CBSE Mathematics Class 12:
Formulas, Theorems & AI Tools
Complete formula reference for calculus, matrices, vectors, 3D geometry, and probability — organised by the sections that carry the most marks in CBSE boards.
Generate Maths flashcards free →Derivatives, Integrals, and Differential Equations
Derivatives — key rules
- d/dx(xⁿ) = nxⁿ⁻¹
- Product: d(uv)/dx = uv' + vu'
- Quotient: (uv' − vu')/v²
- Chain rule: (dy/du)·(du/dx)
- d/dx(eˣ) = eˣ | d/dx(ln x) = 1/x
- d/dx(sin x) = cos x | d/dx(cos x) = −sin x
Integration — standard forms
- ∫xⁿ dx = xⁿ⁺¹/(n+1) + C
- ∫eˣ dx = eˣ + C
- ∫dx/(x²+a²) = (1/a)tan⁻¹(x/a)+C
- ∫dx/√(a²−x²) = sin⁻¹(x/a)+C
- By parts: ∫u·v = u∫v − ∫(u'·∫v)
- Property: ∫₀ᵃf(x)=∫₀ᵃf(a−x) dx
Differential Equations
- Linear: dy/dx + Py = Q
- I.F. = e^(∫P dx)
- Solution: y·I.F. = ∫Q·I.F. dx + C
- Variable sep.: f(y)dy = g(x)dx
- Order = highest derivative
- Degree = power of highest derivative
Forgetting +C on every indefinite integral costs marks. Boards deduct for each missing constant. Also: degree of D.E. is undefined when the equation is not a polynomial in derivatives — e.g. sin(y') has no defined degree.
High-yield formulas for the second half
Matrices & Determinants
- A⁻¹ = adj(A)/|A| (|A| ≠ 0)
- A·adj(A) = |A|·I
- |adj(A)| = |A|^(n−1)
- Cramer: x = D_x/D, y = D_y/D
- |kA| = kⁿ|A| for n×n matrix
Vectors
- a·b = |a||b|cosθ
- |a×b| = |a||b|sinθ
- Area of triangle = ½|a×b|
- Projection of a on b = a·b/|b|
- Triple product: [a b c] = a·(b×c)
- Coplanar iff [a b c] = 0
Probability
- P(A|B) = P(A∩B)/P(B)
- Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/ΣP(Eⱼ)P(A|Eⱼ)
- Binomial: P(X=r) = ⁿCᵣpʳqⁿ⁻ʳ
- Mean = np | Variance = npq
- P(A∪B) = P(A)+P(B)−P(A∩B)
Generate Maths flashcards from your notes in 30 seconds
Paste any NCERT Maths chapter — Calculus, Algebra, Statistics. Get CBSE-style Q&As covering the exact step-by-step patterns boards reward marks for.
Generate flashcards free →How to maximise marks in Maths
Always show working — every step
CBSE Maths rewards method marks. A 4-mark question where you get the formula right but make an arithmetic error can still score 3 marks — but only if you showed the formula and substitution clearly.
Integration by parts — ILATE order
Choose u based on ILATE: Inverse trig → Logarithmic → Algebraic → Trigonometric → Exponential. The first type in the list becomes u. This order works for ~95% of CBSE by-parts questions.
Definite integral property — examiner favourite
∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx is used to simplify almost every tricky definite integral in boards. If you see sin/cos or log in the integrand, try this substitution first.
Bayes' theorem — structured approach
Draw the tree diagram first: prior events on first branches, conditional probabilities on second branches. Then apply the formula. Students who skip the tree lose track of which numerator/denominator to use.