Class 10 Maths Important Formulas – Chapter Wise
⚡ CBSE CLASS 10 BOARD EXAM EDITION 2026
Class 10 Mathematics: Master Formula Deck
Rationalized NCERT Syllabus • Strict Board Presentation Standards • Errorless Quick-Revision Tool
CH 01 Real Numbers
Weightage: 6 Marks
Fundamental Theorem of Arithmetic
Composite Number = p₁ᵃ × p₂ᵇ × p₃ᶜ ...
Every composite number can be uniquely factorized into primes, regardless of the order of factors.
HCF & LCM Product Relation
HCF(a, b) × LCM(a, b) = a × b
Strictly valid for two positive integers. (Never use directly for three numbers).
Board Note
For rational number p/q in simplest form (HCF(p, q) = 1), decimal expansion is terminating if and only if q = 2ᵐ × 5ⁿ.
CH 02 Polynomials
Weightage: 4 Marks
Sum of Zeroes
α + β = −b / a
= −(Coeff. of x) / (Coeff. of x²)
Product of Zeroes
αβ = c / a
= Constant Term / (Coeff. of x²)
Forming Quadratic Polynomial
k [ x² − (α + β)x + αβ ]
Where k is any non-zero real constant.
Algebra Shortcuts
α² + β² = (α + β)² − 2αβ | 1/α + 1/β = (α + β) / (αβ) | (α − β)² = (α + β)² − 4αβ
CH 03 Pair of Linear Equations in Two Variables
Weightage: 6 Marks| Ratio Condition | Graphical Representation | Algebraic Solutions | System Consistency |
|---|---|---|---|
| a₁/a₂ ≠ b₁/b₂ | Intersecting Lines | Unique Solution (Exactly One) | Consistent |
| a₁/a₂ = b₁/b₂ = c₁/c₂ | Coincident (Overlapping) Lines | Infinitely Many Solutions | Consistent (Dependent) |
| a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | Parallel Lines | No Solution | Inconsistent |
CH 04 Quadratic Equations
Weightage: 6-8 Marks
Standard Form & Shreedharacharya Formula
ax² + bx + c = 0 (a ≠ 0)
x = [ −b ± √(b² − 4ac) ] / 2a
Discriminant (D = b² − 4ac) Nature of Roots
• D > 0: Two distinct real roots
• D = 0: Two equal real roots (x = −b / 2a)
• D < 0: No real roots (roots are unreal)
• D = 0: Two equal real roots (x = −b / 2a)
• D < 0: No real roots (roots are unreal)
CH 05 Arithmetic Progressions (AP)
Weightage: 5-6 Marks
n-th General Term (aₙ)
aₙ = a + (n − 1)d
d = aₖ − aₖ₋₁ (Common Difference)
n-th Term from End
aₙ(end) = l − (n − 1)d
where l is the last term of finite AP.
Sum of First n Terms (Sₙ)
Sₙ = (n/2)[ 2a + (n − 1)d ]
Sₙ = (n/2)[ a + l ]
Key Property
If Sₙ is given, the n-th term is obtained by: aₙ = Sₙ − Sₙ₋₁.
CH 06 Triangles
Weightage: 8 Marks
Thales Theorem (BPT)
AD / DB = AE / EC
If line DE ∥ BC intersects AB at D and AC at E. (Also: AB/AD = AC/AE).
Similarity Criteria & Scale Factor
ΔABC ~ ΔPQR ⇒ AB/PQ = BC/QR = AC/PR
Valid Criteria: AA (or AAA), SSS, SAS. Perimeter Ratio = Side Ratio.
CH 07 Coordinate Geometry
Weightage: 6 Marks
Distance Formula
d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]
Distance from Origin: √(x² + y²)
Section Formula (m:n)
x = (m x₂ + n x₁) / (m + n)
y = (m y₂ + n y₁) / (m + n)
Midpoint & Centroid
Mid = ( [x₁+x₂]/2 , [y₁+y₂]/2 )
Centroid = ( [x₁+x₂+x₃]/3 , [y₁+y₂+y₃]/3 )
CH 08 Introduction to Trigonometry
Weightage: 8 Marks
Pythagorean Identities
• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ (sec²θ − tan²θ = 1)
• 1 + cot²θ = cosec²θ (cosec²θ − cot²θ = 1)
Reciprocals & Quotient Rules
• tanθ = sinθ/cosθ | cotθ = cosθ/sinθ
• cosecθ = 1/sinθ | secθ = 1/cosθ
| Ratio (θ) | 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 | Not Defined |
| cosec θ | Not Defined | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | Not Defined |
| cot θ | Not Defined | √3 | 1 | 1/√3 | 0 |
CH 09 Some Applications of Trigonometry
Weightage: 4 Marks
Angle of Elevation
Line of Sight Rises UP
Angle between eye horizontal line and object above the observer.
Angle of Depression
Line of Sight Drops DOWN
Angle of Depression = Alternate Interior Angle of Elevation.
CH 10 Circles
Weightage: 6 Marks
Theorem 10.1: Radius ⟂ Tangent
OP ⟂ Tangent XY
Tangent at any point is perpendicular to radius through point of contact.
Theorem 10.2: External Point Tangents
PA = PB
Tangents drawn from an external point to a circle are equal in length and subtend equal angles at centre.
CH 11 Areas Related to Circles
Weightage: 4 Marks
Length of Sector Arc
l = (θ / 360°) × 2πr
Total Circumference = 2πr
Area of Sector
Area = (θ / 360°) × πr²
Alternative form: (1/2) × l × r
Area of Segment
Area = Sector − Δ Area
= (θ/360°)πr² − (1/2)r² sinθ
CH 12 Surface Areas and Volumes
Weightage: 6 Marks| 3D Solid | Curved / Lateral Area (CSA) | Total Surface Area (TSA) | Volume (V) |
|---|---|---|---|
| Cuboid | 2h(l + b) | 2(lb + bh + hl) | l × b × h |
| Cube | 4a² | 6a² | a³ |
| Cylinder | 2πrh | 2πr(r + h) | πr²h |
| Cone | πrl [ l = √(r² + h²) ] | πr(l + r) | (1/3)πr²h |
| Sphere | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere | 2πr² | 3πr² | (2/3)πr³ |
CH 13 Statistics
Weightage: 7 Marks
Mean (Direct & Assumed)
x̄ = (Σ fᵢxᵢ) / (Σ fᵢ)
x̄ = a + (Σ fᵢdᵢ / Σ fᵢ)
Mode of Grouped Data
l + [ (f₁ − f₀) / (2f₁ − f₀ − f₂) ] × h
f₁ = modal class freq, f₀ = preceding, f₂ = succeeding.
Median of Grouped Data
l + [ (n/2 − cf) / f ] × h
cf = cumulative freq of class preceding median class.
Empirical Relationship
Mode = 3 Median − 2 Mean (Guaranteed board question concept).
CH 14 Probability
Weightage: 4 Marks
Classical Probability
P(E) = n(E) / n(S)
= Favourable Outcomes / Total Elementary Outcomes
Probability Range
0 ≤ P(E) ≤ 1
Sure Event: P = 1 | Impossible Event: P = 0
Complementary Events
P(E) + P(not E) = 1
P(Ē) = 1 − P(E)
