Padikam Lessons Library | Kerala SCERT & CBSE NCERT Textbooks

CBSE NCERTClass 12Mathematics

Mathematics

13 chapters PDF / 94 KB

ORIGINAL STUDY COMPANION PADIKAM CLASS 12 | SCIENCE / COMMERCE Mathematics CBSE / NCERT core lessons | Academic year 2026-27 13 chapter lessons Quick explanations, core points, exam practice and chapter-specific notation in one consistent format. Padikam explanations are independently written. Official textbooks are used as the coverage map, chapter sequence and terminology reference. PADIKAM | CBSE NCERT CLASS 12 1

Contents 01 Relations and Functions 08 Applications of Integrals 02 Inverse Trigonometric Functions 09 Differential Equations 03 Matrices 10 Vector Algebra 04 Determinants 11 Three-dimensional Geometry 05 Continuity and Differentiability 12 Linear Programming 06 Applications of Derivatives 13 Probability 07 Integrals Study rhythm Read the quick explanation, recall the core points aloud, then answer the prompts without looking back. Use the official textbook links for full prescribed readings, activities and diagrams. Textbook and verification links

CHAPTER 01 Relations and Functions Quick understanding Relations classify links between set elements, while functions assign each input exactly one output. Core points Check your learning Test reflexive, symmetric and transitive properties Explain the central idea of Relations and separately. Functions in your own words. One-one concerns repeated outputs; onto concerns Apply one idea from Relations and coverage of the codomain. Functions to a textbook-style example or source. A function has an inverse exactly when it is bijective. Equation and notation (f ∘ g)(x) = f(g(x)) f⁻¹(f(x)) = x SYMBOLS AND UNITS f, g: functions x: element of the relevant domain ∘: composition operator f⁻¹: inverse function; not reciprocal One-line recall Test reflexive, symmetric and transitive properties separately. PADIKAM | CBSE NCERT CLASS 12 3

CHAPTER 02 Inverse Trigonometric Functions Quick understanding Inverse trigonometric functions use restricted domains so each input has one principal-value output. Core points Check your learning Always state the principal-value range. Explain the central idea of Inverse Trigonometric Functions in your own Use identities only after checking domains and signs. words. Graphs are reflections of restricted trigonometric Apply one idea from Inverse graphs in y = x. Trigonometric Functions to a textbook- style example or source. Equation and notation sin⁻¹x + cos⁻¹x = π/2 tan⁻¹x + cot⁻¹x = π/2 SYMBOLS AND UNITS x: real input in the permitted domain π: angle of a half-turn (rad) Inverse-function values are angles, usually in radians One-line recall Always state the principal-value range. PADIKAM | CBSE NCERT CLASS 12 4

CHAPTER 03 Matrices Quick understanding Matrices organise rectangular data and support transformations and systems of equations through rule-based operations. Core points Check your learning Matrix multiplication is generally not commutative. Explain the central idea of Matrices in your own words. Transpose reverses rows and columns. Apply one idea from Matrices to a An inverse exists only for a nonsingular square textbook-style example or source. matrix. Equation and notation (AB)ᵀ = BᵀAᵀ AA⁻¹ = A⁻¹A = I SYMBOLS AND UNITS A, B: compatible matrices ᵀ: transpose A⁻¹: inverse of A I: identity matrix One-line recall Matrix multiplication is generally not commutative. PADIKAM | CBSE NCERT CLASS 12 5

CHAPTER 04 Determinants Quick understanding A determinant is a scalar attached to a square matrix that detects invertibility and supports area and equation solving. Core points Check your learning Row or column operations change determinants in Explain the central idea of Determinants predictable ways. in your own words. A zero determinant means the matrix is singular. Apply one idea from Determinants to a textbook-style example or source. Cofactors and the adjoint construct the inverse. Equation and notation A⁻¹ = adj(A) / |A|, when |A| ≠ 0 Area of triangle = ½|x₁(y₂-y₃)+x₂(y₃-y₁)+x₃(y₁-y₂)| SYMBOLS AND UNITS |A|: determinant of matrix A adj(A): adjoint of A xᵢ, yᵢ: coordinates in the chosen length unit One-line recall Row or column operations change determinants in predictable ways. PADIKAM | CBSE NCERT CLASS 12 6

CHAPTER 05 Continuity and Differentiability Quick understanding Continuity matches limiting and actual values; differentiability gives a locally defined rate of change. Core points Check your learning Differentiability implies continuity at an interior point. Explain the central idea of Continuity and Differentiability in your own words. Use chain, product and quotient rules systematically. Apply one idea from Continuity and Implicit and logarithmic differentiation simplify difficult Differentiability to a textbook-style forms. example or source. Equation and notation f is continuous at a if lim(x→a) f(x) = f(a) d/dx[f(g(x))] = f′(g(x))g′(x) d/dx(eˣ) = eˣ SYMBOLS AND UNITS x, a: real variables f, g: functions f′: derivative with respect to x One-line recall Differentiability implies continuity at an interior point. PADIKAM | CBSE NCERT CLASS 12 7

CHAPTER 06 Applications of Derivatives Quick understanding Derivatives answer questions about rate, slope, monotonicity, tangency and optimum values. Core points Check your learning The sign of f prime determines increasing or Explain the central idea of Applications of decreasing intervals. Derivatives in your own words. Critical points require classification, not automatic Apply one idea from Applications of labelling. Derivatives to a textbook-style example or source. Related-rate problems must keep units and time dependence clear. Equation and notation Slope of tangent = dy/dx Slope of normal = -1 / (dy/dx) At an interior extremum, f′(x) = 0 is a necessary condition SYMBOLS AND UNITS x, y: variables in the problem units dy/dx: rate of y with respect to x f′(x): first derivative One-line recall The sign of f prime determines increasing or decreasing intervals. PADIKAM | CBSE NCERT CLASS 12 8

CHAPTER 07 Integrals Quick understanding Integration reverses differentiation and also accumulates continuously varying quantities. Core points Check your learning An indefinite integral represents a family of Explain the central idea of Integrals in antiderivatives. your own words. Substitution, partial fractions and integration by parts Apply one idea from Integrals to a match different structures. textbook-style example or source. A definite integral is a number with limits and obeys useful symmetry properties. Equation and notation ∫f(x) dx = F(x) + C, where F′(x)=f(x) ∫u dv = uv - ∫v du ∫ₐᵇ f(x)dx = F(b)-F(a) SYMBOLS AND UNITS x: variable of integration C: arbitrary constant a, b: lower and upper limits u, v: differentiable expressions One-line recall An indefinite integral represents a family of antiderivatives. PADIKAM | CBSE NCERT CLASS 12 9

CHAPTER 08 Applications of Integrals Quick understanding Definite integrals calculate areas by summing thin strips while respecting which curve lies above or to the right. Core points Check your learning Sketch the region before choosing limits. Explain the central idea of Applications of Integrals in your own words. Split the interval if curves cross. Apply one idea from Applications of Area is non-negative even when a signed integral is Integrals to a textbook-style example or negative. source. Equation and notation Area = ∫ₐᵇ [upper curve - lower curve] dx Area = ∫cᵈ [right curve - left curve] dy SYMBOLS AND UNITS a, b: x-limits c, d: y-limits Area uses the square of the coordinate length unit One-line recall Sketch the region before choosing limits. PADIKAM | CBSE NCERT CLASS 12 10

CHAPTER 09 Differential Equations Quick understanding A differential equation relates an unknown function to its derivatives, and a solution satisfies that relation over an interval. Core points Check your learning Order is the highest derivative present. Explain the central idea of Differential Equations in your own words. General solutions contain arbitrary constants; particular solutions satisfy given conditions. Apply one idea from Differential Equations to a textbook-style example or Variable-separable and linear forms use different source. standard methods. Equation and notation dy/dx = g(x)h(y) ⇒ ∫dy/h(y) = ∫g(x)dx dy/dx + Py = Q Integrating factor = e^(∫P dx) SYMBOLS AND UNITS x: independent variable y: dependent variable P, Q: functions of x C: integration constant One-line recall Order is the highest derivative present. PADIKAM | CBSE NCERT CLASS 12 11

CHAPTER 10 Vector Algebra Quick understanding Vectors combine magnitude and direction and support geometric calculations through components and products. Core points Check your learning The dot product measures parallel alignment. Explain the central idea of Vector Algebra in your own words. The cross product produces a perpendicular vector. Apply one idea from Vector Algebra to a Resolve vectors in a common basis before textbook-style example or source. operating. Equation and notation a · b = |a||b| cos θ |a × b| = |a||b| sin θ Projection of a on b = (a·b)/|b| SYMBOLS AND UNITS a, b: vectors |a|, |b|: vector magnitudes θ: angle between vectors One-line recall The dot product measures parallel alignment. PADIKAM | CBSE NCERT CLASS 12 12

CHAPTER 11 Three-dimensional Geometry Quick understanding Lines in space are described by a point and direction, and their relationships follow vector geometry. Core points Check your learning Direction ratios need not be unit values. Explain the central idea of Three- dimensional Geometry in your own Skew lines are neither parallel nor intersecting. words. Shortest distance follows a projection on a common Apply one idea from Three-dimensional perpendicular. Geometry to a textbook-style example or source. Equation and notation Line: r = a + λb cos θ = (b₁·b₂)/(|b₁||b₂|) Distance between skew lines = |(a₂-a₁)·(b₁×b₂)| / |b₁×b₂| SYMBOLS AND UNITS r: position vector a, a₁, a₂: fixed position vectors b, b₁, b₂: direction vectors λ: real parameter θ: angle between lines One-line recall Direction ratios need not be unit values. PADIKAM | CBSE NCERT CLASS 12 13

CHAPTER 12 Linear Programming Quick understanding Linear programming optimises a linear objective over a feasible region defined by linear constraints. Core points Check your learning Translate every verbal restriction into an inequality. Explain the central idea of Linear Programming in your own words. The feasible region is the common solution set. Apply one idea from Linear Programming For a bounded polygonal region, test the objective at to a textbook-style example or source. corner points. Equation and notation Optimise Z = ax + by Subject to aᵢx + bᵢy ≤, =, or ≥ cᵢ SYMBOLS AND UNITS x, y: decision variables Z: objective value a, b, aᵢ, bᵢ, cᵢ: given constants One-line recall Translate every verbal restriction into an inequality. PADIKAM | CBSE NCERT CLASS 12 14

CHAPTER 13 Probability Quick understanding Conditional probability updates chance using available information and leads to multiplication, Bayes and random-variable models. Core points Check your learning Independence is stronger than mutual exclusivity. Explain the central idea of Probability in your own words. Bayes theorem reverses a conditional probability using prior information. Apply one idea from Probability to a textbook-style example or source. A probability distribution assigns non-negative probabilities summing to one. Equation and notation P(A|B) = P(A∩B) / P(B) P(A∩B) = P(A|B)P(B) P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / ΣP(Aⱼ)P(B|Aⱼ) For binomial X: P(X=r) = ⁿCᵣpʳqⁿ⁻ʳ SYMBOLS AND UNITS A, B, Aᵢ: events P: probability, dimensionless n: number of trials r: successes p: success probability q = 1-p One-line recall Independence is stronger than mutual exclusivity. PADIKAM | CBSE NCERT CLASS 12 15