Free Formula Sheet · Samacheer Kalvi
10th Standard Mathematics
Complete Formula Sheet — All Chapters
Chapter 01
Relations and Functions
Cartesian Product & Counting
Cartesian Product
A × B = {(a, b) | a ∈ A, b ∈ B}
Number of Elements
n(A × B) = n(A) × n(B)
Number of Relations from A to B
2^(n(A) × n(B))
Number of Functions from A to B
[n(B)]^n(A)
Types of Functions
| Type | Condition |
|---|---|
| One-One (Injective) | f(a) = f(b) ⟹ a = b |
| Onto (Surjective) | Range(f) = Co-domain |
| Bijective | Both one-one and onto |
| Identity Function | f(x) = x |
| Constant Function | f(x) = c for all x |
Composition & Inverse
Composition (apply f first, then g)
(g ∘ f)(x) = g(f(x))
Inverse Function
f(f⁻¹(x)) = f⁻¹(f(x)) = x
Associativity
h ∘ (g ∘ f) = (h ∘ g) ∘ f
NOT Commutative (generally)
g ∘ f ≠ f ∘ g
Equivalence Relation must be: Reflexive (a,a)∈R · Symmetric (a,b)∈R ⟹ (b,a)∈R · Transitive (a,b)∈R & (b,c)∈R ⟹ (a,c)∈R
Chapter 02
Numbers and Sequences
Euclid’s Division & GCD / LCM
Euclid’s Division Lemma
a = bq + r, where 0 ≤ r < b
GCD × LCM Relation
GCD(a,b) × LCM(a,b) = a × b
Arithmetic Progression (AP)
n-th Term
tₙ = a + (n − 1)d
Sum of n Terms — Sₙ
Sₙ = (n/2) [2a + (n − 1)d]
Sum (first & last term known)
Sₙ = (n/2)(a + l)
Common Difference
d = tₙ − tₙ₋₁
Geometric Progression (GP)
n-th Term
tₙ = a · rⁿ⁻¹
Sum of n Terms (r ≠ 1)
Sₙ = a(rⁿ − 1) / (r − 1)
Sum (when r < 1)
Sₙ = a(1 − rⁿ) / (1 − r)
Common Ratio
r = tₙ / tₙ₋₁
Special Summation Formulas
Sum of first n natural numbers
Σn = n(n+1) / 2
Sum of squares
Σn² = n(n+1)(2n+1) / 6
Sum of cubes
Σn³ = [n(n+1) / 2]²
Modular Arithmetic: a ≡ b (mod m) means m divides (a − b). If a ≡ b and c ≡ d (mod m), then a+c ≡ b+d and ac ≡ bd (mod m).
Chapter 03
Algebra
Remainder & Factor Theorems
Remainder Theorem
Remainder of p(x) ÷ (x−a) = p(a)
Factor Theorem
(x − a) is a factor ⟺ p(a) = 0
Quadratic Equation — ax² + bx + c = 0
Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
Discriminant Δ = b² − 4ac
Δ > 0 → 2 real distinct roots
Δ = 0 → 2 equal roots
Δ < 0 → no real roots
Δ = 0 → 2 equal roots
Δ < 0 → no real roots
Sum of Roots (α + β)
α + β = −b / a
Product of Roots (αβ)
αβ = c / a
Form Equation from Roots
x² − (α+β)x + αβ = 0
α² + β²
(α + β)² − 2αβ
Algebraic Identities
| Identity | Expansion |
|---|---|
| (a + b)² | a² + 2ab + b² |
| (a − b)² | a² − 2ab + b² |
| (a + b)(a − b) | a² − b² |
| (a + b)³ | a³ + 3a²b + 3ab² + b³ |
| (a − b)³ | a³ − 3a²b + 3ab² − b³ |
| a³ + b³ | (a + b)(a² − ab + b²) |
| a³ − b³ | (a − b)(a² + ab + b²) |
| (a + b + c)² | a² + b² + c² + 2ab + 2bc + 2ca |
Matrices
Addition (same order)
(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ
Scalar Multiplication
(kA)ᵢⱼ = k · aᵢⱼ
Transpose
(Aᵀ)ᵢⱼ = Aⱼᵢ
(AB)ᵀ
= BᵀAᵀ
Chapter 04
Geometry
Similarity Theorems
Basic Proportionality (Thales) Theorem
AD / DB = AE / EC
Line parallel to BC cuts AB & AC proportionally
Ratio of Areas of Similar Triangles
Area₁ / Area₂ = (l₁ / l₂)²
Square of the ratio of corresponding sides
Pythagoras & Related
Pythagoras Theorem
Hypotenuse² = Side₁² + Side₂²
Altitude on Hypotenuse
BD² = AD × DC
Leg on Hypotenuse
AB² = AD × AC
Converse
If c² = a² + b² → ∠C = 90°
Circle Properties
Equal Tangent Lengths
PA = PB (from external point P)
Intersecting Chords
PA × PB = PC × PD
Secant–Tangent
PT² = PA × PB
Inscribed Angle
Inscribed Angle = ½ × Central Angle
Angle in Semicircle
= 90°
Tangent–Chord Angle
= ½ × intercepted arc
Similarity Criteria: AA – Two angles equal · SSS – All three sides proportional · SAS – Two sides proportional and included angle equal
Chapter 05
Coordinate Geometry
Distance, Section & Midpoint
Distance Formula — A(x₁,y₁) to B(x₂,y₂)
d = √[(x₂−x₁)² + (y₂−y₁)²]
Section Formula (internal, ratio m:n)
P = ( (mx₂+nx₁)/(m+n) , (my₂+ny₁)/(m+n) )
Midpoint Formula
M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )
Centroid of Triangle
G = ( (x₁+x₂+x₃)/3 , (y₁+y₂+y₃)/3 )
Area of Triangle & Collinearity
Area with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃)
Area = (1/2) |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
Points are collinear if Area = 0
Straight Line Equations
| Form | Equation | Notes |
|---|---|---|
| Slope-Intercept | y = mx + c | m = slope, c = y-intercept |
| Point-Slope | y − y₁ = m(x − x₁) | Through point (x₁, y₁) |
| Two-Point | (y−y₁)/(y₂−y₁) = (x−x₁)/(x₂−x₁) | |
| Intercept Form | x/a + y/b = 1 | a = x-int, b = y-int |
| General Form | ax + by + c = 0 | Slope = −a/b |
Slope, Parallel & Perpendicular
Slope (m)
m = (y₂−y₁) / (x₂−x₁) = tan θ
Parallel Lines
m₁ = m₂
Perpendicular Lines
m₁ × m₂ = −1
Distance from point (x₀,y₀) to line ax + by + c = 0: d = |ax₀ + by₀ + c| / √(a² + b²)
Chapter 06
Trigonometry
Basic Ratios (SOH CAH TOA)
sin θ
Opp / Hyp
cos θ
Adj / Hyp
tan θ
Opp / Adj = sin θ / cos θ
cosec θ
1 / sin θ
sec θ
1 / cos θ
cot θ
1 / tan θ = cos θ / sin θ
Standard Values Table
| Ratio | 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 | ∞ |
Pythagorean Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Complementary Angles & Height–Distance
Complementary Angle Relations
sin(90°−θ) = cosθ | cos(90°−θ) = sinθ
tan(90°−θ) = cotθ | cot(90°−θ) = tanθ
sec(90°−θ) = cosecθ | cosec(90°−θ) = secθ
tan(90°−θ) = cotθ | cot(90°−θ) = tanθ
sec(90°−θ) = cosecθ | cosec(90°−θ) = secθ
Height from Angle of Elevation (α)
h = d × tan α
d = horizontal distance from observer to object
Chapter 07
Mensuration
Plane Figures
| Figure | Area | Perimeter |
|---|---|---|
| Square (side a) | a² | 4a |
| Rectangle (l, b) | l × b | 2(l + b) |
| Triangle (base b, h) | (1/2) × b × h | a + b + c |
| Equilateral △ (side a) | (√3/4)a² | 3a |
| Rhombus (d₁, d₂) | (1/2)d₁d₂ | 4a |
| Trapezium (a, b, h) | (1/2)(a + b) × h | Sum of all sides |
| Circle (radius r) | πr² | 2πr |
| Sector (r, θ°) | (θ/360) × πr² | l + 2r where l = (θ/360)×2πr |
Solid Figures — CSA · TSA · Volume
| Solid | CSA | TSA | Volume |
|---|---|---|---|
| Cube (a) | 4a² | 6a² | a³ |
| Cuboid (l, b, h) | 2(l+b)h | 2(lb+bh+lh) | lbh |
| Cylinder (r, h) | 2πrh | 2πr(h+r) | πr²h |
| Cone (r, h, l) | πrl | πr(l+r) | (1/3)πr²h |
| Sphere (r) | 4πr² | 4πr² | (4/3)πr³ |
| Hemisphere (r) | 2πr² | 3πr² | (2/3)πr³ |
| Frustum (R, r, h, l) | π(R+r)l | π(R+r)l + πR² + πr² | (πh/3)(R²+Rr+r²) |
Slant Heights: Cone → l = √(r²+h²) | Frustum → l = √(h²+(R−r)²) | π ≈ 22/7 ≈ 3.14159 | √2 ≈ 1.414 | √3 ≈ 1.732
Chapter 08
Statistics and Probability
Measures of Central Tendency
Mean — Direct Method
x̄ = Σfx / Σf
Mean — Assumed Mean Method
x̄ = A + (Σfd / Σf)
d = x − A (deviation from assumed mean A)
Mean — Step Deviation Method
x̄ = A + (Σft / Σf) × c
t = (x−A)/c, c = class width
Median (grouped data)
M = l + [(N/2 − cf) / f] × h
l = lower bound · cf = cumulative freq before · f = median class freq · h = class width
Mode (grouped data)
Z = l + [(f₁−f₀) / (2f₁−f₀−f₂)] × h
f₁ = modal class · f₀ = prev class · f₂ = next class
Empirical Relation
Mode ≈ 3 Median − 2 Mean
Measures of Dispersion
Standard Deviation σ — Ungrouped
σ = √[ Σ(x−x̄)² / n ]
Standard Deviation σ — Grouped
σ = √[ Σf(x−x̄)² / Σf ]
Coefficient of Variation (CV)
CV = (σ / x̄) × 100%
Lower CV → more consistent data
Variance
Variance = σ²
Probability
Classical Probability
P(A) = n(A) / n(S)
n(S) = total sample space outcomes
Complementary Event
P(Ā) = 1 − P(A)
Addition — Mutually Exclusive
P(A ∪ B) = P(A) + P(B)
Addition — General Rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Key Probability Facts: 0 ≤ P(A) ≤ 1 | P(impossible event) = 0 | P(sure event) = 1 | P(A) + P(Ā) = 1
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