CBSE Class 12 Maths Formulas 2023 For Term 1 & Term 2 | Free Download PDF

CBSE Class 12 Maths Formulas 2023 For Term 1 & Term 2: 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 of CBSE Class 12 Maths Formulas which we have prepared for you. The way we have presented the formulas will clear all your confusion or doubts. You can even download the Class 12 Maths Formulas PDF files from this article.

Download Chapter-Wise CBSE Class 12 Maths Formulas 2023 For Term 1 & Term 2

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

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 = I_X and fog = I_Y. 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=sin−1x⇒x=sinyy=sin−1x⇒x=siny
2. x=siny⇒y=sin−1xx=siny⇒y=sin−1x
3. sin−11x=cosec−1xsin−11x=cosec−1x
4. cos−11x=sec−1xcos−11x=sec−1x
5. tan−11x=cot−1xtan−11x=cot−1x
6. cos−1(−x)=π−cos−1xcos−1(−x)=π−cos−1x
7. cot−1(−x)=π−cot−1xcot−1(−x)=π−cot−1x
8. sec−1(−x)=π−sec−1xsec−1(−x)=π−sec−1x
9. sin−1(−x)=−sin−1xsin−1(−x)=−sin−1x
10. tan−1(−x)=−tan−1xtan−1(−x)=−tan−1x
11. cosec−1(−x)=−cosec−1xcosec−1(−x)=−cosec−1x
12. tan−1x+cot−1x=π2tan−1x+cot−1x=π2
13. sin−1x+cos−1x=π2sin−1x+cos−1x=π2
14. cosec−1x+sec−1x=π2cosec−1x+sec−1x=π2
15. tan−1x+tan−1y=tan−1x+y1−xytan−1x+tan−1y=tan−1x+y1−xy
16. 2tan−1x=sin−12x1+x2=cos−11−x21+x22tan−1x=sin−12×1+x2=cos−11−x21+x2
17. 2tan−1x=tan−12x1−x22tan−1x=tan−12×1−x2
18. tan−1x+tan−1y=π+tan−1(x+y1−xy)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 = [a_ij]_{m × m} will be known as diagonal matrix if a_ij = 0, when i ≠ j.
4. A = [a_ij]_{n × n} is a scalar matrix if a_ij = 0, when i ≠ j, a_ij = k, (where k is some constant); and i = j.
5. A = [a_ij]_{n × n} is an identity matrix, if a_ij = 1, when i = j and a_ij = 0, when i ≠ j.
6. A zero matrix will contain all its elements as zero.
7. A = [a_ij] = [b_ij] = B if and only if:
   • (i) A and B are of the same order
   • (ii) a_ij = b_ij for all the certain values of i and j

CBSE Class 12 Maths Formulas | Elementary Operations 

1. Some basic operations of matrices:
   • (i) kA = k[a_ij]_{m × n} = [k(a_ij)]_{m × n}
   • (ii) – A = (– 1)A
   • (iii) A – B = A + (– 1)B
   • (iv) A + B = B + A
   • (v) (A + B) + C = A + (B + C); where A, B and C all are of the same order
   • (vi) k(A + B) = kA + kB; where A and B are of the same order; k is constant
   • (vii) (k + l)A = kA + lA; where k and l are the constant
2. If A = [a_ij]_{m × n} and B = [b_jk]_{n × p}, then
   AB = C = [c_ik]_{m × p} ; where c_ik = ∑nj=1aijbjk∑j=1naijbjk
   • (i) A.(BC) = (AB).C
   • (ii) A(B + C) = AB + AC
   • (iii) (A + B)C = AC + BC
3. If A= [a_ij]_{m × n}, then A’ or AT = [a_ji]_{n × m}
   • (i) (A’)’ = A
   • (ii) (kA)’ = kA’
   • (iii) (A + B)’ = A’ + B’
   • (iv) (AB)’ = B’A’
4. Some elementary operations:
   • (i) R_i ↔ R_j or C_i ↔ C_j
   • (ii) R_i → kR_i or C_i → kC_i
   • (iii) R_i → R_i + kR_j or C_i → C_i + kC_j
5. A is said to known as a symmetric matrix if A′ = A
6. A is said to be the skew-symmetric matrix if A′ = –A

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

• 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|

• Area of triangle with vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃) is:
• ∆ = 1212∣∣∣∣x1x2x3y1y2y3111∣∣∣∣|x1y11x2y21x3y31|
• Cofactor of aij of given by A_ij = (– 1)^{i+ j} M_ij
• If A = ⎡⎣⎢a11a21a31a12a22a32a13a23a33⎤⎦⎥[a11a12a13a21a22a23a31a32a33], then adj A = ⎡⎣⎢A11A12A13A21A22A23A31A32A33⎤⎦⎥[A11A21A31A12A22A32A13A23A33] ; where A_ij is the cofactor of a_ij.
• A−1=1|A|(adjA)A−1=1|A|(adjA)
• 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]
• 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)b−af′(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

• 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
• 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=k∫f(x)dx∫kf(x)dx=k∫f(x)dx
  • (iii) ∫[k1f1(x)+k2f2(x)+…+knfn(x)]dx=

k1∫f1(x)dx+k2∫f2(x)dx+…+kn∫fn(x)dx∫[k1f1(x)+k2f2(x)+…+knfn(x)]dx=k1∫f1(x)dx+k2∫f2(x)dx+…+kn∫fn(x)dx

• First fundamental theorem of integral calculus: Let the area function be defined as: A(x)=∫xaf(x)dxA(x)=∫axf(x)dx for all x≥ax≥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: 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,n≠−1∫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. ∫dx1−x2√=sin−1x+C∫dx1−x2=sin−1x+C
9. ∫dx1−x2√=−cos−1x+C∫dx1−x2=−cos−1x+C
10. ∫dx1+x2=tan−1x+C∫dx1+x2=tan−1x+C
11. ∫dx1+x2=−cot−1x+C∫dx1+x2=−cot−1x+C
12. ∫exdx=ex+C∫exdx=ex+C
13. ∫axdx=axloga+C∫axdx=axloga+C
14. ∫dxxx2−1√=sec−1x+C∫dxxx2−1=sec−1x+C
15. ∫dxxx2−1√=−cosec−1x+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|cosecx−cotx|+C∫cosecxdx=log|cosecx−cotx|+C

CBSE Class 12 Maths Formulas – Integrals (Special Functions)

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

• ∫x2−a2−−−−−−√dx=x2x2−a2−−−−−−√−a22log∣∣x+x2−a2−−−−−−√∣∣+C∫x2−a2dx
• =x2x2−a2−a22log|x+x2−a2|+C
• ∫x2+a2−−−−−−√dx=x2x2+a2−−−−−−√+a22log∣∣x+x2+a2−−−−−−√∣∣+C∫x2+a2dx=

x2x2+a2+a22log|x+x2+a2|+C

• ∫a2−x2−−−−−−√dx=x2a2−x2−−−−−−√+a2sin−1xa+C∫a2−x2dx=x2a2−x2+a2sin−1xa+C
• ax2+bx+c=a[x2+bax+ca]=a[(x+b2a)2+(ca−b24a2)]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 m−nna→−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 x2−x1PQ,y2−y1PQ,z2−z1PQx2−x1PQ,y2−y1PQ,z2−z1PQ where
   • PQ=(x2−x1)2+(y2−y1)2+(z2−z1)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: x−x1l=y−y1m=z−z1nx−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→+λb1→r→=a1→+λb1→ and r⃗ =a2→+μb2→r→=a2→+μb2→ is:
   • ∣∣∣(b1→×b2→).(a2→−a1→)|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 ×(a2→−a1→)|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:
   • ∣∣∣∣x−x1x2−x1x3−x1y−y1y2−y1y3−y1z−z1z2−z1z3−z1∣∣∣∣=
   • 0|x−x1y−y1z−z1x2−x1y2−y1z2−z1x3−x1y3−y1z3−z1|=0
9. The two lines r⃗ =a1→+λb1→r→=a1→+λb1→ and r⃗ =a2→+μb2→r→=a2→+μb2→ are coplanar if:
   • (a2→−a1→).(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+C21√A22+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: |d−a⃗ .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)=E∩FP(F),P(F)≠0P(E|F)=E∩FP(F),P(F)≠0
2. Total Probability: Let E₁ , E₂ , …. , E_n be the partition of a sample space and A be any event; then,
   • P(A) = P(E₁) P (A|E₁) + P (E₂) P (A|E₂) + … + P (E_n) . P(A|E_n)
3. Bayes Theorem: If E₁ , E₂ , …. , E_n are events contituting in a sample space S; then,
   • P(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

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FAQ- CBSE Class 12 Maths Formulas 2023 For Term 1 & Term 2