# Chapter-wise Class 12 Mathematics Formula Cheat Sheet

## Maths Formulas for Class 12: Learn Important Formulas

As soon as the CBSE announces the Class 12 board exam date sheet, it creates nervousness among students. Many start to find a solution that can help them complete the syllabus in the remaining time. Many students also experience fear and anxiety.  Mathematical Anxiety is a term students might not be familiar with but the majority of the students experience it.

Understanding CBSE Class 12 Maths Formulas is the key that can significantly help in reducing anxiety. This list of important class 12 maths all formulas chapter-wise can help in scoring at least passing marks since every step helps in scoring.  This list will also help during competitive exams like IIT-JEE and CUET preparation and is aligned with the CBSE Class 12 Math Syllabus.

S.No. Table Of Content
1 Important CBSE Class 12 Mathematics Formulas
2 List of CBSE Class 12 Chapter Wise Maths Formulas Part-I
3 List of CBSE Class 12 Chapter Wise Maths Formulas Part-II
4 Important CBSE Class 12 Mathematics Formulas: Examples
5 Important CBSE Class 12 Mathematics Formulas: Applications
6 How to Learn Important CBSE Class 12 Mathematics Formulas

## Important CBSE Class 12 Mathematics Formulas

These are some of the most important formulas that can help in quick revision for CBSE Class 12 board exams. The formulas are listed as per the marks weightage given in the CBSE Syllabus for Class 12 Maths.

1. ∫ f(x) dx = F(x) + C
2. ∫ ln(x) dx = x ln(x) - x + C
3. ∫cos x dx = sin x + C
4. ∫ sin x dx = -cos x + C
5. ∫ sec2 x dx = tan x + C
6. ∫ cosec2 x dx = -cot x + C
7. ∫ sec x tan x dx = sec x + C
8. ∫ cosec x cot x dx = - cosec x + C
9. (A.B )= |P| |Q| cos θ ( Dot Product )
10. (A × B )= |P| |Q| sin θ (Cross Product)
11. k (A + B )= kA + kB
12. tan-1x + cot-1x = π / 2
13. sin-1x + cos-1 x = π / 2
14. cos-1(-x) = π - cos-1x
15. Distance between two points P(x1, y1, z1) and Q(x2, y2, z2): PQ = √ ((x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2
16. P(A ∩ B) = P(A) P(B | A).
17. P(A | B) = P(A ∩ B) / P(B), provided P(B) ≠ 0.

## List of CBSE Class 12 Chapter Wise Maths Formulas

Below are given CBSE Class 12 Maths Formulas for all chapters PDFs and formula list.

### Unit-I: Relations and Functions

#### 1. Relations and Functions

Chapter 1 Relation and Functions makes 4 marks in the final board examinations. Below are given relations and functions Class 12 formulas.

Relations

• Cartesian Product of Two Non-empty Sets: A × B = {(a, b) : a ∈ A and b ∈ B}
• Cartesian Product of One-empty Sets: A × B = φ
• Inverse of Relation: R-1 ={(b, a) : (a, b) ∈ R}

where, Domain of R = Range of R-1 and Range of R = Domain of R-1.

• Universal Relation: R = A × A i.e. reflexive, symmetric, and transitive

Reflexive- if (a, a) ∈ R, for every a ∈ A,

Symmetric- if (a1, a2) ∈ R implies that (a2, a1) ∈ R , for all a1, a2∈ A,

Transitive- if (a1, a2) ∈ R and (a2, a3) ∈ R  implies that (a1, a3) ∈ R  for all a1, a2, a3 ∈A.

• Identity Relation: R = {(a, a) : a ∈ A
• Empty Relation: , R = φ or R = { }

Functions

Let f: X → R and g: X → R be any two real functions, where X ⊆ R. then, we define

• (f ± g): X → R by (f ± g)(x) = f(x) ± g(x), for all x ∈ X.
• (f.g) : X → R by (f.g)(x) = f(x).g(x), for all x ∈ X.
• (f/g) :X → R by (f/g)(x) = f(x)/g(x) provided g(x) ≠ 0, x ∈ X.

Let f: X → R and a be a real number. Then, we define (αf ): X → R by (αf )(x) = af(x), for all x ∈ X.

• (f + g)(x) = f(x) + g(x) ; x ϵ X
• (f – g)(x) = f(x) – g(x)
• (f . g)(x) = f(x).g(x)
• (kf)(x) = k(f(x)) where k is a real number
• {f/g}(x) = f(x)/g(x), g(x)≠0

#### 2. Inverse Trigonometric Functions

Chapter 2 Inverse Trigonometric makes 4 marks in the final board examinations. Below are the links for inverse trigonometry formulas for class 12 PDF download.

Function Domain Range
y = sin-1 x [–1, 1] [−π/2,π/2]
y = cos-1 x [–1, 1] [0,π]
y = cosec-1 x R – (–1, 1 [−π/2, π/2] – {0}
y = sec-1 x R – (–1, 1) [0,π] – {π/2, π/2}
y = tan-1 x R (−π/2, π/2)
y = cot-1 x R (0,π)

• sin(sin-1 x)=x
• sin-1 (sin x) = x
• cos(cos-1 x) = x
• cos-1 (cos x) = x
• tan(tan-1 x) = x
• tan-1 (tan x) = x
• sec(sec-1x) = x
• sec-1(sec x) = x
• cosec-1(cosec x) = x
• cosec(cosec-1 x) = x
• cot-1(cot x) = x
• cot(cot-1x) = x
• sin-1 (1/x) = cosec-1 x, x ≥ 1 or x ≤ -1
• cos-1 (1/x) = sec-1 x, x ≥ 1 or x ≤ -1
• tan-1 (1/x) = cot-1 x, x > 0
• sin-1 (-x) = -sin-1 (x), x ∈ [-1, 1]
• tan-1 (-x) = -tan-1 (x), x ∈ R
• cosec-1 (-x) = -cosec-1 (x), |x| ≥1
• cos-1 (-x) = π – cos-1 (x), x ∈ [-1, 1]
• sec-1 (-x) = π – sec-1 (x), |x| ≥1
• cot-1 (-x) = π – cot-1 (x), x ∈ R
• sin-1 x + cos-1 x = π/2
• tan-1 x + cot-1 x = π/2
• cosec-1 x + sec-1 x = π/2
• tan-1 x + tan-1 y = tan-1 {(x+y)/(1−xy)}
• tan-1 x – tan-1 y = tan-1 {(x-y)/(1+xy)}
• sin-1 x  + sin-1 y = sin-1 [x√(1-y2)+y√(1-x2)]
• sin-1 x – sin-1 y = sin-1 [x√(1-y2)-y√(1-x2)]
• cos-1 x + cos-1 y = cos-1 [xy-√(1-x2)√(1-y2)]
• cos-1 x – cos-1 y = cos-1 [xy+√(1-x2)√(1-y2)]
• cot-1 x + cot-1 y = cot-1 [(xy-1)/(x+y)]
• cot-1 x + cot-1 y = cot-1 [(xy+1)/(y-x)]
• 2tan-1 x = sin-1 (2x/1+x2)
• 2tan-1 x = cos-1 (1-x2/1+x2)
• 2tan-1 x = tan-1 (2x/1-x2)
• 2sin-1 x = sin-1 (2x√(1+x2))
• 2cos-1 x = sin-1 (2x√(1-x2))
• sin-1 x = cos-1 √(1-x2) = tan-1 {x/x√(1-x2)} = cot-1 {√(1-x2)/x}
• cos-1 x = sin-1 √(1-x2) = tan-1 {√(1-x2)/x} = cot-1 {x/√(1-x2)}
• tan-1 x = sin-1 {x/√(1-x2)} = cos-1 {x/√(1+x2)} = sec-1 √(1+x2) = cosec-1 {√(1+x2)/x}

### ‍Unit-II: Algebra

#### 1. Matrices

Chapter 3 Matrics makes 5 marks in the final board examinations. Below are the links for matrics formulas for class 12 PDF download.

• Two matrices A = [aij] and B = [bij] are said to be equal if

They are of the same order; Each element of A is equal to the corresponding element of B, i.e., aij = bij, for all i and j

• Order of Matrix: In m × n matrix when m = n
• Diagonal Matrix: if aij = 0, when i ≠ j
• Scalar Matrix:  if aij = 0, when i ≠ j aij = k (k=constant),and i = j
• Identity Matrix: if aij = 1, when i = j when i = j and aij = 0, when i ≠ j.
• Zero Matrix: When all elements are 0
• Column Matrix: [A]n×1
• Row Matrix: [A]1×n
• Addition of Matrix: A + B = [aij +bij]m×n, 1 ≤ i ≤ m, 1 ≤ j ≤ n
• Subtraction of Matrix: A – B = [aij – bij]m×n, 1 ≤ i ≤ m, 1 ≤ j ≤ n
• Multiplication of a matrix: if A = [aij]m×n, then kA = [kaij]m×n
• Multiplication of Matrices: if the number of columns in matrix A is equal to the number of rows in matrix B.

#### ‍2. Determinants

Chapter 4 Determinants make 5 marks in the final board examinations. Below are the links for determinants formulas for class 12 PDF download.

• |A| = a11
• |A| =  a11a22 – a12a21
• Area of a Triangle Using Determinants: 1/2 |x1(y2 – y3) + x2(y3 – y1) + x3(y1 – y2)|
• Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and jth column in which element aij lies. The minor of aij is denoted by Mi
• The cofactor of an element aij, denoted by Aij, is defined by, Aij = (–1)i + j Mij where Mij is the minor of aij.
• Adjoint of a Matrix: The transpose of the matrix [Aij] is called the adjoint of the matrix A
• |adj A| = |A|n – 1
• |A (adj A)| = |A|n
• A(adj A) = |A|n – 2A
• Inverse of a Matrix: A– 1 = 1/|a|.(adj A)
• AA– 1 = 1
• Solutions of System of Linear Equations Using Inverse of a Matrix

AX = B

• Case I: If |A| ≠ 0, i.e., if A is a non-singular matrix, then A– 1 exists; X = A– 1B
• Case II: If |A| = 0, i.e., if A is a singular matrix, then  A– 1 does not exist.
• If (adj A)B ≠ O, then the solution does not exist and the given system of equations is inconsistent.
• If (adj A)B = O, then either no solution exist or infinitely many solutions exist, and accordingly, the given system of equations will be inconsistent or consistent, respectively.

### ‍Unit-III: Calculus

#### 1. Continuity and Differentiability

Chapter 5 Continuity and Differentiability make 9 marks in the final board examinations. Below are the links for the continuity and differentiability class 12 formulas PDF download.

Continuity:

• y=f(x) is continuous at x=a if the graph of the function y=f(x) is continuous (without any break) at x = a.
• f(x) is said to be continuous at a point x = a when: f(a) exists i.e. f(a) is finite, definite and real.

limx→a f(x) exists. lim⁡𝑥→𝑎 𝑓(𝑥)=𝑓(𝑎)

• function f(x) is continuous at x = a if

limh→0 f(a+h)= limh→0 f(a-h)= (a)

• Continuity of a Function in a Closed Interval: limx→a-0 f(x)= f(a) and limx→b-0 f(x)= (b)
• Continuity of a Function in an Open Interval: open interval (a,b) if it is continuous at every point in  (a,b)
• Discontinuity: if f and g are continuous functions, then

(f ± g) (x) = f(x) ± g(x) is continuous(f . g) (x) = f(x) . g(x) is continuous{f/g)(x)=f(x)/g(x)

• Chain Rule: If f = v o u, t = u (x), and if both dt/dx and dv/dx exist, then: df/dx = dv/dt. dt/dx
• Addition Rule: (u±v)′ = u′ ± v’
• Product Rule: (uv)′ = u′v + uv’
• Mean Value Theorem: f′(c) = (f(b)−f(a))/(b−a)
• Rolle’s Theorem: If f: [a, b] → R is continuous and differentiable on (a, b) whereas f(a) = f(b) then there exists some c in (a, b) such that f ′(c) = 0.
• Lagrange’s Mean Value Theorem: If f: [a, b] → R is continuous and differentiable on (a, b) then there exists some c in (a, b) such that f ′(c) = (f(b)-f(a))/b-a

• d (f+g)/dx = d (f)/dx + d (g)/dx
• d (f-g)/dx = d (f)/dx - d (g)/dx
• d (f.g)/dx = f.(d (g)/dx) + g.(d (f)/dx)
• d (f/g)/dx ={ g.(d (f)/dx) + f.(d (g)/dx)}/g2

#### ‍2. Applications of Derivatives

Chapter 6 Applications of Derivatives got 4 marks in the final board examinations. Below are the links for the applications of derivatives class 12 formulas PDF download.

• Instantaneous rate of change= f'(a)=limh→0f(a+h)−f(a)hf′(a) = limh→0f(a+h)−f(a)h
• Average rate of change= ∆y = f(x + ∆x) – f(x)

If the value of y increases with an increase in the value of x, then ∆y and ∆x are both positive and hence dy/dx is positiveIf the value of y decreases with an increase in the value of x, then ∆y is negative and ∆x is positive. So, dy/dx is negative.

• Equation of the function of the tangent: L(x) = f(a) + f'(a)(x−a)
• Tangent line to the curve: m = (y2−y1)/(x2−x1)
• Slope of the normal line to a curve of a function y = f(x) at a point (x1,y1)(x2,y2):  n = -1/m = – 1/ f'(x)
• Equation of the normal line to the curve: −1/m = (y2−y1)/(x2−x1)
• Maxima, Minima, and Point of Inflection:

Maxima when the slope or f’(x) changes its sign from +ve to -ve as we move via point c. And f(c) is the maximum value.Minima when the slope or f’(x) changes its sign from -ve to +ve as we move via point c. And f(c) is the minimum value.Point C is called the Point of inflection when the sign of slope or sign of the f’(x) doesn’t change as we move via c.

• Increasing and Decreasing Functions on an Interval
• strictly increasing on I, if for all x1, x2 ∈ I, we have x1 < x2 ⇒ f(x1) < f(x2).
• increasing on I, if for all x1, x2 ∈ I, we have x1 < x2 ⇒ f(x1) ≤ f(x2).
• strictly decreasing on I, if for all x1, x2 ∈ I, we have x1 < x2 ⇒ f(x1) > f(x2).
• decreasing on I, if for all x1, x2∈ I, we have x1 < x2 ⇒ f (x1) ≥ f (x2).

#### ‍3. Integrals

Chapter 7 Integrals got 9 marks in the final board examinations. Below are the links for the class 12 integration formulas.

• ∫f(x) dx = F(x) + C
• ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
• ∫[k1f1(x) + k2 f2(x)+…+ knfn(x)] dx = k1 ∫f1(x) dx + k2 ∫ f2(x) dx+…+ kn ∫fn(x) dx
• ∫tan x dx = log|sec x| + C
• ∫cot x dx = log|sin x| + C
• ∫sec x dx =log|sec x + tan x| + C
• ∫cosec x dx = log|cosec x − cot x| + C
• ∫xndx = xn+1/(n+1) + C
• ∫cos x dx = sin x + C
• ∫sin x dx = −cos x + C
• ∫sec2x dx = tan x + C
• ∫cosec2x dx = −cot x + C
• ∫sec x.tan x dx = sec x + C
• ∫cosec x.cot x dx = −cosec x + C
• ∫exdx = ex + C
• ∫axdx = axlogea + C
• ∫1/x dx = log|x| + C
• A(x) = ∫axf(x)dx for all x ≥ a then, A’ (x) = f (x) for every x ∈ [a, b]
• baf(x)dx = [F(x) + C]ba = F(b)−F(a)

#### ‍4. Applications of the Integrals

Chapter 8 Applications of the Integrals got 6 marks in the final board examinations. Below are the links for the class 12 applications of the integrals formulas.

• Area = ∫bay.dx  =∫baf(x).dx
• Area = ∫dcx.dx  =∫dcφ(y).dy
• Area = ∫ba[f(x)−g(x)].dx
• Area=∫ba[f(x)−g(x)].dx, + ∫bc[g(x)−f(x)].dx

#### ‍5. Differential Equations

Chapter 9 Differential Equations got 6 marks in the final board examinations. Below are the links for the class 12 differential equations formulas.

### ‍Unit-IV: Vectors and Three-Dimensional Geometry

#### 1. Vectors

Chapter 10 Vectors got 7 marks in the final board examinations. Below are the links for the vector algebra class 12 formulas.

• A + B = B + A (Commutative Law)
• A + (B + C) = (A + B) + C (Associative Law)
• (A • B )= |P| |Q| cos θ ( Dot Product )
• (A × B )= |P| |Q| sin θ (Cross Product)
• k (A + B )= kA + kB
• A + 0 = 0 + A (Additive Identity)

#### ‍2. Three - dimensional Geometry

Chapter 11 Three-dimensional geometry got 7 marks in the final board examinations. Below are the links for the 3d geometry class 12 formulas.

• Distance Formula: AB= √(x2-x1)2 + (y2-y1)2 + (z2-z1)2
• Distance Formula: A(x, y, z) from the origin O(0, 0, 0) = √x2 + (y2-y2 + z2
• Section Formula:

Case 1: When R divides PQ internally in the ratio m: n, then

Case 2: When R divides PQ externally in the ratio m: n, then,

• Incentre of triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is,

((ax1+bx2+cx3)/a+b+c),((ay1+by2+cy3)/a+b+c),((az1+bz2+cz3)/a+b+c)Direction Cosines of a Line:

• l2 + m2 + n2 = 1
• cos2 α + cos2 β + cos2 γ = 1

Direction Ratios of a Line:

• Direction ratios of a line

l/a = m/b = n/c

• Direction Cosines

l= ± [ a/ √a2+b2+c2)m= ± [ b/ √a2+b2+c2)n= ± [ c/ √a2+b2+c2)Angle between two line segments: cosθ = | (a1a2 +  b1b2 + c1c2)/√a12+b12+c1 √a22+b22+c2|Two lines will be perpendicular if= a1a2 +  b1b2 + c1c2 = 0Two lines will be parallel if= a1/a2 +  b1/b2 + c1/c2 = 0Projection of a line segment on a line:| (l(x2-x1) + m(y2-y1) + n(z2-z1)|Equation of a Plane

• General form: ax+by+cz+d = 0, where a, b, c are not all zero, a, b, c, d ∈ R.
• Normal form: lx+my+nz = p
• Plane through the point (x1, y1, z1): a(x-x1)+b(y-y1)+c(z-z1) = 0

Equation of a Plane

• General form: ax+by+cz+d = 0, where a, b, c are not all zero, a, b, c, d ∈ R.
• Normal form: lx+my+nz = p
• Plane through the point (x1, y1, z1): a(x-x1)+b(y-y1)+c(z-z1) = 0

Planes Parallel to Axes

• Plane Parallel to X-axis is by + cz + d = 0
• Plane Parallel to Y-axis is ax + cz + d = 0
• Plane Parallel to Z-axis is ax + by + d = 0

### ‍Unit-V: Linear Programming

#### 1. Linear Programming

Chapter 12 Linear Programming got 7 marks in the final board examinations. Below are the links for the 12 maths formula for linear programming

• Theorem 1: Let R be the feasible region (convex region) for LPP and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
• Theorem 2: Let R be the feasible region (convex region) for LPP and let Z = ax + by be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum values on R and each of these occurs at the corner point (vertex) of the feasible region.

### Unit-VI: Probability

#### 1. Probability

Chapter 13 Probability got 8 marks in the final board examinations. Below are the links for the probability formulas class 12.

• Addition Theorem for Two Events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Conditional Probability:P(A|B) = P(A ∩ B)/P(B) (for P(B) ≠ 0).

P(S|F) = P(F|F) = 1P(F)≠0: P((A ∪ B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F)P(E′|F) = 1 − P(E|F)

• Multiplication Rule: P(E ∩ F) = P(E) P(F|E) = P(F) P(E|F)
• P(E ∩ F ∩ G) = P(E) P(F|E) P(G|(E ∩ F)) = P(E) P(F|E) P(G|EF)
• Independent Events: P (E ∩ F) = P (E).P(F)
• Baye’s Theorem: (P(Ei|A)= P(Ei) P (A|Ei)) /Σnj=1P(Ej) P (A|Ej))
• Theorem of Total Probability: P(A) = P(E1) P (A|E1) + P (E2) P (A|E2) + … + P (En) . P(A|En)
• Random Variables and their Probability Distributions:Σni=1 Pi = 1
• Bernoulli Trials and Binomial Distribution:P (X = x) = P(x) = nCx qn-x px

## Important CBSE Class 12 Mathematics Formulas: Examples

Q1. Find:  ∫sec3θ dθ

Sol. Use Integration by parts and tan2θ=sec2θ-1

∫sec3θ dθ= ∫secθ d(tanθ)

= secθ tanθ - ∫tanθ d(secθ)

= secθ tanθ - ∫secθ tan2θ dθ

= secθ tanθ - ∫secθ (sec2θ - 1) dθ

=secθ tanθ - ∫secθ dθ - ∫sec3θ dθ

∴ 2∫sec3θ dθ

=secθ tanθ - ∫secθ dθ

Q2. Find, dy/dx, if y = (cos x)x+ cos–1√x is given.

Sol. y = (cos x)x+ cos–1√x

dy/dx = d((cos x)x+ cos–1√x)/dx

//The derivative of arccos x is given by -1/√(1-x2) where -1 < x < 1.

Apply the Sum/Difference Rule: (f ± g)’=f’ ± g’

d((cos x)x/dx + d(arccos√x)/dx

f’= d((cos x)x/dx

= cosx(x) (ln(cos(x)) - xtan(x))

g’= d(arccos√x)/dx

= -1/(2√x √1-x

cosx(x) (ln(cos(x)) - xtan(x)) -1/(2√x √1-x

## Important CBSE Class 12 Mathematics Formulas: Applications

Many students often wonder if the formulas they are supposed to learn will be helpful to them in real-life situations. As much as they all laugh at it, many quantities in Physics and magnitude calculation are done using algebraic formulas, and navigation, and insurance risks calculations are also done using these formulas.

1. Class 12 Trigonometry Formula is useful in geography and astronomy. These formulas are really helpful in estimating slopes, buildings, and building design.
2. Class 12 Calculus (Continuity and Differentiability, Applications of Derivatives, Integrals, Applications of the Integrals, and Differential Equations) is useful in Physics.
3. Class 12 Calculus can help in calculating elements like the centre of mass, and the object’s velocity. It can also help calculate the mass moment of inertia.
4. Vector Algebra Class 12 Formulas can help find equipotential surfaces, electromagnetic fields, and much more.
5. Class 12 relations and functions can also help economics in the case of demand and supply.

## How to Learn Important CBSE Class 12 Mathematics Formulas

All formulas of maths class 12 chapter-wise PDF can be utilised effectively with these tips and tricks can help in memorising and scoring well in board exams.

1. Understand the situations in which that formula can be applied and try to solve as many questions as possible. Regular practice will help in understanding the concept better and remembering the formula.
2. Students can make color-coded notes or flashcards to memorise the formulas. It will also help in quick revision during CBSE board exams.
3. Illustrating the formulas or using mind maps is one of the effective ways to understand.