Class 10 Maths - Detailed Notes (All Chapters)
Chapter 1: Real Numbers
Key Concepts: Euclid’s Division Lemma, Fundamental Theorem of Arithmetic, LCM, HCF, rational and irrational numbers, decimals.
- Euclid's Division Lemma: For any two integers a and b, there exist unique integers q and r such that a = bq + r, 0 ≤ r < b.
- Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes, and this factorization is unique except for the order.
- Important results: HCF(a, b) × LCM(a, b) = a × b.
- Decimal expansions: Rational numbers have terminating or repeating decimal expansions; irrational numbers are non-terminating and non-repeating.
- Examples: Prove √2 is irrational, find HCF using Euclid’s method, convert decimals to fractions.
Chapter 2: Polynomials
Key Concepts: Types of polynomials, zeroes of polynomials, relationship between coefficients and zeroes, division algorithm.
- Types: Linear, quadratic, cubic polynomials.
- Zeroes: α is a zero of p(x) if p(α) = 0.
- Relations for quadratic: α + β = –b/a, αβ = c/a.
- Division Algorithm: If p(x) and g(x) are polynomials, then p(x) = g(x) × q(x) + r(x), deg r(x) < deg g(x).
- Examples: Find zeroes, verify relations, use synthetic division.
Chapter 3: Pair of Linear Equations in Two Variables
Key Concepts: Graphical and algebraic methods of solution, consistency, types of solutions.
- Methods: Substitution, elimination, cross-multiplication.
- Consistency: Unique solution (intersecting lines), infinitely many solutions (coincident lines), no solution (parallel lines).
- Applications: Word problems on ages, speed-distance, investments.
- Examples: Solve systems, interpret graphs, analyze special cases.
Chapter 4: Quadratic Equations
Key Concepts: Standard form, discriminant, nature of roots, factorization, quadratic formula, completing the square.
- Nature of roots: D = b² – 4ac; D > 0 → real and distinct, D = 0 → real and equal, D < 0 → complex.
- Formula: x = [-b ± √D]/2a.
- Applications: Area, motion, number problems.
- Examples: Solve by factorization, formula, complete square method.
Chapter 5: Arithmetic Progressions (AP)
Key Concepts: Definition, nth term, sum of n terms, applications.
- nth term: an = a + (n–1)d.
- Sum of n terms: Sn = n/2 [2a + (n–1)d] or Sn = n/2 (a + l).
- Applications: Finding missing terms, real-life problems.
- Examples: Calculate term positions, find sums, apply to puzzles.
Chapter 6: Triangles
Key Concepts: Similarity, criteria of similarity (AAA, SSS, SAS), Pythagoras theorem, areas of similar triangles.
- Basic Proportionality Theorem (Thales): A line parallel to one side of a triangle divides the other two sides proportionally.
- Pythagoras theorem: In a right-angled triangle, (hypotenuse)² = (base)² + (height)².
- Applications: Heights, distances, geometric proofs.
- Examples: Prove similarity, calculate lengths, area ratios.
Chapter 7: Coordinate Geometry
Key Concepts: Distance formula, section formula, area of triangle.
- Distance formula: √[(x₂–x₁)² + (y₂–y₁)²]
- Section formula: (mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)
- Area of triangle: ½ |x₁(y₂–y₃) + x₂(y₃–y₁) + x₃(y₁–y₂)|
- Examples: Find midpoints, centroids, distance between points.
Chapter 8: Introduction to Trigonometry
Key Concepts: Trigonometric ratios, identities, values, complementary angles.
- Ratios: sin, cos, tan, cosec, sec, cot.
- Identities: sin²θ + cos²θ =1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ.
- Values: sin 30°=½, sin 45°=1/√2, sin 60°=√3/2, etc.
- Applications: Height, distance, angle measurement.
Chapter 9: Some Applications of Trigonometry
Key Concepts: Heights and distances, angle of elevation and depression.
- Concept: Use trigonometric ratios to find unknown heights, distances.
- Examples: Measure tower height, find width of river, calculate angle of depression.
Chapter 10: Circles
Key Concepts: Tangent to a circle, number of tangents from a point.
- Properties: Tangent at any point perpendicular to the radius; lengths of tangents drawn from an external point are equal.
- Examples: Draw tangents, find lengths, solve geometric problems.
Chapter 11: Constructions
Key Concepts: Division of a line segment, construction of tangents to a circle.
- Steps: Divide segment in given ratio; construct tangents using point on circle or external point.
- Examples: Practice compass-ruler methods for accurate construction.
Chapter 12: Areas Related to Circles
Key Concepts: Perimeter, area, sector, segment of circle.
- Formulas: Circumference = 2πr; area = πr²; sector area = (θ/360°)×πr².
- Applications: Finding remaining areas, field shapes, design problems.
Chapter 13: Surface Areas and Volumes
Key Concepts: CSA, TSA, volume of cube, cuboid, cylinder, cone, sphere, hemisphere, frustum.
- Formulas: TSA_cube =6a²; TSA_cylinder=2πr(h + r); V_cone=⅓πr²h, etc.
- Examples: Solve for curved surfaces, combined solids, water filling problems.
Chapter 14: Statistics
Key Concepts: Mean, median, mode of grouped data; cumulative frequency.
- Formulas: Mean by direct, assumed mean, step deviation; median using cumulative frequency table.
- Examples: Find central tendencies, draw ogive curves.
Chapter 15: Probability
Key Concepts: Classical probability, outcomes, events.
- Formula: P(E) = number of favourable outcomes / total outcomes.
- Examples: Card problems, dice, coins, daily life examples.