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NCERT Formula Reference
All Class 12 Mathematics Formulas
NCERT-aligned formula cheat sheet for all 13 Class 12 Mathematics chapters — from relations and functions through probability. Bookmark this page for quick revision before board exams and JEE.
1
Full chapter notesRelations and Functions
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Relations and Functions — Formulas
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Sets & Functions
- n(A × B) = n(A) × n(B)
- One-one:f(x₁) = f(x₂) ⇒ x₁ = x₂
- Onto:Range(f) = Codomain
- Composition:(f ∘ g)(x) = f(g(x))
2
Full chapter notesInverse Trigonometric Functions
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Inverse Trigonometric Functions — Formulas
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Inverse Trig Identities
- sin⁻¹x + cos⁻¹x = π/2, x ∈ [−1,1]
- tan⁻¹x + cot⁻¹x = π/2, x ∈ ℝ
- sin⁻¹(−x) = −sin⁻¹x ; tan⁻¹(−x) = −tan⁻¹x
- cos⁻¹(−x) = π − cos⁻¹x ; cot⁻¹(−x) = π − cot⁻¹x
- sin⁻¹x = tan⁻¹(x/√(1−x²))
- cos⁻¹x = tan⁻¹(√(1−x²)/x)
3
Full chapter notesMatrices
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Matrices — Formulas
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Matrix Algebra
- (A ± B)ᵀ = Aᵀ ± Bᵀ
- (kA)ᵀ = kAᵀ ; (AB)ᵀ = BᵀAᵀ
- (Aᵀ)ᵀ = A
- (A⁻¹)ᵀ = (Aᵀ)⁻¹
- A⁻¹ = (1/|A|) adj(A), |A| ≠ 0
- |AB| = |A||B| ; |Aᵀ| = |A|
- adj(AB) = adj(B) adj(A)
4
Full chapter notesDeterminants
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Determinants — Formulas
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Determinants
- 2 × 2:|A| = ad − bc for [[a,b],[c,d]]
- |A| = 0 if two rows/columns are identical or proportional
- If a row/column is a sum of two, |A| splits into sum of two determinants
5
Full chapter notesContinuity and Differentiability
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Continuity and Differentiability — Formulas
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Continuity & Derivatives
- Continuous at a:lim(x→a⁻) f(x) = f(a) = lim(x→a⁺) f(x)
- Derivative:f′(a) = lim[h→0] [f(a+h) − f(a)]/h
- (xⁿ)′ = nxⁿ⁻¹
- (sin x)′ = cos x ; (cos x)′ = −sin x
- (tan x)′ = sec²x ; (cot x)′ = −cosec²x
- (sec x)′ = sec x tan x ; (cosec x)′ = −cosec x cot x
- (eˣ)′ = eˣ ; (aˣ)′ = aˣ ln a
- (log x)′ = 1/(x ln a) ; (ln x)′ = 1/x
6
Full chapter notesApplications of Derivatives
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Applications of Derivatives — Formulas
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Applications of Derivatives
- Product:(uv)′ = u′v + uv′
- Quotient:(u/v)′ = (u′v − uv′)/v²
- Chain:[f(g(x))]′ = f′(g(x))·g′(x)
- f″(x) = d²y/dx²
- Tangent:slope = f′(a); y − y₁ = f′(a)(x − x₁)
- Normal:slope = −1/f′(a)
- Max/min:f′(a)=0; f″(a)<0 → max, f″(a)>0 → min
7
Full chapter notesIntegrals
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Integrals — Formulas
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Standard Integrals
- ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
- ∫ (1/x) dx = ln|x| + C
- ∫ eˣ dx = eˣ + C ; ∫ aˣ dx = aˣ/(ln a) + C
- ∫ sin x dx = −cos x + C ; ∫ cos x dx = sin 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
- ∫ dx/(1+x²) = tan⁻¹x + C
- ∫ dx/√(1−x²) = sin⁻¹x + C
8
Full chapter notesApplications of Integrals
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Applications of Integrals — Formulas
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Area & Volume
- Area under curve:A = ∫ₐᵇ f(x) dx
- Between curves:A = ∫ₐᵇ |f(x) − g(x)| dx
- Circle area = πr² ; Triangle = ½ × base × height
- Cylinder volume = πr²h
- Cone volume = (1/3)πr²h
- Sphere volume = (4/3)πr³
9
Full chapter notesDifferential Equations
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Differential Equations — Formulas
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Differential Equations
- Separable:dy/dx = f(x)g(y) ⇒ ∫dy/g(y) = ∫f(x)dx + C
- Homogeneous:Substitute y = vx
- Linear:dy/dx + P(x)y = Q(x)
- IF = e^(∫P dx)
- Solution: y·IF = ∫Q·IF dx + C
10
Full chapter notesVector Algebra
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Vector Algebra — Formulas
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Vector Algebra
- Dot product:a⃗·b⃗ = |a⃗||b⃗|cos θ
- Cross product:|a⃗×b⃗| = |a⃗||b⃗|sin θ
- |a⃗ + b⃗|² = |a⃗|² + |b⃗|² + 2a⃗·b⃗
- |a⃗×b⃗|² = |a⃗|²|b⃗|² − (a⃗·b⃗)²
11
Full chapter notesThree Dimensional Geometry
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Three Dimensional Geometry — Formulas
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3D Geometry
- Direction cosines:l² + m² + n² = 1
- Angle between lines:cos θ = |l₁l₂ + m₁m₂ + n₁n₂|
- Distance:PQ = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
- Line: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c
- Plane: ax + by + cz + d = 0
12
Full chapter notesLinear Programming
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Linear Programming — Formulas
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Linear Programming
- Objective function: Z = ax + by
- Optimal value occurs at a corner point of the feasible region
- Constraints: linear inequalities + x ≥ 0, y ≥ 0
13
Full chapter notesProbability
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Probability — Formulas
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Probability
- P(E) = favourable outcomes / total outcomes
- P(A∪B) = P(A) + P(B) − P(A∩B)
- P(A|B) = P(A∩B) / P(B)
- Independent: P(A∩B) = P(A)P(B)
- Bayes:P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ)
- E(X) = Σ xᵢpᵢ ; Var(X) = E(X²) − [E(X)]²
- Binomial: P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ
Each chapter links to full notes, solved examples, and MCQs on Rishtaara.