& b_{1}& c_{1}\\ a_{2}& b_{2}& c_{2} \end{bmatrix}\).

*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 \(\frac{x_2-x_1}{PQ}\:,\:\frac{y_2-y_1}{PQ}\:,\frac{z_2-z_1}{PQ}\) where- PQ=\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

- Equation of a line through a point (x
_{1}, y_{1}, z_{1}) and having direction cosines l, m, n is: \(\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}\) - The vector equation of a line which passes through two points whose position vectors \(\vec{a}\) and \(\vec{b}\) is \(\vec{r}=\vec{a}+\lambda (\vec{b}-\vec{a})\)
- The shortest distance between \(\vec{r}=\vec{a_1}+\lambda\: \vec{b_1}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b_2}\) is:
- \(\left | \frac{(\vec{b_1}\times \vec{b_2}).(\vec{a_2}-\vec{a_1})}{|\vec{b_1}\times \vec{b_2}|} \right |\)

- The distance between parallel lines \(\vec{r}=\vec{a_1}+\lambda\: \vec{b}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b}\) is
- \(\left | \frac{\vec{b}\times (\vec{a_2}-\vec{a_1})}{|\vec{b}|} \right |\)

- The equation of a plane through a point whose position vector is \(\vec{a}\) and perpendicular to the vector \(\vec{N}\) is \((\vec{r}-\vec{a})\:.\:\vec{N}=0\)
- Equation of a plane perpendicular to a given line with direction ratios A, B, C and passing through a given point (x
_{1}, y_{1}, z_{1}) is A (x – x_{1}) + B (y – y_{1}) + C (z – z_{1}) = 0 - The equation of a plane passing through three non-collinear points (x
_{1}, y_{1}, z_{1}); (x_{2}, y_{2}, z_{2}) and (x_{3}, y_{3}, z_{3}) is:- \(\begin{vmatrix} x-x_1& y-y_1& z-z_1\\ x_2-x_1& y_2-y_1& z_2-z_1\\ x_3-x_1& y_3-y_1& z_3-z_1 \end{vmatrix}=0\)

- The two lines \(\vec{r}=\vec{a_1}+\lambda\: \vec{b_1}\) and \(\vec{r}=\vec{a_2}+\mu \: \vec{b_2}\) are coplanar if:
- \((\vec{a_2}-\vec{a_1})\:.\:(\vec{b_1}\times \vec{b_2})=0\)

- The angle φ between the line \(\vec{r}=\vec{a}+\lambda\: \vec{b}\) and the plane \(\vec{r}\:.\:\hat{n}=d\) is given by:
- \(sin\:\phi =\left |\frac{\vec{b}\:.\:\hat{n}}{|\vec{b}||\hat{n}|} \right |\)

- The angle θ between the planes A
_{1}x + B_{1}y + C_{1}z + D_{1}= 0 and A_{2}x + B_{2}y + C_{2}z + D_{2}= 0 is given by:- \(cos\:\theta =\left | \frac{A_1\:A_2+B_1\:B_2+C_1\:C_2}{\sqrt{A_1^2+B_1^2+C_1^2}\:\sqrt{A_2^2+B_2^2+C_2^2}} \right |\)

- The distance of a point whose position vector is \(\vec{a}\) from the plane \(\vec{r}\:.\:\hat{n}=d\) is given by: \(\left | d-\vec{a}\:.\:\hat{n} \right |\)
- The distance from a point (x
_{1}, y_{1}, z_{1}) to the plane Ax + By + Cz + D = 0:- \(\left | \frac{Ax_1+By_1+Cz_1+D}{\sqrt{A^2+B^2+C^2}} \right |\)

*Definition/Properties*

- The conditional probability of an event E holds the value of the occurrence of the event F as:
- \(P(E\:|\:F)=\frac{E\cap F}{P(F)}\:,\:P(F)\neq 0\)

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

- P(A) = P(E
**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}

*Please Note: If you are having difficulties accessing these formulas on your mobile, try opening the Desktop site on your mobile in your mobile’s browser settings.*

Here we have provided the most frequently asked questions (FAQs) related to NCERT Class 12 Maths formulas:

*Ques: Give a list of basic maths formulas used in CBSE class 12th?*

Ans: A list containing the basic maths formulas that have been introduced for the students of class 12th have been listed in this above article. The basic list of topics include:

1. Algebra

2. Matrices

3. Geometry

4. Linear Programming

5. Calculus

6. Probability

*Questions: How many formulas can you find in class 12 CBSE Maths.*

Ans: It is nearly impossible to keep track of all formulas in the Maths book for class 12 CBSE. Each theory and concept in the book has one or more formulas that can be used to solve the mathematical problem. The difficulty of mathematics at the school level is made more difficult by the increasing number of formulas.

*Ques: Why is it important to use mathematical formulas?*

Ans: Mathematical formulas are crucial because they help in solving mathematical problems with ease. These mathematical formulas are essential for solving problems within a specified time frame and in an efficient way. The generalized form of mathematical formulas is used to solve mathematical problems. All we have to do is add the value of the entities into the formula and speed up the process.

*Ques: What is the formula used for the trigonometric ratio integration?*

Ans: ∫sin (x) dx = -Cos x + C

∫cos(x) dx = Sin x + C

∫sec^2x dx = tan x + C, etc.

*Ques: What is the formula used for the exponential function integration?*

Ans: If an exponential function is integrated, the function will remain unchanged with a constant being added to it.

Hence, ∫e^x dx = e^x + Constant ©

*Ques: Where do I find the complete class 12 Mathematics formula in the NCERT book.*

Ans: Students will find the complete list of formulas in the article on the embibe platform free of charge. You can also find links to Class 12 Math Solutions, notes, practice papers, mock exams, important questions, etc. This article contains the complete NCERT Solutions Class 12 Mathematics, as well as links to the NCERT Book Class 12 Mathematical & other important study materials.

*Ques – Which solution is best for NCERT Class 12 Mathematics?*

Ans: Embibe allows students to find 100% accurate solutions for NCERT Class 12 Mathematics. This article provides the solution, which was solved by embibe’s math teachers. Embibe provides solutions to all questions in the Class 12 Mathematics textbook. We have taken the CBSE Board guidelines into consideration and used the most recent NCERT book for Class 12 Mathematics.

Now you know all the maths formulas for class 12. We hope you found this article helpful. These Class 12 Maths formulas can be used to solve problems. The problem can be solved for free**Class 12 Maths questions**Embibe will be a great resource. These resources can be used to your advantage and you will master the subject.