Class 12 Maths Formulas Handbook
A single quick-reference page for every Class 12 Maths chapter’s key formulas and results, updated as each chapter’s NCERT Solutions go live on the site. Bookmark this page — it grows chapter by chapter through the year.
Chapter 1: Relations and Functions
- Reflexive: (a,a)∈R for all a∈A.
- Symmetric: (a,b)∈R ⇒ (b,a)∈R.
- Transitive: (a,b)∈R, (b,c)∈R ⇒ (a,c)∈R.
- Equivalence relation = reflexive + symmetric + transitive; partitions A into equivalence classes.
- One-one (injective): f(x1)=f(x2) ⇒ x1=x2.
- Onto (surjective): range(f) = co-domain.
- Bijective = one-one + onto = invertible.
- Composition: (gof)(x)=g(f(x)); if f,g are bijective, gof is bijective and (gof)⁻¹=f⁻¹og⁻¹.
- Binary operation on A: a function *:A×A→A. Commutative: a*b=b*a. Associative: (a*b)*c=a*(b*c). Identity e: a*e=e*a=a. Inverse of a: element b with a*b=b*a=e.
Chapter 2: Inverse Trigonometric Functions
- Principal value branches: sin⁻¹x: [−π/2,π/2]; cos⁻¹x: [0,π]; tan⁻¹x: (−π/2,π/2).
- sin⁻¹x+cos⁻¹x=π/2; tan⁻¹x+cot⁻¹x=π/2.
- tan⁻¹x+tan⁻¹y=tan⁻¹[(x+y)/(1−xy)] (xy<1).
- sin⁻¹(sin x)=x only for x∈[−π/2,π/2] (principal domain).
Related NCERT Content for Class 12 Maths Chapter 2
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Chapter 3: Matrices
- Matrix order: m×n (rows×columns).
- Multiplication AB defined only if cols(A)=rows(B); not commutative.
- Transpose A’: rows↔columns. Symmetric: A’=A. Skew-symmetric: A’=−A.
- Any square matrix = symmetric part + skew-symmetric part.
Related NCERT Content for Class 12 Maths Chapter 3
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Chapter 4: Determinants
- 2×2 determinant: |A|=ad−bc.
- Singular matrix: |A|=0 (no inverse).
- A⁻¹=adj(A)/|A|, |A|≠0.
- Matrix method: AX=B ⇒ X=A⁻¹B.
- Area of triangle via determinant formula.
Related NCERT Content for Class 12 Maths Chapter 4
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Chapter 5: Continuity and Differentiability
- Continuous at a: LHL=RHL=f(a).
- Differentiable ⇒ continuous (not vice versa, e.g. |x| at x=0).
- Chain rule: [f(g(x))]’=f'(g(x))g'(x).
- d/dx(ex)=ex; d/dx(log x)=1/x.
- Rolle’s Theorem: f(a)=f(b) ⇒ ∃c with f'(c)=0.
Related NCERT Content for Class 12 Maths Chapter 5
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Chapter 6: Applications of Derivatives
- Increasing: f'(x)≥0; decreasing: f'(x)≤0.
- Tangent: y−y₀=f'(x₀)(x−x₀); Normal slope=−1/f'(x₀).
- Δy≈f'(x)Δx (approximation via differentials).
- Second derivative test: f”<0 ⇒ local max; f”>0 ⇒ local min.
Related NCERT Content for Class 12 Maths Chapter 6
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Chapter 7: Integrals
- ∫xndx=xn+1/(n+1)+C; ∫1/x dx=log|x|+C; ∫exdx=ex+C.
- Integration by parts: ∫u dv=uv−∫v du (ILATE order).
- ∫abf(x)dx=F(b)−F(a) (Fundamental Theorem).
Related NCERT Content for Class 12 Maths Chapter 7
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Chapter 8: Applications of Integrals
- Area under curve: ∫ab|f(x)|dx.
- Area between curves: ∫ab[f(x)−g(x)]dx (upper−lower).
- Circle x²+y²=a²: area=πa².
Related NCERT Content for Class 12 Maths Chapter 8
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Chapter 9: Differential Equations
- Order=highest derivative; degree=power of highest-order derivative.
- Variable separable: f(y)dy=g(x)dx.
- Homogeneous: substitute y=vx.
- Linear dy/dx+Py=Q: IF=e∫P dx, solution y·IF=∫Q·IF dx+C.
Related NCERT Content for Class 12 Maths Chapter 9
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Chapter 10: Vectors
- |a|=√(a₁²+a₂²+a₃²); unit vector=a/|a|.
- Dot product: a·b=|a||b|cosθ; perpendicular ⇔ dot=0.
- Cross product: a×b=|a||b|sinθ &ncirc;; parallel ⇔ cross=0.
- Triangle area=½|a×b|.
Related NCERT Content for Class 12 Maths Chapter 10
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Chapter 11: Three-Dimensional Geometry
- Direction cosines: l²+m²+n²=1.
- Line: r=a+λb; Cartesian (x−x₁)/a=(y−y₁)/b=(z−z₁)/c.
- Plane: r·n=d; Cartesian ax+by+cz=d.
- Point-to-plane distance: |ax₁+by₁+cz₁−d|/√(a²+b²+c²).
Related NCERT Content for Class 12 Maths Chapter 11
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Chapter 12: Linear Programming
- Objective function: Z=ax+by (maximize/minimize).
- Feasible region: common region satisfying all linear constraints.
- Optimal Z occurs at a corner point (vertex) of the feasible region.
Related NCERT Content for Class 12 Maths Chapter 12
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Chapter 13: Probability
- Conditional: P(A|B)=P(A∩B)/P(B).
- Independence: P(A∩B)=P(A)P(B).
- Bayes theorem: P(Ei|A)=[P(Ei)P(A|Ei)]/ΣP(Ej)P(A|Ej).
- Binomial: P(X=r)=nCrprqn−r.
Related NCERT Content for Class 12 Maths Chapter 13
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How to Use this Handbook
Each chapter section above lists only the formulas and definitions you need for quick recall before an exam or while solving problems — for full worked examples, see that chapter’s NCERT Solutions and Extra Questions (HOTS) posts, linked from the class hub page.
Frequently Asked Questions
Q: Does this handbook cover the full Class 12 Maths syllabus?
A: It covers all 13 chapters of the Class 12 Maths syllabus (2026-27): Relations and Functions, Inverse Trigonometric Functions, Matrices, Determinants, Continuity and Differentiability, Applications of Derivatives, Integrals, Applications of Integrals, Differential Equations, Vectors, Three-Dimensional Geometry, Linear Programming, and Probability.
Q: Is this handbook enough to revise for exams on its own?
A: It’s a fast recall tool for formulas and definitions — pair it with the chapter’s Revision Notes post (condensed explanations) and the Solutions/Extra Questions posts (worked practice) for full exam prep.
