Class 10 Maths: Chapter-wise Guide and Formula Sheet
Use this guide for quick revision before a test. Read each chapter summary, learn its formulas, then check your understanding with the MCQ test. Each chapter is explained in plain language so you know what the examiner expects.
Real Numbers
This chapter covers Euclid's division lemma, the Fundamental Theorem of Arithmetic and irrational numbers. Most questions ask for HCF and LCM by prime factorisation, proofs that numbers like √2 and √3 are irrational, and whether a fraction has a terminating decimal. A fraction p/q terminates only when q has no prime factors except 2 and 5.
Key formulas and facts:
- a = bq + r, 0 ≤ r < b
- HCF × LCM = product of the two numbers
Polynomials
You learn how zeros of a polynomial relate to its coefficients and how the graph of a quadratic cuts the x-axis. Remember that a quadratic has at most two zeros, and practise forming a polynomial from a given sum and product of zeros.
Key formulas and facts:
- Sum of zeros = −b/a, product = c/a
- Polynomial: x² − (sum)x + product
Linear Equations in Two Variables
Solve by substitution, elimination or cross-multiplication and decide consistency from the ratios of coefficients. Word problems on age, speed and money are common, so train yourself to form equations carefully.
Key formulas and facts:
- a₁/a₂ ≠ b₁/b₂: one solution
- a₁/a₂ = b₁/b₂ = c₁/c₂: infinitely many solutions
- a₁/a₂ = b₁/b₂ ≠ c₁/c₂: no solution
Quadratic Equations
Solve by factorisation, completing the square or the quadratic formula. The discriminant tells the nature of roots without solving. Always check that a root makes sense in word problems.
Key formulas and facts:
- x = [−b ± √(b² − 4ac)] / 2a
- D > 0 two real roots; D = 0 equal roots; D < 0 no real roots
Arithmetic Progressions
An AP has a constant difference between terms. Identify a and d first, then pick the nth-term or sum formula. For sum problems, check whether the last term is given.
Key formulas and facts:
- aₙ = a + (n − 1)d
- Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
Triangles
Focus on similarity criteria, the basic proportionality theorem and Pythagoras theorem. Areas of similar triangles are in the ratio of the squares of their sides.
Key formulas and facts:
- AB/AC = BD/DC when AD bisects angle A (internal bisector)
- Pythagoras: h² = p² + b²
- Area ratio = (side ratio)²
Coordinate Geometry
Everything starts from the distance formula. Use the section formula for dividing a line in a given ratio and the area formula for triangles and collinearity.
Key formulas and facts:
- d = √[(x₂−x₁)² + (y₂−y₁)²]
- Section: ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
- Area = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Trigonometry and its Applications
Memorise the table of values for 0°, 30°, 45°, 60° and 90°, then use identities to simplify. In height and distance problems, draw a neat figure and mark the angle of elevation or depression first.
Key formulas and facts:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ
- tan θ = sin θ / cos θ
Circles, Areas and Constructions
A tangent is perpendicular to the radius at the point of contact, and tangents from an external point are equal. Area problems use sectors and segments.
Key formulas and facts:
- Tangent length = √(d² − r²)
- Circumference = 2πr; Area = πr²
- Sector area = (θ/360) × πr²; arc = (θ/360) × 2πr
Surface Areas, Volumes, Statistics and Probability
For solids, decide whether you need curved or total surface area. In statistics, learn mean, median and mode for grouped data. Probability values always lie between 0 and 1.
Key formulas and facts:
- Cylinder: V = πr²h, CSA = 2πrh
- Cone: V = ⅓πr²h, CSA = πrl
- Sphere: V = 4/3 πr³, SA = 4πr²
- Mean = Σfx/Σf; Mode = l + [(f₁−f₀)/(2f₁−f₀−f₂)] × h
- P(E) = favourable outcomes / total outcomes