Chapter Notes›Class 12 Mathematics

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Class 12 MathematicsChapter Notes

13 chapters · Definitions, key points, formulas & exam tips · Updated 2025-26

Ch 1 Ch 2 Ch 3 Ch 4 Ch 5 Ch 6 Ch 7 Ch 8 Ch 9 Ch 10 Ch 11 Ch 12 Ch 13

Ch 1

Relations and Functions

Key Definitions

Relation: A subset of the Cartesian product A × B. Represents a connection between elements of two sets.

Function: A relation where every element of domain has exactly one image in codomain.

Bijective Function: A function that is both injective (one-one) and surjective (onto).

Key Points to Remember

→Reflexive: (a,a) ∈ R for all a ∈ A.
→Symmetric: (a,b) ∈ R ⟹ (b,a) ∈ R.
→Transitive: (a,b) ∈ R and (b,c) ∈ R ⟹ (a,c) ∈ R.
→Equivalence relation: reflexive + symmetric + transitive.
→One-one (injective): f(a) = f(b) ⟹ a = b.
→Onto (surjective): every element of codomain has at least one pre-image.
→Invertible function must be bijective.

Exam Tips

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To prove a function is one-one: assume f(x₁) = f(x₂) and show x₁ = x₂.

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To prove onto: let y be in codomain, find x in domain such that f(x) = y.

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Composition of functions: (f∘g)(x) = f(g(x)).

Ch 2

Inverse Trigonometric Functions

Key Definitions

Principal Value: The value of inverse trig function that lies in the defined principal range.

Key Points to Remember

→Range of sin⁻¹: [−π/2, π/2]. Range of cos⁻¹: [0, π]. Range of tan⁻¹: (−π/2, π/2).
→sin⁻¹(sin x) = x only if x ∈ [−π/2, π/2].
→sin⁻¹(−x) = −sin⁻¹(x). cos⁻¹(−x) = π − cos⁻¹(x).
→sin⁻¹x + cos⁻¹x = π/2. tan⁻¹x + cot⁻¹x = π/2.
→2tan⁻¹x = sin⁻¹(2x/1+x²) = cos⁻¹(1−x²/1+x²) = tan⁻¹(2x/1−x²).

Exam Tips

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Always check domain before writing inverse trig value.

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Identity problems: use compound angle formulas for tan⁻¹.

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Simplification: convert to single inverse trig function.

Ch 3

Matrices

Key Definitions

Matrix: A rectangular array of numbers arranged in rows and columns. Order = m × n (m rows, n columns).

Symmetric Matrix: A square matrix where A = Aᵀ (transpose equals original). All diagonal matrices are symmetric.

Skew-Symmetric Matrix: A = −Aᵀ. Diagonal elements must be zero. Every square matrix = symmetric + skew-symmetric.

Key Points to Remember

→Addition: defined only when orders are same. A + B = B + A (commutative).
→Multiplication: A × B defined only when columns of A = rows of B. Order of AB = (rows of A) × (columns of B).
→Matrix multiplication is NOT commutative: AB ≠ BA in general.
→Transpose: (AB)ᵀ = BᵀAᵀ. (A + B)ᵀ = Aᵀ + Bᵀ.
→Identity matrix I: AI = IA = A. Null matrix O: AO = OA = O.
→Inverse: A⁻¹ exists only if A is square and non-singular (det ≠ 0). AA⁻¹ = I.

Exam Tips

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Order of product AB: if A is m×n and B is n×p, AB is m×p. Check compatibility first.

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For 2×2: A⁻¹ = (1/det A) × [d, −b; −c, a] for A = [a,b;c,d].

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Idempotent matrix: A² = A. Involutory: A² = I.

Ch 4

Determinants

Key Definitions

Determinant: A scalar value computed from a square matrix. |A| denotes determinant of A. Non-zero determinant means matrix is invertible.

Cofactor: Cij = (−1)^(i+j) × Mij where Mij is the minor (determinant of submatrix obtained by deleting row i, column j).

Adjoint: Transpose of the cofactor matrix. adj(A) = Cᵀ. A⁻¹ = adj(A)/|A|.

Key Points to Remember

→Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): Δ = ½|det of 3×3 matrix with 1s in third column|.
→Singular matrix: |A| = 0 — no inverse exists.
→Properties: det(AB) = det(A) × det(B). det(Aᵀ) = det(A). det(kA) = kⁿ det(A) for n×n matrix.
→Row operations on determinant: swapping rows changes sign; multiplying row by k multiplies det by k.
→System Ax = b: unique solution if |A| ≠ 0. Cramer's rule: xᵢ = |Aᵢ|/|A|.
→Consistent system: |A| ≠ 0 → unique solution. |A| = 0 → either no solution or infinitely many.

Formulas & Equations

A⁻¹ = adj(A)/|A|

Area of Δ = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

det of 2×2: |a b; c d| = ad − bc

Exam Tips

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Expand along the row/column with most zeros — saves calculation.

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For 3×3: expand along first row using cofactors C₁₁, C₁₂, C₁₃.

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Collinear points: area of triangle = 0 (use determinant method).

Ch 5

Continuity and Differentiability

Key Definitions

Continuity: f is continuous at x = a if lim(x→a) f(x) = f(a). Three conditions: limit exists, f(a) defined, they are equal.

Differentiability: f is differentiable at x = a if the limit [f(a+h)−f(a)]/h exists as h→0. Differentiable ⟹ continuous, but not vice versa.

Rolle's Theorem: If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then ∃ c ∈ (a,b) such that f'(c) = 0.

Key Points to Remember

→Chain rule: d/dx[f(g(x))] = f'(g(x)) × g'(x).
→Implicit differentiation: differentiate both sides with respect to x; use dy/dx for y terms.
→Logarithmic differentiation: use when function is of the form uᵛ or product of many terms.
→Mean Value Theorem: f'(c) = [f(b)−f(a)]/(b−a) for some c ∈ (a,b).
→Second derivative test: f''(c) < 0 → local max at c; f''(c) > 0 → local min at c.
→Standard derivatives: d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(sin⁻¹x) = 1/√(1−x²).

Formulas & Equations

d/dx(xⁿ) = nxⁿ⁻¹

d/dx(sin⁻¹x) = 1/√(1−x²)

d/dx(tan⁻¹x) = 1/(1+x²)

Parametric: dy/dx = (dy/dt)/(dx/dt)

Exam Tips

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For continuity at piecewise boundary: check LHL = RHL = f(a).

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Differentiability at corner points of modulus functions: always check using first principles.

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y = xˣ: take log both sides → ln y = x ln x → differentiate implicitly.

Ch 6

Application of Derivatives

Key Definitions

Rate of Change: dy/dx = rate of change of y with respect to x. If x and y are functions of t, use chain rule: dy/dt = (dy/dx)(dx/dt).

Increasing Function: f is increasing on (a,b) if f'(x) > 0 for all x ∈ (a,b).

Tangent and Normal: Tangent slope at P = f'(x₀). Normal is perpendicular to tangent: slope of normal = −1/f'(x₀).

Key Points to Remember

→Maxima/minima (first derivative test): f'(x) changes from +ve to −ve at c → local max; −ve to +ve → local min.
→Critical points: f'(x) = 0 or f'(x) undefined.
→Absolute max/min on [a,b]: evaluate f at critical points + endpoints; compare values.
→Approximation: Δy ≈ dy = f'(x) Δx (linear approximation).
→Increasing: f'(x) > 0. Decreasing: f'(x) < 0. Neither (stationary): f'(x) = 0.
→Point of inflection: f''(x) = 0 and sign of f'' changes.

Exam Tips

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Optimization: form the objective function, find derivative, solve for critical points.

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Check endpoints for absolute extrema on a closed interval.

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Equation of tangent at (x₁,y₁): y − y₁ = m(x − x₁) where m = dy/dx at (x₁,y₁).

Ch 7

Integrals

Key Definitions

Indefinite Integral: ∫f(x)dx = F(x) + C where F'(x) = f(x). C is constant of integration.

Definite Integral: ∫ₐᵇ f(x)dx = F(b) − F(a) (Fundamental Theorem of Calculus). Gives a specific numerical value.

Integration by Parts: ∫u dv = uv − ∫v du. Choose u using ILATE rule: Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential.

Key Points to Remember

→Standard integrals: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ −1). ∫1/x dx = ln|x| + C.
→Substitution: if integrand has f(g(x))g'(x), substitute t = g(x).
→Partial fractions: for rational functions — decompose into simpler fractions before integrating.
→∫eˣ[f(x) + f'(x)]dx = eˣf(x) + C (standard result for integration by parts with eˣ).
→Properties of definite integrals: ∫ₐᵇ f = −∫ᵦₐ f. ∫ₐᵃ f = 0. ∫₀²ᵃ f = ∫₀ᵃ f + ∫₀ᵃ f(2a−x).
→If f is even: ∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx. If f is odd: ∫₋ₐᵃ f(x)dx = 0.

Formulas & Equations

∫sin x dx = −cos x + C

∫cos x dx = sin x + C

∫eˣ dx = eˣ + C

∫1/(x²+a²)dx = (1/a)tan⁻¹(x/a) + C

Exam Tips

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ILATE rule for integration by parts: choose u in order I → L → A → T → E.

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For ∫√(a²−x²)dx and ∫√(x²±a²)dx: use standard formula directly — memorise these.

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Limit substitution in definite integrals: when substituting t = g(x), change limits too.

Ch 8

Application of Integrals

Key Definitions

Area under a Curve: Area between curve y = f(x) and x-axis from x = a to x = b: A = ∫ₐᵇ |f(x)| dx.

Area between Two Curves: A = ∫ₐᵇ [f(x) − g(x)] dx when f(x) ≥ g(x) on [a, b].

Key Points to Remember

→For area below x-axis: f(x) < 0, so take absolute value: A = |∫ₐᵇ f(x)dx|.
→Find intersection points of two curves: solve f(x) = g(x) to get limits of integration.
→Area of standard regions: circle x²+y²=r² has area πr² (∫₋ᵣʳ √(r²−x²)dx = πr²/2).
→When curve and line enclose region: split if needed at intersection points.
→Area can be computed along x-axis (horizontal strips: dx) or y-axis (vertical strips: dy).

Formulas & Equations

A = ∫ₐᵇ [f(x) − g(x)]dx (upper − lower)

A = ∫_c^d [h(y) − k(y)]dy (right − left, integrating along y)

Exam Tips

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Always sketch the region first — identify which curve is on top.

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If curves cross inside [a,b]: split integral at crossing point.

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Common exam: area enclosed by parabola and line — find intersection points first.

Ch 9

Differential Equations

Key Definitions

Order: Order of the highest derivative in the equation. e.g., y'' + 3y' + 2y = 0 has order 2.

Degree: Power of the highest order derivative (after making the equation polynomial in derivatives). Defined only for polynomial DEs.

General Solution: Solution containing arbitrary constants (equal in number to the order of the DE).

Key Points to Remember

→Variable separable: rearrange to get g(y)dy = f(x)dx, then integrate both sides.
→Homogeneous DE: f(x,y) where f(λx,λy) = λⁿf(x,y). Substitute y = vx (v = y/x).
→Linear DE: dy/dx + P(x)y = Q(x). Integrating factor: IF = e^(∫P dx). Solution: y × IF = ∫Q × IF dx.
→Formation: differentiate y = f(x, C) as many times as constants, then eliminate constants.
→Particular solution: substitute initial condition into general solution to find constant.

Formulas & Equations

Linear DE solution: y × e^(∫P dx) = ∫Q × e^(∫P dx) dx + C

Variable separable: ∫g(y)dy = ∫f(x)dx + C

Exam Tips

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Identify the type: variable separable → homogeneous → linear first order.

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For homogeneous DEs: after y = vx substitution, get a separable DE in v and x.

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Integrating factor: e^(∫P(x)dx) — simplify P(x) first before integrating.

Ch 10

Vector Algebra

Key Definitions

Position Vector: Vector from origin O to point P. Written as OP⃗ or p⃗. Components: (x, y, z) = xî + yĵ + zk̂.

Dot Product: a⃗·b⃗ = |a||b|cosθ. Scalar result. a⃗·b⃗ = 0 ⟹ perpendicular vectors.

Cross Product: a⃗×b⃗ = |a||b|sinθ n̂. Vector result. |a⃗×b⃗| = area of parallelogram. a⃗×b⃗ = 0 ⟹ parallel vectors.

Key Points to Remember

→Magnitude: |a⃗| = √(a₁² + a₂² + a₃²). Unit vector: â = a⃗/|a⃗|.
→Section formula (internal division): r⃗ = (mb⃗ + na⃗)/(m+n).
→î·î = 1, î·ĵ = 0. î×î = 0, î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ.
→Projection of a⃗ on b⃗ = (a⃗·b⃗)/|b⃗|.
→Scalar triple product: [a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗). Volume of parallelepiped = |[a⃗ b⃗ c⃗]|.
→Coplanar vectors: scalar triple product = 0.

Formulas & Equations

a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃

cosθ = (a⃗·b⃗)/(|a⃗||b⃗|)

|a⃗×b⃗| = |a⃗||b⃗|sinθ

Area of Δ = ½|a⃗×b⃗|

Exam Tips

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Cross product using determinant: a⃗×b⃗ = |î ĵ k̂; a₁ a₂ a₃; b₁ b₂ b₃|.

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Angle between vectors: use dot product formula — always yields angle in [0°, 180°].

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Parallel vectors: a⃗×b⃗ = 0 (cross product is zero vector).

Ch 11

Three Dimensional Geometry

Key Definitions

Direction Cosines: Cosines of angles made by a line with positive x, y, z axes: l, m, n. l² + m² + n² = 1.

Direction Ratios: Any numbers proportional to direction cosines. If DRs are a, b, c then l = a/√(a²+b²+c²) etc.

Skew Lines: Lines in 3D that are neither parallel nor intersecting. Shortest distance between them: d = |(b₂−b₁)·(d₁×d₂)|/|d₁×d₂|.

Key Points to Remember

→Equation of line through (x₁,y₁,z₁) with DRs a,b,c: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c.
→Vector form of line: r⃗ = a⃗ + λb⃗ (a⃗ = position vector of point, b⃗ = direction vector).
→Equation of plane: ax + by + cz = d where (a,b,c) is normal vector.
→Distance of point (x₁,y₁,z₁) from plane ax+by+cz+d=0: |ax₁+by₁+cz₁+d|/√(a²+b²+c²).
→Angle between two planes: cosθ = |n₁⃗·n₂⃗|/(|n₁⃗||n₂⃗|) where n₁, n₂ are normal vectors.
→Angle between line and plane: sinθ = |b⃗·n⃗|/(|b⃗||n⃗|).

Formulas & Equations

Distance between parallel planes ax+by+cz=d₁ and d₂: |d₁−d₂|/√(a²+b²+c²)

Plane through 3 points: use determinant form

Line perpendicular to plane: DRs = normal vector of plane

Exam Tips

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Foot of perpendicular from point to line: parametrise line, minimise distance squared.

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Image of point in plane: foot of perpendicular is midpoint of point and its image.

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Coplanar lines: check if (a₂−a₁)·(b₁×b₂) = 0 (scalar triple product).

Ch 12

Linear Programming

Key Definitions

Objective Function: The linear function to be maximised or minimised. Z = ax + by.

Feasible Region: The region satisfying all constraints. Corner points of this region are candidates for optimal solution.

Optimal Solution: The maximum or minimum value of Z occurs at a corner (vertex) of the feasible region — Corner Point Theorem.

Key Points to Remember

→Feasible region is convex and bounded by constraint lines.
→Method: graph constraints → identify feasible region → find corner points → evaluate Z at each corner.
→Unbounded feasible region: maximum may not exist (check if Z can be increased indefinitely).
→If feasible region is empty: no solution (problem is infeasible).
→Multiple optimal solutions: Z is same at two adjacent corners — all points on that edge are optimal.
→Class 12 syllabus: problems in two variables only.

Exam Tips

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Convert inequalities to equalities to draw boundary lines. Use intercept method: set x=0 for y-intercept, y=0 for x-intercept.

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To determine which side of line to shade: substitute (0,0) in inequality and check truth.

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Evaluate Z at every corner point — don't skip any corner.

Ch 13

Probability

Key Definitions

Conditional Probability: P(A|B) = P(A∩B)/P(B). Probability of A given B has occurred. Defined when P(B) > 0.

Independent Events: P(A∩B) = P(A)·P(B). Knowing B occurred does not affect probability of A. Note: mutually exclusive ≠ independent.

Bayes' Theorem: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ). Updates probability of hypothesis given observed evidence.

Key Points to Remember

→Multiplication theorem: P(A∩B) = P(A)·P(B|A) = P(B)·P(A|B).
→Total Probability: P(A) = Σ P(Eᵢ)·P(A|Eᵢ) where E₁, E₂, … partition the sample space.
→Binomial distribution: X ~ B(n,p). P(X=r) = ⁿCᵣ × pʳ × (1−p)ⁿ⁻ʳ.
→Mean of binomial distribution = np. Variance = npq (where q = 1−p).
→Random variable: variable that takes numerical values based on outcome of random experiment.
→Probability distribution: lists all values of X with their probabilities. Sum of probabilities = 1.

Formulas & Equations

P(A|B) = P(A∩B)/P(B)

Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ)

Binomial: P(X=r) = ⁿCᵣ pʳ qⁿ⁻ʳ

E(X) = np, Var(X) = npq

Exam Tips

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Bayes' theorem problems: always list all hypotheses (prior probabilities) and likelihoods first, then apply the formula.

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Independent vs mutually exclusive: two events can't be both (unless one has zero probability).

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For binomial: confirm n (trials), p (success probability), r (required successes).

Frequently Asked Questions

Are these notes based on 2025-26 CBSE syllabus for Class 12 Mathematics?

Yes. All chapter notes here are based on the latest 2025-26 CBSE syllabus for Class 12 Mathematics. Deleted topics are clearly marked so you focus only on what will be tested in your board exam.

How to study Class 12 Mathematics notes effectively for board exams?

Read each chapter's notes once to build understanding. Then close the notes and try to recall every key point, definition, and formula from memory. Anything you miss is your weak area — revisit only those points. This active recall method takes less time and retains far more than re-reading.

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