Class 10 Board Revision Notes: Maths Chapter-wise Key Points

These Class 10 Maths revision notes cover the key formulas, theorems and exam tips for every chapter of the NCERT Class 10 Maths textbook, all on one page. Use them for quick revision after you have practised the exercises.

Chapter 1: Real Numbers

• Fundamental Theorem of Arithmetic: every composite number can be written as a product of primes in a unique way (apart from the order).
• For two numbers: HCF × LCM = product of the numbers.
• √2, √3 and √5 are irrational. Proofs use contradiction.
• Exam tip: prime factorisation questions are quick marks. See our LCM and HCF guide.

Chapter 2: Polynomials

• A quadratic polynomial ax² + bx + c has at most 2 zeroes.
• Sum of zeroes: α + β = −b/a. Product of zeroes: αβ = c/a.
• A quadratic with zeroes α and β: x² − (α + β)x + αβ
• Graphically, the zeroes are where the graph cuts the x-axis.

Chapter 3: Pair of Linear Equations in Two Variables

For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:

Condition	Graph	Solutions
a₁/a₂ ≠ b₁/b₂	Intersecting lines	Exactly one (consistent)
a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Parallel lines	None (inconsistent)
a₁/a₂ = b₁/b₂ = c₁/c₂	Coincident lines	Infinitely many

Methods: substitution and elimination.

Chapter 4: Quadratic Equations

• Standard form: ax² + bx + c = 0, where a ≠ 0
• Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a
• Discriminant D = b² − 4ac

D	Nature of roots
D > 0	Two distinct real roots
D = 0	Two equal real roots
D < 0	No real roots

Chapter 5: Arithmetic Progressions

• nth term: aₙ = a + (n − 1)d
• Sum of n terms: Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l), where l is the last term
• Common difference: d = a₂ − a₁
• Exam tip: in word problems, first write down a, d and n clearly.

Chapter 6: Triangles

• Similar triangles have equal corresponding angles and proportional corresponding sides.
• Basic Proportionality Theorem (Thales): a line drawn parallel to one side of a triangle divides the other two sides in the same ratio.
• Similarity criteria: AAA (or AA), SSS and SAS.
• Exam tip: practise the BPT proof — it is often asked.

Chapter 7: Coordinate Geometry

• Distance formula: √[(x₂ − x₁)² + (y₂ − y₁)²]
• Section formula (ratio m : n): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n))
• Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)

Chapter 8: Introduction to Trigonometry

• sin θ = P/H, cos θ = B/H, tan θ = P/B
• cosec, sec and cot are the reciprocals of sin, cos and tan.
• Identities: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = cosec²θ

θ	0°	30°	45°	60°	90°
sin	0	1/2	1/√2	√3/2	1
cos	1	√3/2	1/√2	1/2	0
tan	0	1/√3	1	√3	not defined

Trick: for sin, write √0/2, √1/2, √2/2, √3/2, √4/2 for 0°, 30°, 45°, 60°, 90°. Cos is the same list in reverse.

Chapter 9: Some Applications of Trigonometry

• Angle of elevation: looking up from the horizontal.
• Angle of depression: looking down from the horizontal.
• Draw a clear figure first — most questions use tan θ = height/distance.

Chapter 10: Circles

• A tangent is perpendicular to the radius at the point of contact.
• The lengths of the two tangents drawn from an external point to a circle are equal.
• Exam tip: both theorems are frequently asked as proofs.

Chapter 11: Areas Related to Circles

• Area of a sector = (θ/360°) × πr²
• Length of an arc = (θ/360°) × 2πr
• Area of a segment = area of the sector − area of the triangle

Chapter 12: Surface Areas and Volumes

Problems combine two solids (for example, a cone on a hemisphere). For combined solids, add only the visible surface areas, but add the full volumes. All formulas are in our area and perimeter formulas guide.

Chapter 13: Statistics

• Mean (direct method): x̄ = Σfᵢxᵢ / Σfᵢ
• Mode: l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
• Median: l + [(n/2 − cf) / f] × h

Here l = lower limit of the modal or median class, h = class size, f₁ = frequency of the modal class, f₀ and f₂ = frequencies of the classes before and after it, cf = cumulative frequency of the class before the median class, and f = frequency of the median class.

Chapter 14: Probability

• P(E) = number of favourable outcomes ÷ total number of outcomes
• 0 ≤ P(E) ≤ 1, and P(E) + P(not E) = 1
• A deck of cards has 52 cards: 4 suits of 13 each, with 12 face cards (J, Q, K in each suit).
• Two dice give 36 possible outcomes.

Last-Week Revision Strategy

1. Read these notes daily for 15 minutes.
2. Make your own formula sheet in your handwriting.
3. Solve at least one full sample paper every two days. See how to use sample papers.
4. Revisit every question you got wrong in earlier papers.

Frequently Asked Questions

How many chapters are there in Class 10 Maths NCERT?
The current NCERT textbook has 14 chapters. Check the latest syllabus for any changes.

Which chapters carry the most marks in Class 10 Maths?
Algebra and geometry-related units usually carry significant weight. Check the unit-wise marks in your board's current syllabus.

What is the formula of the discriminant?
D = b² − 4ac, for the quadratic equation ax² + bx + c = 0.

Are these notes enough for the board exam?
They are for quick revision. Solve all NCERT exercises and examples first, then use these notes to revise.