📘 CBSE Class 10 Maths Complete Formula Cheat Sheet (2026)
Ultimate Board Exam Revision Handbook! Includes ALL 14 Chapters, Formula Memorization, and Topper Hacks.
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1. Real Numbers
📐 Important Formulas & Concepts
• Euclid's Division Lemma: a = bq + r
• Golden Formula: HCF(a,b) × LCM(a,b) = a × b
• Decimal Expansion: Terminating if denominator has only 2 & 5.
• Golden Formula: HCF(a,b) × LCM(a,b) = a × b
• Decimal Expansion: Terminating if denominator has only 2 & 5.
- Fundamental Theorem of Arithmetic: Every composite number has a unique prime factorization.
- Irrationality Proof: Prove √2, √3, √5 are irrational (VVIP for Board Exam).
- Irrational numbers are non-terminating & non-repeating.
⚠️ Common Mistake
HCF × LCM formula is ONLY valid for two numbers, not for three! Also, students forget to simplify fractions before checking decimal expansion.
⚡ Exam Tip
To check terminating decimal: Denominator after simplification should contain only 2 or 5. Prime factorization questions are highly repeated in boards.
2. Polynomials
📐 Algebraic Identities & Quadratic Zeroes
• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• a² - b² = (a + b)(a - b)
• (a + b)³ = a³ + b³ + 3ab(a + b)
• a³ - b³ = (a - b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² - ab + b²)
For quadratic polynomial ax² + bx + c:
• Sum of zeroes (α + β) = -b/a
• Product of zeroes (αβ) = c/a
• Formation of Polynomial: k[x² - (α + β)x + αβ]
• Sum of zeroes (α + β) = -b/a
• Product of zeroes (αβ) = c/a
• Formation of Polynomial: k[x² - (α + β)x + αβ]
⚡ Memory Trick
"Sum is minus middle coefficient."
3. Pair of Linear Equations in 2 Variables
📊 Consistency Conditions (VVIP TABLE)
| Condition | Graphical Rep. | Algebraic Solution | Consistency |
|---|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting Lines | Exactly One (Unique) | Consistent |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident Lines | Infinitely Many | Consistent (Dependent) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel Lines | No Solution | Inconsistent |
💡 Exam Tip
Cross-multiplication method is often the fastest in boards, but verify relations using the standard form carefully observing the signs (+/-).
4. Quadratic Equations
📐 Quadratic Formula & Nature of Roots
Standard form: ax² + bx + c = 0
Quadratic Formula (Shreedharacharya):
x = [-b ± √(b² - 4ac)] / 2a
Quadratic Formula (Shreedharacharya):
x = [-b ± √(b² - 4ac)] / 2a
📊 Discriminant (D = b² - 4ac)
| D > 0 | Real & Distinct Roots |
| D = 0 | Real & Equal Roots |
| D < 0 | No Real Roots |
⚠️ Common Mistake
Forgetting the denominator 2a entirely in the formula.
5. Arithmetic Progressions (AP)
📐 AP Formulas
Let first term = a, common difference = d
• nth term: aₙ = a + (n - 1)d
• Sum of n terms: Sₙ = n/2 [2a + (n - 1)d]
• Short Sum Formula: Sₙ = n/2 [a + l] (where l is last term)
• Relation: aₙ = Sₙ - Sₙ₋₁
• nth term: aₙ = a + (n - 1)d
• Sum of n terms: Sₙ = n/2 [2a + (n - 1)d]
• Short Sum Formula: Sₙ = n/2 [a + l] (where l is last term)
• Relation: aₙ = Sₙ - Sₙ₋₁
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6. Triangles
📐 Important Theorems
- BPT (Thales Theorem): If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
- Similarity Criteria: AAA, SAS, SSS.
- Area Theorem: Area₁/Area₂ = (Corresponding Side₁ / Corresponding Side₂)²
⚠️ Warning
Write similarity matching letters carefully. If ΔABC ~ ΔPQR, then AB/PQ = BC/QR = AC/PR. Do not mix the order!
7. Coordinate Geometry
📍 Distance & Section Formula
• Distance: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
• Distance from origin: √(x² + y²)
• Section Formula (Internally):
x = (m₁x₂ + m₂x₁) / (m₁ + m₂)
y = (m₁y₂ + m₂y₁) / (m₁ + m₂)
• Mid-Point: ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Distance from origin: √(x² + y²)
• Section Formula (Internally):
x = (m₁x₂ + m₂x₁) / (m₁ + m₂)
y = (m₁y₂ + m₂y₁) / (m₁ + m₂)
• Mid-Point: ((x₁ + x₂)/2, (y₁ + y₂)/2)
💡 Exam Tip
Distance formula is a board favorite. Always double-check your negative signs during subtraction.
8. Introduction to Trigonometry
📐 Trigonometric Ratios & Identities
• sin²θ + cos²θ = 1
• sec²θ - tan²θ = 1
• cosec²θ - cot²θ = 1
• tanθ = sinθ / cosθ
• sec²θ - tan²θ = 1
• cosec²θ - cot²θ = 1
• tanθ = sinθ / cosθ
Trick: Pandit Badri Prasad / Har Har Bole
(Sin=P/H, Cos=B/H, Tan=P/B)
📊 Standard Values Table
| Angle | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | N.D. |
9. Some Applications of Trigonometry
📏 Heights and Distances
- Angle of Elevation: Looking UP at an object.
- Angle of Depression: Looking DOWN at an object (Always equal to alternate interior angle of elevation).
- Formula: tan θ = Height / Distance
💡 Exam Trick
90% of Board questions in this chapter are solved using tan θ = Perpendicular/Base. Draw the diagram first!
10. Circles & 11. Areas Related to Circles
⭕ Key Theorems (Circles)
- The tangent at any point of a circle is perpendicular to the radius through the point of contact.
- VVIP: The lengths of tangents drawn from an external point to a circle are equal.
⚠️ Mistake: Students draw tangent touching at two points.
📐 Sector and Segment
• Circumference: 2πr
• Area of Circle: πr²
• Area of Sector: (θ/360°) × πr²
• Length of Arc: (θ/360°) × 2πr
• Area of Segment: Area of Sector - Area of Triangle
• Area of Circle: πr²
• Area of Sector: (θ/360°) × πr²
• Length of Arc: (θ/360°) × 2πr
• Area of Segment: Area of Sector - Area of Triangle
12. Surface Areas and Volumes
| Solid Shape | Curved/Lateral Surface Area (CSA/LSA) | Total Surface Area (TSA) | Volume (V) |
|---|---|---|---|
| Cube (side a) | 4a² | 6a² | a³ |
| Cuboid (l, b, h) | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
| Cylinder (r, h) | 2πrh | 2πr(r + h) | πr²h |
| Cone (r, h, l) | πrl [Slant l = √(r² + h²)] |
πr(r + l) | (1/3)πr²h |
| Sphere (r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (r) | 2πr² | 3πr² | (2/3)πr³ |
13. Statistics & 14. Probability
📊 Statistics Formulas
• Mean (Direct): Σ(fᵢxᵢ) / Σfᵢ
• Mean (Assumed): a + (Σfᵢdᵢ / Σfᵢ)
• Median: l + [((N/2) - cf) / f] × h
• Mode: l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
• Empirical Formula (VVIP):
3 × Median = Mode + 2 × Mean
• Mean (Assumed): a + (Σfᵢdᵢ / Σfᵢ)
• Median: l + [((N/2) - cf) / f] × h
• Mode: l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h
• Empirical Formula (VVIP):
3 × Median = Mode + 2 × Mean
🎲 Probability Rules
• P(E) = Fav. Outcomes / Total Outcomes
• Range: 0 ≤ P(E) ≤ 1
• P(Sure Event) = 1
• P(Impossible Event) = 0
• Complementary: P(E) + P(not E) = 1
• Range: 0 ≤ P(E) ≤ 1
• P(Sure Event) = 1
• P(Impossible Event) = 0
• Complementary: P(E) + P(not E) = 1
🃏 Deck of Cards Trick:
Total 52 cards = 26 Red (Hearts, Diamonds) + 26 Black (Spades, Clubs). Face cards = 12 (J, Q, K in 4 suits).
Total 52 cards = 26 Red (Hearts, Diamonds) + 26 Black (Spades, Clubs). Face cards = 12 (J, Q, K in 4 suits).
🔥 BONUS: MASTER REVISION SECTION
Secret strategies, Exam hacks, and High-Weightage concepts.
⭐ Top Board Exam Concepts
- Proof of Irrationality (√3, √5).
- Finding unknown frequencies (x, y) in Median/Mean.
- BPT Theorem Proof & Application.
- Lengths of Tangents from external point proof.
- Roots of Quadratic Eq. using D=0 to find 'k'.
- Trigonometry Proofs from NCERT Ex 8.3/8.4.
⚠️ Common Board Mistakes
| Sign errors | Recheck calculations |
| Missing units | Always write cm²/cm³ (0.5 marks deduct) |
| Trig mix-up | Learn ratios visually |
| Stats error | Taking cf of median class instead of preceding |
🧠 Golden Memory Tricks
| Formula | Trick |
|---|---|
| AP nth term | Start + jumps |
| Quadratic Formula | "Minus b plus-minus root..." |
| Trigonometry (Sine) | Perpendicular / Hypotenuse |
| Sector Area | Angle/360 × circle area |
📅 7-Day Board Revision Strategy
| Day | Focus Area | Action Item |
|---|---|---|
| Day 1 | Algebra (Poly, Linear, Quadratic, AP) | Practice quadratic formula and word problems. |
| Day 2 | Trigonometry & Applications | Memorize identity proofs and height/distance diagrams. |
| Day 3 | Geometry (Triangles & Circles) | Write proofs of BPT and Tangent theorems 3 times. |
| Day 4 | Mensuration (Areas & Volumes) | Use this cheat sheet to memorize the complete formula table. |
| Day 5 | Coordinate Geo & Real Numbers | Solve irrationality proofs and section formula questions. |
| Day 6 | Statistics & Probability | Practice one missing frequency question for Mean/Median. |
| Day 7 | Full Mock Test | Attempt a 3-hour sample paper. Review mistakes. |
📝 Final Board Exam Hacks
- Formula revision gives fastest marks.
- Geometry diagrams improve presentation.
- Write theorem names properly.
- Underline final answers.
- Attempt easy questions first.
- Keep last 15 mins for checking.
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