# CBSE Class 12 Maths Formulas

CBSE Class 12 Maths Formulas: Class 12 is one of those phases which determine the future of the students. A student comes across the first turning point just after completing Class 10’s board exam and the next turning point is Class 12.

The performances of the students will decide the eligibility in the various colleges. A student will be able to pursue a subject of his choice without any problem if he is aware of the techniques or strategies. In this blog, we are going to be dealing with an effective method that will be beneficial for the subject of Maths.

Maths is that kind of subject which cannot be avoided when it comes to building your career. You have to deal with these Maths no matter what you study and a solid structure is something you need to take your preparation to the next level.

Getting hold of the appropriate resources is a necessary thing to do before modifying the methods of preparation for the final Maths exam. The sections which we are covering here are associated with the essential aspects of CBSE Class 12 Maths Formulas.

Here in this blog, you will get to see the list of chapters along with the important formulas. Read all the parts with a focused mind.

## Important: CBSE Class 12 Maths Formulas

If you can manage to get CBSE Class 12 Maths Formulas PDF file you will understand the relevance of following the important Class 12 Maths Formulas.

Most of the students struggle to find the right formula for a particular problem and for this reason you need to refer to the compiled version which we have prepared for you. The way we have presented the formulas will clear all your confusion or doubts. You can even ask for the Class 12 Maths Formulas PDF files from your friends or relatives to understand the basics concepts which are involved in each chapter.

In many ways, the formula deals with the entire concept, and a compiled version can help the students to work on the areas which require immediate attention. You can even refer to Download CBSE Class 12 Maths Formulas PDF files which are available on the various blogs. The teachers from various schools provide various suggestions and over there you can see the formulas which are important for your final Maths exam.

Looking for the Maths Formulas For Class 12 Download files is a must while practicing various problems and the best way to score good marks in this subject is by making a note of the important formulas. Download CBSE Class 12 Maths Formulas PDF to bring changes in your methods of preparation.

The CBSE Class 12 Maths Formulas from these chapters have been mentioned below. Here is the list of chapters-

• Chapter 1 Relations and Functions
• Chapter 2 Inverse Trigonometric Functions
• Chapter 3 Matrices
• Chapter 4 Determinants
• Chapter 5 Continuity and Differentiability
• Chapter 6 Applications of Derivatives
• Chapter 7 Integrals
• Chapter 8 Applications of Integrals
• Chapter 9 Differential Equations
• Chapter 10 Vector Algebra
• Chapter 11 Three dimensional Geometry
• Chapter 12 Linear Programming
• Chapter 13 Probability

### CBSE 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 & CBSE Class 12 Maths Formulas

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  performed on a set A is a function  from A × A to A.
6. An element e ∈ X possess the identity element for binary operation  : X × X → X, if a  e = a = e  a; ∀ a ∈ X.
7. An element a ∈ X shows the invertible property for binary operation  : X × X → X, if there exists b ∈ X such that a  b = e = b  a where e is said to be the identity for the binary operation . The element b is called the inverse of a and is denoted by a–1.
8. An operation  on X is said to be commutative if a  b = b  a; ∀ a, b in X.
9. An operation  on X is said to associative if (a  b)  c = a  (b  c); ∀ a, b, c in X.

### CBSE 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] [−π2,π2][−π2,π2] y = cos-1 x [–1, 1] [0,π][0,π] y = cosec-1 x R – (–1, 1) [−π2,π2][−π2,π2] – {0} y = sec-1 x R – (–1, 1) [0,π][0,π] – {π2π2} y = tan-1 x R (−π2,π2)(−π2,π2) y = cot-1 x R (0,π)(0,π)

CBSE Class 12 Maths Formulas

1. y=sin1xx=sinyy=sin−1x⇒x=siny
2. x=sinyy=sin1xx=siny⇒y=sin−1x
3. sin11x=cosec1xsin−11x=cosec−1x
4. cos11x=sec1xcos−11x=sec−1x
5. tan11x=cot1xtan−11x=cot−1x
6. cos1(x)=πcos1xcos−1(−x)=π−cos−1x
7. cot1(x)=πcot1xcot−1(−x)=π−cot−1x
8. sec1(x)=πsec1xsec−1(−x)=π−sec−1x
9. sin1(x)=sin1xsin−1(−x)=−sin−1x
10. tan1(x)=tan1xtan−1(−x)=−tan−1x
11. cosec1(x)=cosec1xcosec−1(−x)=−cosec−1x
12. tan1x+cot1x=π2tan−1x+cot−1x=π2
13. sin1x+cos1x=π2sin−1x+cos−1x=π2
14. cosec1x+sec1x=π2cosec−1x+sec−1x=π2
15. tan1x+tan1y=tan1x+y1xytan−1x+tan−1y=tan−1x+y1−xy
16. 2tan1x=sin12x1+x2=cos11x21+x22tan−1x=sin−12×1+x2=cos−11−x21+x2
17. 2tan1x=tan12x1x22tan−1x=tan−12×1−x2
18. tan1x+tan1y=π+tan1(x+y1xy)tan−1x+tan−1y=π+tan−1(x+y1−xy); xy > 1; x, y > 0

### CBSE Class 12 Maths Formulas: 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 elements 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

CBSE Class 12 Maths Formulas | 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 = nj=1aijbjk∑j=1naijbjk
• (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

### CBSE 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.

CBSE Class 12 Maths Formulas

1. Determinant of a matrix A=⎡⎣⎢a1a2a3b1b2b3c1c2c3⎤⎦⎥A=[a1b1c1a2b2c2a3b3c3] can be expanded as:
|A| = ∣∣∣∣a1a2a3b1b2b3c1c2c3∣∣∣∣=a1∣∣∣b2b3c2c3∣∣∣b1∣∣∣a2a3c2c3∣∣∣+c1∣∣∣a2a3b2b3∣∣∣|a1b1c1a2b2c2a3b3c3|=a1|b2c2b3c3|−b1|a2c2a3c3|+c1|a2b2a3b3|
2. Area of triangle with vertices (x1, y1), (x2, y2) and (x3, y3) is:
∆ = 1212∣∣∣∣x1x2x3y1y2y3111∣∣∣∣|x1y11x2y21x3y31|
3. Cofactor of aij of given by Aij = (– 1)i+ j Mij
4. If A = ⎡⎣⎢a11a21a31a12a22a32a13a23a33⎤⎦⎥[a11a12a13a21a22a23a31a32a33], then adj A = ⎡⎣⎢A11A12A13A21A22A23A31A32A33⎤⎦⎥[A11A21A31A12A22A32A13A23A33] ; where Aij is the cofactor of aij.
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=⎡⎣⎢a1a2a3b1b2b3c1c2c3⎤⎦⎥[a1b1c1a2b2c2a3b3c3], X = ⎡⎣⎢xyz⎤⎦⎥[xyz] and B = ⎡⎣⎢d1d2d3⎤⎦⎥[d1d2d3]
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.

### CBSE Class 12 Maths Formulas: 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±g)(x)=f(x)±g(x)(f±g)(x)=f(x)±g(x) is continuous.
• (ii) (f.g)(x)=f(x).g(x)(f.g)(x)=f(x).g(x) is continuous.
• (iii) fg(x)=f(x)g(x)fg(x)=f(x)g(x) (whenever g(x)0g(x)≠0 is continuous.
3. Chain Rule: If f = v o u, t = u (x) and if both dtdxdtdx and dvdxdvdx exists, then:
dfdx=dvdt.dtdxdfdx=dvdt.dtdx
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(c)=f(b)f(a)baf′(c)=f(b)−f(a)b−a

CBSE Class 12 Maths Formulas

Given below are the CBSE Class 12 Maths Formulas for standard derivatives:

 Derivative Formulas ddx(sin−1x)ddx(sin−1x) 11−x2√11−x2 ddx(cos−1x)ddx(cos−1x) −11−x2√−11−x2 ddx(tan−1x)ddx(tan−1x) 11+x211+x2 ddx(cot−1x)ddx(cot−1x) −11+x2−11+x2 ddx(sec−1x)ddx(sec−1x) 1x1−x2√1×1−x2 ddx(cosec−1x)ddx(cosec−1x) −1x1−x2√−1×1−x2 ddx(ex)ddx(ex) exex ddx(logx)ddx(logx) 1x1x

### CBSE Class 12 Maths Formulas: Integrals

Definition/Properties

1. Integration is the inverse process of differentiation. Suppose, ddxF(x)=f(x)ddxF(x)=f(x); then we can write f(x)dx=F(x)+C∫f(x)dx=F(x)+C
2. Properties of indefinite integrals:
• (i) [f(x)+g(x)]dx=f(x)dx+g(x)dx∫[f(x)+g(x)]dx=∫f(x)dx+∫g(x)dx
• (ii) For any real number k, kf(x)dx=kf(x)dx∫kf(x)dx=k∫f(x)dx
• (iii) [k1f1(x)+k2f2(x)++knfn(x)]dx=k1f1(x)dx+k2f2(x)dx++knfn(x)dx∫[k1f1(x)+k2f2(x)+…+knfn(x)]dx=k1∫f1(x)dx+k2∫f2(x)dx+…+kn∫fn(x)dx
3. First fundamental theorem of integral calculus: Let the area function be defined as: A(x)=xaf(x)dxA(x)=∫axf(x)dx for all xax≥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: ddxF(x)=f(x)ddxF(x)=f(x) for every x falling in the domain of f; then,
baf(x)dx=[F(x)+C]ba=F(b)F(a)∫abf(x)dx=[F(x)+C]ab=F(b)−F(a)

CBSE Class 12 Maths Formulas – Standard Integrals

1. xndx=xn+1n+1+C,n1∫xndx=xn+1n+1+C,n≠−1. Particularly, dx=x+C)∫dx=x+C)
2. cosxdx=sinx+C∫cosxdx=sinx+C
3. sinxdx=cosx+C∫sinxdx=−cosx+C
4. sec2xdx=tanx+C∫sec2xdx=tanx+C
5. cosec2xdx=cotx+C∫cosec2xdx=−cotx+C
6. secxtanxdx=secx+C∫secxtanxdx=secx+C
7. cosecxcotxdx=cosecx+C∫cosecxcotxdx=−cosecx+C
8. dx1x2=sin1x+C∫dx1−x2=sin−1x+C
9. dx1x2=cos1x+C∫dx1−x2=−cos−1x+C
10. dx1+x2=tan1x+C∫dx1+x2=tan−1x+C
11. dx1+x2=cot1x+C∫dx1+x2=−cot−1x+C
12. exdx=ex+C∫exdx=ex+C
13. axdx=axloga+C∫axdx=axloga+C
14. dxxx21=sec1x+C∫dxxx2−1=sec−1x+C
15. dxxx21=cosec1x+C∫dxxx2−1=−cosec−1x+C
16. 1xdx=log|x|+C∫1xdx=log|x|+C

CBSE Class 12 Maths Formulas – Partial Fractions

 Partial Fraction Formulas px+q(x−a)(x−b)px+q(x−a)(x−b) Ax−a+Bx−b,a≠bAx−a+Bx−b,a≠b px+q(x−a)2px+q(x−a)2 Ax−a+B(x−b)2Ax−a+B(x−b)2 px2+qx+r(x−a)(x−b)(x−c)px2+qx+r(x−a)(x−b)(x−c) Ax−a+Bx−b+Cx−cAx−a+Bx−b+Cx−c px2+qx+r(x−a)2(x−b)px2+qx+r(x−a)2(x−b) Ax−a+B(x−a)2+Cx−bAx−a+B(x−a)2+Cx−b px2+qx+r(x−a)(x2+bx+c)px2+qx+r(x−a)(x2+bx+c) Ax−a+Bx+Cx2+bx+cAx−a+Bx+Cx2+bx+c

CBSE Class 12 Maths Formulas – Integration by Substitution

1. tanxdx=log|secx|+C∫tanxdx=log|secx|+C
2. cotxdx=log|sinx|+C∫cotxdx=log|sinx|+C
3. secxdx=log|secx+tanx|+C∫secxdx=log|secx+tanx|+C
4. cosecxdx=log|cosecxcotx|+C∫cosecxdx=log|cosecx−cotx|+C

CBSE Class 12 Maths Formulas – Integrals (Special Functions)

1. dxx2a2=12alog∣∣xax+a∣∣+C∫dxx2−a2=12alog|x−ax+a|+C
2. dxa2x2=12alog∣∣a+xax∣∣+C∫dxa2−x2=12alog|a+xa−x|+C
3. dxx2+a2=1atan1xa+C∫dxx2+a2=1atan−1xa+C
4. dxx2a2=log∣∣x+x2a2−−−−−−√∣∣+C∫dxx2−a2=log|x+x2−a2|+C
5. dxx2+a2=log∣∣x+x2+a2−−−−−−√∣∣+C∫dxx2+a2=log|x+x2+a2|+C
6. dxx2a2=sin1xa+C∫dxx2−a2=sin−1xa+C

CBSE Class 12 Maths 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}
f1(x).f2(x)=f1(x)f2(x)dx[ddxf1(x).f2(x)dx]dx∫f1(x).f2(x)=f1(x)∫f2(x)dx−∫[ddxf1(x).∫f2(x)dx]dx
2. ex[f(x)+f(x)]dx=exf(x)dx+C∫ex[f(x)+f′(x)]dx=∫exf(x)dx+C

CBSE Class 12 Maths Formulas – Special Integrals

1. x2a2−−−−−−√dx=x2x2a2−−−−−−√a22log∣∣x+x2a2−−−−−−√∣∣+C∫x2−a2dx
2. =x2x2−a2−a22log|x+x2−a2|+C
3. x2+a2−−−−−−√dx=x2x2+a2−−−−−−√+a22log∣∣x+x2+a2−−−−−−√∣∣+C∫x2+a2dx=x2x2+a2+a22log|x+x2+a2|+C
4. a2x2−−−−−−√dx=x2a2x2−−−−−−√+a2sin1xa+C∫a2−x2dx=x2a2−x2+a2sin−1xa+C
5. ax2+bx+c=a[x2+bax+ca]=a[(x+b2a)2+(cab24a2)]ax2+bx+c=a[x2+bax+ca]=a[(x+b2a)2+(ca−b24a2)]

### CBSE Class 12 Maths Formulas: Application Of Integrals

Read out the CBSE Class 12 Maths Formulas for the mentioned chapter below:

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=baydx=baf(x)dxArea=∫abydx=∫abf(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=dcxdy=dcϕ(y)dyArea=∫cdxdy=∫cdϕ(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=ba[f(x)g(x)]dx,where,f(x)g(x)in[a,b]Area=∫ab[f(x)−g(x)]dx,where,f(x)≥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=ca[f(x)g(x)]dx,+bc[g(x)f(x)]dxArea=∫ac[f(x)−g(x)]dx,+∫cb[g(x)−f(x)]dx

### CBSE 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:
OP−→−(=r⃗ )=xi^+yj^+zk^OP→(=r→)=xi^+yj^+zk^
2. The scalar product of two given vectors a⃗ a→ and b⃗ b→ having angle θ between them is defined as:
• a⃗ .b⃗ =|a⃗ ||b⃗ |cosθa→.b→=|a→||b→|cosθ
3. The position vector of a point R dividing a line segment joining the points P and Q whose position vectors a⃗ a→ and b⃗ b→ are respectively, in the ratio m : n is given by:
• (i) internally: na⃗ +mb⃗ m+nna→+mb→m+n
• (ii) externally: na⃗ mb⃗ mnna→−mb→m−n

CBSE Class 12 Maths Formulas

If two vectors a⃗ a→ and b⃗ b→ are given in its component forms as a^=a1i^+a2j^+a3k^a^=a1i^+a2j^+a3k^ and b^=b1i^+b2j^+b3k^b^=b1i^+b2j^+b3k^ and λ as the scalar part; then:

• (i) a⃗ +b⃗ =(a1+b1)i^+(a2+b2)j^+(a3+b3)k^a→+b→=(a1+b1)i^+(a2+b2)j^+(a3+b3)k^ ;
• (ii) λa⃗ =(λa1)i^+(λa2)j^+(λa3)k^λa→=(λa1)i^+(λa2)j^+(λa3)k^ ;
• (iii) a⃗ .b⃗ =(a1b1)+(a2b2)+(a3b3)a→.b→=(a1b1)+(a2b2)+(a3b3)
• (iv) and a⃗ ×b⃗ =⎡⎣⎢i^a1a2j^b1b2k^c1c2⎤⎦⎥a→×b→=[i^j^k^a1b1c1a2b2c2].

### CBSE 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 that 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.

CBSE Class 12 Maths Formulas

1. The Direction cosines of a line joining two points P (x1 , y1 , z1) and Q (x2 , y2 , z2) are x2x1PQ,y2y1PQ,z2z1PQx2−x1PQ,y2−y1PQ,z2−z1PQ where
• PQ=(x2x1)2+(y2y1)2+(z2z1)2−−−−−−−−−−−−−−−−−−−−−−−−−−−−√(x2−x1)2+(y2−y1)2+(z2−z1)2
2. Equation of a line through a point (x1 , y1 , z1 ) and having direction cosines l, m, n is: xx1l=yy1m=zz1nx−x1l=y−y1m=z−z1n
3. The vector equation of a line which passes through two points whose position vectors a⃗ a→ and b⃗ b→ is r⃗ =a⃗ +λ(b⃗ a⃗ )r→=a→+λ(b→−a→)
4. The shortest distance between r⃗ =a1+λb1r→=a1→+λb1→ and r⃗ =a2+μb2r→=a2→+μb2→ is:
• ∣∣∣(b1×b2).(a2a1)|b1×b2|∣∣∣|(b1→×b2→).(a2→−a1→)|b1→×b2→||
5. The distance between parallel lines r⃗ =a1+λb⃗ r→=a1→+λb→ and r⃗ =a2+μb⃗ r→=a2→+μb→ is
• ∣∣∣b⃗ ×(a2a1)|b⃗ |∣∣∣|b→×(a2→−a1→)|b→||
6. The equation of a plane through a point whose position vector is a⃗ a→ and perpendicular to the vector N⃗ N→ is (r⃗ a⃗ ).N⃗ =0(r→−a→).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:
• ∣∣∣∣xx1x2x1x3x1yy1y2y1y3y1zz1z2z1z3z1∣∣∣∣=0|x−x1y−y1z−z1x2−x1y2−y1z2−z1x3−x1y3−y1z3−z1|=0
9. The two lines r⃗ =a1+λb1r→=a1→+λb1→ and r⃗ =a2+μb2r→=a2→+μb2→ are coplanar if:
• (a2a1).(b1×b2)=0(a2→−a1→).(b1→×b2→)=0
10. The angle φ between the line r⃗ =a⃗ +λb⃗ r→=a→+λb→ and the plane r⃗ .n^=dr→.n^=d is given by:
• sinϕ=∣∣∣b⃗ .n^|b⃗ ||n^|∣∣∣sinϕ=|b→.n^|b→||n^||
11. The angle θ between the planes A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 is given by:
• cosθ=∣∣∣A1A2+B1B2+C1C2A21+B21+C21A22+B22+C22∣∣∣cosθ=|A1A2+B1B2+C1C2A12+B12+C12A22+B22+C22|
12. The distance of a point whose position vector is a⃗ a→ from the plane r⃗ .n^=dr→.n^=d is given by: |da⃗ .n^||d−a→.n^|
13. The distance from a point (x1 , y1 , z1) to the plane Ax + By + Cz + D = 0:
• ∣∣∣Ax1+By1+Cz1+DA2+B2+C2∣∣∣|Ax1+By1+Cz1+DA2+B2+C2|

### CBSE 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)=EFP(F),P(F)0P(E|F)=E∩FP(F),P(F)≠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(Ei|A)=P(Ei)P(A|Ei)nj=1P(Ej)P(A|Ej)P(Ei|A)=P(Ei)P(A|Ei)∑j=1nP(Ej)P(A|Ej)
4. Var (X) = E (X2) – [E(X)]2

### Get Other Highly Important Essentials for CBSE Class 12 Maths Exam

We have covered the detailed guide on CBSE Class 12 Maths Formula 2021. You should have proper CBSE 12th study material to excel at the level of preparation in the correct way. Feel free to ask any questions.

### 1. Can we get all the important math formulas for class 12?

Yes, you can get all the important math formulas for class 12 from our above mention articles.

### 2. Do I need to memorize Maths formulas for class 12?

Yes, it is very important to remember to solve CBSE 12 Mat questions, that can be solved directly through formulas.

### 3. What are the formulas of CBSE class 12 Maths?

The formulas are Arithmetics formulas, Trigonometric formulas, Standard Integrals formulas, and so on, which you can get from the above blog.

### 4. What are the most important CBSE Class 12 Maths Formulas?

You can get all the important formulas of CBSE Class 12 Maths Formulas that are very important to remember to solve those questions from the above blog.

### 5. How to memorize the formulas of CBSE Class 12 Maths Formulas?

It is very easy to memorize the CBSE Class 12 Maths formula, you can go on solving the same question at one time which is solved by the same formula or you only read and write the formula number of times.

### 6. Are all the questions of Class 12 Maths solved with formulas?

No, all mathematics questions are not solved by formulas, some questions are solved with basic concepts, but you should at least memorize the basic formulas.

### 7. How many formulas should I memorize for Class 12 Maths?

You should memorize for CBSE Class 12 Maths Formulas at least given at our articles.