Notes for Class 12 Mathematics 



Relations and Functions

Relations are subsets of Cartesian products, while functions map each element to a unique output. Key concepts include types of relations (reflexive, symmetric, transitive), types of functions (one-one, onto, inverse), and composition of functions. Study steps: Understand domain, codomain, range, practice function properties, and solve NCERT exercises 1.1–1.4.

Formulas:

  • Composition: (f ∘ g)(x) = f(g(x))
  • Inverse function: If f: A → B is bijective, f⁻¹ exists
  • One-one: f(x₁) = f(x₂) ⇒ x₁ = x₂

Example:
NCERT Exercise 1.2, Q1: Check if f(x) = 2x + 1 is one-one. Solution: If f(x₁) = f(x₂), then 2x₁ + 1 = 2x₂ + 1, so x₁ = x₂. Hence, f is one-one.

Study Steps:

  1. Learn types of relations and functions.
  2. Practice composition and inverse functions.
  3. Solve problems on injectivity and surjectivity.
  4. Complete NCERT exercises and RD Sharma for extra practice.

Inverse Trigonometric Functions

Inverse trigonometric functions (e.g., sin⁻¹, cos⁻¹) are defined for specific domains and ranges. Key concepts: Properties, identities, and solving equations. Study steps: Memorize domains, ranges, principal values, and solve NCERT exercises 2.1–2.2.

Formulas:

  • sin⁻¹(-x) = -sin⁻¹x
  • sin⁻¹x + cos⁻¹x = π/2
  • 2sin⁻¹x = sin⁻¹(2x√(1-x²)), |x| ≤ 1/√2

Example:
NCERT Exercise 2.2, Q1: Find value of tan⁻¹(1). Solution: tan⁻¹(1) = π/4, as tan(π/4) = 1.

Study Steps:

  1. Memorize principal value ranges.
  2. Practice simplifying expressions using identities.
  3. Solve trigonometric equations.
  4. Complete NCERT exercises and RS Aggarwal.

Matrices

Matrices are rectangular arrays with operations like addition, multiplication, and determinants. Key concepts: Types (row, column, square), transpose, inverse. Study steps: Learn matrix operations, find inverses, and solve NCERT exercises 3.1–3.4.

Formulas:

  • Matrix multiplication: (AB)ij = Σ(aik × bkj)
  • Determinant (2×2): |A| = ad - bc
  • Inverse (2×2): A⁻¹ = (1/|A|) × adj(A)

Example:
NCERT Exercise 3.2, Q1: Find determinant of A = [[1, 2], [3, 4]]. Solution: |A| = (1×4) - (2×3) = 4 - 6 = -2.

Study Steps:

  1. Understand matrix types and operations.
  2. Practice finding determinants and inverses.
  3. Solve system of equations using matrices.
  4. Complete NCERT exercises and RD Sharma.

Determinants

Determinants are scalars associated with square matrices, used in solving equations and finding inverses. Key concepts: Properties, minors, cofactors, adjoint. Study steps: Learn properties, practice determinant calculations, and solve NCERT exercises 4.1–4.6.

Formulas:

  • Determinant (3×3): a(ei - fh) - b(di - fg) + c(dh - eg)
  • Adjoint: adj(A) = transpose of cofactor matrix
  • Inverse: A⁻¹ = adj(A)/|A|

Example:
NCERT Exercise 4.2, Q1: Find |A| for A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. Solution: |A| = 1(45-48) - 2(36-42) + 3(32-35) = -3 + 12 - 9 = 0.

Study Steps:

  1. Memorize determinant properties.
  2. Practice 3×3 determinant calculations.
  3. Solve inverse and equation problems.
  4. Complete NCERT exercises and RS Aggarwal.

Continuity and Differentiability

Continuity ensures no breaks in a function’s graph, differentiability implies smoothness. Key concepts: Continuity test, derivative rules, chain rule. Study steps: Learn rules, practice differentiation, and solve NCERT exercises 5.1–5.8.

Formulas:

  • Continuity at x = c: lim(x→c) f(x) = f(c)
  • Chain rule: d/dx [f(g(x))] = f'(g(x)) × g'(x)
  • Product rule: d/dx (uv) = u'v + uv'

Example:
NCERT Exercise 5.2, Q1: Check continuity of f(x) = x² at x = 1. Solution: lim(x→1) x² = 1, f(1) = 1, so continuous.

Study Steps:

  1. Learn continuity and differentiability definitions.
  2. Practice derivative rules (product, quotient).
  3. Solve continuity problems.
  4. Complete NCERT exercises and RD Sharma.

Application of Derivatives

Derivatives are used for tangents, normals, increasing/decreasing functions, and maxima/minima. Study steps: Learn applications, solve optimization problems, and practice NCERT exercises 6.1–6.5.

Formulas:

  • Slope of tangent: f'(x)
  • Increasing: f'(x) > 0, Decreasing: f'(x) < 0
  • Maxima/Minima: f'(x) = 0, check f''(x)

Example:
NCERT Exercise 6.5, Q1: Find maximum of f(x) = x³ - 6x² + 9x + 15. Solution: f'(x) = 3x² - 12x + 9 = 0, x = 1, 3. f''(1) < 0 (maxima), max at x = 1.

Study Steps:

  1. Learn tangent and normal equations.
  2. Practice monotonicity and extrema problems.
  3. Solve optimization word problems.
  4. Complete NCERT exercises and Oswaal papers.

Integrals

Integration is the reverse of differentiation, covering indefinite and definite integrals. Key concepts: Standard integrals, substitution, integration by parts. Study steps: Memorize standard forms, practice techniques, and solve NCERT exercises 7.1–7.11.

Formulas:

  • ∫xⁿ dx = x^(n+1)/(n+1) + C
  • Integration by parts: ∫uv dx = u∫v dx - ∫(u' ∫v dx) dx
  • Definite integral: ∫[a,b] f(x) dx = F(b) - F(a)

Example:
NCERT Exercise 7.2, Q1: Find ∫x² dx. Solution: ∫x² dx = x³/3 + C.

Study Steps:

  1. Memorize standard integrals.
  2. Practice substitution and by parts.
  3. Solve definite integral problems.
  4. Complete NCERT exercises and RD Sharma.

Application of Integrals

Integrals are used to find areas under curves and between curves. Study steps: Learn area formulas, practice curve sketching, and solve NCERT exercises 8.1–8.2.

Formulas:

  • Area under curve: ∫[a,b] f(x) dx
  • Area between curves: ∫[a,b] [f(x) - g(x)] dx

Example:
NCERT Exercise 8.1, Q1: Find area under y = x², x = 0 to x = 2. Solution: ∫[0,2] x² dx = [x³/3][0,2] = 8/3.

Study Steps:

  1. Learn area calculation methods.
  2. Practice sketching curves.
  3. Solve area between curves problems.
  4. Complete NCERT exercises and RS Aggarwal.

Differential Equations

Differential equations involve derivatives, with types like separable variables and linear equations. Study steps: Learn solution methods, practice applications, and solve NCERT exercises 9.1–9.6.

Formulas:

  • Separable: ∫f(x) dx = ∫g(y) dy
  • Linear: dy/dx + Py = Q, solution: y × IF = ∫Q × IF dx, IF = e^∫P dx

Example:
NCERT Exercise 9.4, Q1: Solve dy/dx = y/x. Solution: ∫dy/y = ∫dx/x, ln|y| = ln|x| + C, y = kx.

Study Steps:

  1. Understand types of differential equations.
  2. Practice separable variables method.
  3. Solve linear equations.
  4. Complete NCERT exercises and RD Sharma.

Vector Algebra

Vectors involve magnitude, direction, dot product, and cross product. Study steps: Learn operations, practice projections, and solve NCERT exercises 10.1–10.4.

Formulas:

  • Dot product: a · b = |a||b|cosθ
  • Cross product: |a × b| = |a||b|sinθ
  • Magnitude: |a| = √(x² + y² + z²)

Example:
NCERT Exercise 10.2, Q1: Find dot product of a = 2i + 3j, b = i + j. Solution: a · b = (2×1) + (3×1) = 5.

Study Steps:

  1. Learn vector operations.
  2. Practice dot and cross products.
  3. Solve angle and projection problems.
  4. Complete NCERT exercises and Oswaal papers.

Three Dimensional Geometry

This topic covers lines, planes, and their angles in 3D space. Study steps: Learn equations, practice angle calculations, and solve NCERT exercises 11.1–11.3.

Formulas:

  • Line: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
  • Plane: ax + by + cz + d = 0
  • Angle between lines: cosθ = |(l₁l₂ + m₁m₂ + n₁n₂)|/√(l₁²+m₁²+n₁²)√(l₂²+m₂²+n₂²)

Example:
NCERT Exercise 11.2, Q1: Find equation of line through (1, 2, 3) with direction ratios 1, 2, 3. Solution: (x-1)/1 = (y-2)/2 = (z-3)/3.

Study Steps:

  1. Memorize line and plane equations.
  2. Practice angle and distance problems.
  3. Solve coplanarity questions.
  4. Complete NCERT exercises and RS Aggarwal.

Linear Programming

Linear programming optimizes a linear objective function subject to constraints. Study steps: Learn graphical method, solve optimization problems, and practice NCERT exercises 12.1–12.2.

Formulas:

  • Objective function: z = ax + by
  • Constraints: ax + by ≤ c (or ≥, =)

Example:
NCERT Exercise 12.1, Q1: Maximize z = 3x + 2y, subject to x + y ≤ 4, x ≥ 0, y ≥ 0. Solution: Graph constraints, find vertices, max z = 12 at (4, 0).

Study Steps:

  1. Understand feasible region concept.
  2. Practice graphing constraints.
  3. Solve maximization/minimization problems.
  4. Complete NCERT exercises and Oswaal papers.

Probability

Probability covers conditional probability, Bayes’ theorem, and random variables. Study steps: Learn probability rules, practice distributions, and solve NCERT exercises 13.1–13.5.

Formulas:

  • Conditional: P(A|B) = P(A ∩ B)/P(B)
  • Bayes’ theorem: P(A|B) = P(B|A)P(A)/P(B)
  • Variance: Var(X) = E(X²) - [E(X)]²

Example:
NCERT Exercise 13.2, Q1: If P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.2, find P(A|B). Solution: P(A|B) = 0.2/0.5 = 0.4.

Study Steps:

  1. Understand probability axioms.
  2. Practice Bayes’ theorem problems.
  3. Solve random variable questions.
  4. Complete NCERT exercises and RS Aggarwal.

General Preparation Tips

  1. NCERT Focus: Solve all NCERT exercises and examples, as CBSE exams heavily rely on them.
  2. Practice Regularly: Use RD Sharma, RS Aggarwal, and Oswaal sample papers for diverse problems.
  3. Formula Sheet: Create a consolidated formula sheet for quick revision.
  4. Mock Tests: Take timed tests to enhance speed and accuracy.
  5. Resources: Visit cbse.nic.in for syllabus updates, check X posts for tips, and use YouTube for concept videos.

If you need specific notes, additional examples, or a chart (e.g., trigonometric identities or integration formulas), let me know! For resources or pricing

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