UP Board Class 10 Maths Chapter Wise Notes 2027 - All 15 Chapters with Formulas

In short: UP Board Class 10 Maths 2027 follows the NCERT syllabus with 15 chapters across Algebra, Geometry, Trigonometry, Mensuration, Statistics, and Number Systems. The 70-mark written paper covers Real Numbers (HCF, LCM, irrationality proofs), Polynomials (zeroes and coefficients relationship), Linear Equations (graphical and algebraic methods), Quadratic Equations (factorisation, quadratic formula), Arithmetic Progressions (nth term, sum of n terms), Triangles (similarity theorems, BPT), Coordinate Geometry (distance, section, midpoint formulas), Introduction to Trigonometry (ratios, identities), Applications of Trigonometry (heights and distances), Circles (tangent properties, theorems), Constructions (tangents, similar triangles), Areas Related to Circles (sector, segment areas), Surface Areas and Volumes (combined solids, conversion), Statistics (mean, median, mode for grouped data), and Probability (classical definition, cards, dice problems).

The UP Board Maths Paper: What You Need to Know

UP Board Class 10 Maths follows the NCERT textbook — the same 15 chapters as CBSE. The exam carries 70 marks for the written paper plus 30 marks for internal assessment. The paper pattern includes both objective questions (20 marks) and descriptive questions (50 marks), making it essential to prepare for both quick-recall and detailed problem-solving.

Algebra alone carries approximately 20 marks — that is nearly 30% of your written paper from just four chapters. If you are weak in Maths, start with the how to study maths when you hate maths guide before diving into these notes. For the detailed marks breakdown, refer to the UP Board Class 10 Maths marks weightage guide.

Complete Chapter List

Chapter-Wise Notes

Chapter 1: Real Numbers

• Core concepts: Euclid's Division Lemma (a = bq + r, where 0 ≤ r < b) to find HCF. Fundamental Theorem of Arithmetic — every composite number can be expressed as a product of primes uniquely. Proving irrationality of √2, √3, √5 using contradiction method. Decimal expansion of rational numbers (terminating vs non-terminating recurring).
• Key formulas: HCF × LCM = Product of two numbers. A number has a terminating decimal expansion if the denominator (in lowest form) has only 2 and 5 as prime factors.
• Must-know proofs: Proof that √2 is irrational (proof by contradiction). This is a guaranteed 3-mark question.
• Exam tip: HCF/LCM problems using prime factorisation are easy 2-3 mark questions. Always verify your answer: HCF × LCM should equal the product of the two numbers.

Chapter 2: Polynomials

• Core concepts: Zeroes of a polynomial. Relationship between zeroes and coefficients of quadratic polynomial: for ax² + bx + c, sum of zeroes = -b/a, product of zeroes = c/a. Division algorithm for polynomials. Finding a quadratic polynomial when zeroes are given.
• Key formulas: Sum of zeroes (α + β) = -b/a. Product of zeroes (αβ) = c/a. Polynomial = k[x² - (α + β)x + αβ].
• Exam tip: Questions on finding zeroes graphically and verifying the relationship between zeroes and coefficients are common 3-mark questions. Practice forming polynomials when given sum and product of zeroes.

Chapter 3: Pair of Linear Equations in Two Variables

• Core concepts: Graphical method — consistent (unique solution or infinitely many) and inconsistent (no solution) systems. Algebraic methods — substitution, elimination, and cross-multiplication. Conditions: unique solution (a₁/a₂ ≠ b₁/b₂), no solution (a₁/a₂ = b₁/b₂ ≠ c₁/c₂), infinite solutions (a₁/a₂ = b₁/b₂ = c₁/c₂). Word problems converting to linear equations.
• Exam tip: Word problems carry 4-5 marks and appear every year. Common types: age problems, speed-distance-time problems, fraction problems. Practice converting the English statement into two equations, then solve. Master at least 2 algebraic methods thoroughly.

Chapter 4: Quadratic Equations

• Core concepts: Standard form ax² + bx + c = 0. Solution by factorisation (splitting the middle term). Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. Nature of roots using discriminant D = b² - 4ac: D > 0 (two distinct real roots), D = 0 (two equal real roots), D < 0 (no real roots).
• Key formulas: Quadratic formula and discriminant. Sum of roots = -b/a, Product of roots = c/a.
• Exam tip: This chapter is tested with at least one 3-5 mark word problem. Common word problems: finding two consecutive numbers, area problems, speed problems. Always check the discriminant first in "nature of roots" questions — it saves time.

Chapter 5: Arithmetic Progressions

• Core concepts: AP definition — constant common difference. nth term: aₙ = a + (n-1)d. Sum of first n terms: Sₙ = n/2[2a + (n-1)d] or Sₙ = n/2(a + l) where l is the last term. Finding number of terms, common difference, or first term when other values are given.
• Key formulas: aₙ = a + (n-1)d, Sₙ = n/2[2a + (n-1)d], Sₙ = n/2(a + l). Also: aₙ = Sₙ - Sₙ₋₁.
• Exam tip: Word problems on AP (salary increments, saving patterns, seating arrangements) are 4-5 mark questions. When asked "how many terms," set up the equation and solve for n. Always check that n is a positive integer.

Chapter 6: Triangles

• Core concepts: Similar triangles — AAA, AA, SSS, SAS similarity criteria. Basic Proportionality Theorem (BPT/Thales theorem): if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Converse of BPT. Pythagoras Theorem and its converse. Areas of similar triangles — ratio of areas equals ratio of squares of corresponding sides.
• Key theorems: BPT proof (important 4-5 mark question), Pythagoras theorem proof.
• Exam tip: Theorem proofs carry 4-5 marks. Practice writing the BPT proof and Pythagoras theorem proof step by step with diagrams. Application problems using similarity and Pythagoras are also common. This is the highest-weightage Geometry chapter.

Chapter 7: Coordinate Geometry

• Core concepts: Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]. Section formula: ((m₁x₂+m₂x₁)/(m₁+m₂), (m₁y₂+m₂y₁)/(m₁+m₂)). Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2). Area of triangle: ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|. Collinearity condition — area of triangle = 0.
• Exam tip: This is a formula-based chapter. Memorize the four formulas and practice 25-30 problems. Common question types: finding distance between two points, finding the ratio in which a point divides a line segment, proving collinearity, and finding the area of a triangle.

Chapter 8: Introduction to Trigonometry

• Core concepts: Trigonometric ratios (sin, cos, tan, cosec, sec, cot) for angles 0°, 30°, 45°, 60°, 90°. Trigonometric identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = cosec²θ. Complementary angle relationships: sin(90°-θ) = cosθ, etc.
• Key table: The standard angle values table (sin 0° = 0, sin 30° = ½, sin 45° = 1/√2, sin 60° = √3/2, sin 90° = 1) must be memorized completely.
• Exam tip: Identity-based proof questions carry 3-4 marks. Start from the more complex side (usually LHS) and simplify to the other side. Convert everything to sin and cos if stuck. Practice at least 30 identity proofs. See smart study strategies for effective techniques.

Chapter 9: Some Applications of Trigonometry

• Core concepts: Heights and distances problems. Angle of elevation (looking up) and angle of depression (looking down). Line of sight, horizontal line. Problems involving two observation points (two-triangle problems).
• Key approach: Draw the figure, identify the right triangle, write the trigonometric ratio, and solve.
• Exam tip: Always draw a neat figure first — it helps you identify which ratio to use. Common problem types: height of a tower/building, distance of a ship from a lighthouse, height when angle changes as observer moves. These carry 3-4 marks.

Chapter 10: Circles

• Core concepts: Tangent to a circle — a line touching the circle at exactly one point. Theorem: tangent at any point is perpendicular to the radius through the point of contact. Number of tangents from an external point = 2, and they are equal in length. Theorem proofs.
• Key theorem: The lengths of tangents drawn from an external point to a circle are equal (proof is a 3-4 mark question).
• Exam tip: Theorem proof is frequently asked. Application problems combining tangent properties with Pythagoras theorem are common 3-mark questions.

Chapter 11: Constructions

• Core concepts: Division of a line segment in a given ratio. Construction of a triangle similar to a given triangle (scale factor). Construction of tangents to a circle from an external point. Construction of tangents when centre is given/not given.
• Exam tip: Construction questions carry 3-4 marks. Use a sharp pencil, compass, and ruler. Write all construction steps alongside the figure. Do not erase construction arcs — examiners check them.

Chapter 12: Areas Related to Circles

• Core concepts: Area of circle = πr². Circumference = 2πr. Area of sector = (θ/360°)πr². Length of arc = (θ/360°)2πr. Area of segment = Area of sector - Area of triangle. Combined figures — area of shaded regions by adding/subtracting.
• Exam tip: Combined figure problems (finding area of shaded region) carry 3-4 marks. Break the complex figure into known shapes. Always use π = 22/7 unless told otherwise, and leave the answer in simplified fraction form if needed.

Chapter 13: Surface Areas and Volumes

• Core concepts: Surface area and volume of combined solids (cylinder + cone, hemisphere + cylinder, etc.). Conversion of one solid into another — when a solid is melted and recast, volume remains constant. Frustum of a cone — formulas for volume, CSA, and total surface area.
• Key formulas: Volume of frustum = πh/3(r₁² + r₂² + r₁r₂). CSA of frustum = π(r₁+r₂)l where l = √[h² + (r₁-r₂)²].
• Exam tip: Conversion problems ("a cone is melted and recast into smaller spheres — how many?") are 3-4 mark questions. Set up: Volume of original = n × Volume of each small solid. Combined solid problems require identifying the component shapes correctly.

Chapter 14: Statistics

• Core concepts: Mean of grouped data — direct method, assumed mean method, step deviation method. Median of grouped data using the formula: Median = l + [(n/2 - cf)/f] × h. Mode of grouped data: Mode = l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h. Ogive (cumulative frequency curve) — less than and more than type.
• Exam tip: Statistics questions are formula-based and scoring. The step deviation method for mean is fastest. Median formula questions carry 3-4 marks — make the cumulative frequency table carefully. Always double-check your class intervals and frequencies.

Chapter 15: Probability

• Core concepts: Classical (theoretical) probability: P(E) = Number of favourable outcomes / Total outcomes. P(E) + P(not E) = 1. Probability of impossible event = 0, certain event = 1. Common experiments: coins, dice, playing cards (52 cards = 4 suits × 13 cards each).
• Key facts: Total outcomes for 1 die = 6, 2 dice = 36. Total playing cards = 52 (26 red, 26 black, 4 suits of 13 each, 12 face cards).
• Exam tip: Playing card problems are the most commonly asked. Know the deck composition. For dice problems, list all outcomes systematically. These are easy 3-4 mark questions if you know the basics. Check the UP Board model paper for typical question patterns.

How to Use These Notes

UP Board Maths requires a problem-solving approach. Reading theory alone will not help — you must practice problems daily. Here is the recommended study method:

1. Understand before memorizing: Read the NCERT chapter explanation first. Understand why a formula works before memorizing it. This helps you apply it to unfamiliar problems.

2. Build a formula sheet: Create a single-page formula sheet for each chapter. Review it daily before starting practice. Over time, you will memorize formulas through repeated use.

3. Solve NCERT exercises completely: Every NCERT exercise problem is a potential exam question. Solve all textbook exercises and examples. Do not skip the examples — they teach the method.

4. Practice previous year papers: Solve at least 5 previous year UP Board papers under timed conditions. This builds speed and shows you the actual difficulty level. Use the time management strategy for paper-solving.

5. Focus on high-weightage chapters: Algebra (Chapters 3, 4, 5) and Geometry (Chapter 6) together carry 20+ marks. Spend 40% of your Maths study time on these chapters.

6. Tackle the objective section strategically: The 20-mark objective section needs quick mental math. Practice mental calculations and learn shortcut methods for common problem types.

7. Write step-by-step solutions: In the descriptive section, examiners give marks for each step. Never skip steps, even if you can do the calculation mentally. Write: Given → To find → Formula → Substitution → Answer with unit.

Priority Chapters by Weightage

Frequently Asked Questions

How many chapters are there in UP Board Class 10 Maths?

There are 15 chapters in UP Board Class 10 Maths, organized into six units: Number Systems (Ch 1), Algebra (Ch 2-5), Coordinate Geometry (Ch 7), Geometry (Ch 6, 10, 11), Trigonometry (Ch 8-9), Mensuration (Ch 12-13), and Statistics & Probability (Ch 14-15). All chapters follow the NCERT textbook.

Which is the most important chapter in UP Board Maths?

Quadratic Equations (Chapter 4) and Pair of Linear Equations (Chapter 3) are the most important chapters, each carrying 5-6 marks. Combined with Arithmetic Progressions (Chapter 5), the Algebra unit alone accounts for about 20 marks out of 70. Triangles (Chapter 6) is the most important Geometry chapter.

How to score full marks in UP Board Maths?

Focus on step-by-step solution writing — examiners award marks for each step. Complete all NCERT exercises, memorize all formulas, practice 30+ previous year questions per chapter, and solve at least 5 full mock papers under timed conditions. In the objective section, be careful with signs and decimal points. In constructions, write the steps clearly alongside the drawing.

Is NCERT enough for UP Board Maths?

For most students aiming for 50-60+ out of 70, NCERT textbook exercises and examples are sufficient. The UP Board paper directly follows NCERT content. However, if you are targeting 65+, supplement with NCERT Exemplar problems for additional practice on challenging question types.

What formulas should I memorize for Maths?

The critical formulas are: Quadratic formula (x = (-b ± √D)/2a), AP formulas (aₙ = a + (n-1)d, Sₙ = n/2[2a + (n-1)d]), Distance and Section formulas, all trigonometric identities, area of sector and segment, volume and surface area of combined solids and frustum, and the median and mode formulas for grouped data. Create a single formula sheet and revise it daily.

How should I handle word problems in Maths?

Word problems appear in Chapters 3, 4, and 5 and carry 10-15 marks combined. The strategy: (1) Read the problem twice, (2) identify what is unknown (assign variables), (3) form the equation from the given conditions, (4) solve the equation, (5) verify the answer makes sense in context. Practice at least 30 word problems from previous year papers.

Related Guides

• UP Board Class 10 Maths Chapter-Wise Marks Weightage 2027
• UP Board Class 10 Important Questions 2027
• UP Board Class 10 Model Paper 2027 — All Subjects with Marking Scheme
• UP Board Class 10 Last 20 Days Subject-Wise Strategy
• How to Study Maths When You Hate Maths
• Board Exam Time Management — 3 Hour Strategy

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