Formulas & Key Concepts

Euclid's Division Lemma

a = bq + r, where 0 ≤ r < b

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of primes (unique up to order)

HCF × LCM Property

HCF(a, b) × LCM(a, b) = a × b

Irrational Proof Strategy

If p divides a², then p divides a (p is prime)

Used to prove √2, √3, √5 are irrational

Terminating Decimal Condition

p/q terminates if q = 2ⁿ × 5ᵐ (and only if)

Zeroes of a Quadratic (Sum)

α + β = -b/a

For ax² + bx + c

Zeroes of a Quadratic (Product)

αβ = c/a

Quadratic from Zeroes

p(x) = x² - (α + β)x + αβ

Zeroes of a Cubic (Sum)

α + β + γ = -b/a

For ax³ + bx² + cx + d

Zeroes of a Cubic (Sum of products)

αβ + βγ + γα = c/a

Zeroes of a Cubic (Product)

αβγ = -d/a

Division Algorithm

p(x) = g(x) × q(x) + r(x)

deg r(x) < deg g(x)

General Form

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

Unique Solution

a₁/a₂ ≠ b₁/b₂

Consistent — exactly one solution

No Solution

a₁/a₂ = b₁/b₂ ≠ c₁/c₂

Inconsistent

Infinite Solutions

a₁/a₂ = b₁/b₂ = c₁/c₂

Consistent — dependent

Cross-Multiplication

x / (b₁c₂ - b₂c₁) = y / (c₁a₂ - c₂a₁) = 1 / (a₁b₂ - a₂b₁)

Standard Form

ax² + bx + c = 0, where a ≠ 0

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

Discriminant

D = b² - 4ac

Two Distinct Real Roots

D > 0

Two Equal Real Roots

D = 0 → x = -b/2a

No Real Roots

D < 0

Sum of Roots

α + β = -b/a

Product of Roots

αβ = c/a

General Term (nth term)

aₙ = a + (n - 1)d

a = first term, d = common difference

Sum of n Terms (Form 1)

Sₙ = n/2 [2a + (n - 1)d]

Sum of n Terms (Form 2)

Sₙ = n/2 (a + l)

l = last term

Common Difference

d = aₙ - aₙ₋₁

nth Term from Sum

aₙ = Sₙ - Sₙ₋₁ (for n ≥ 2)

Sum of First n Natural Numbers

S = n(n + 1)/2

BPT (Thales' Theorem)

If DE ∥ BC in △ABC, then AD/DB = AE/EC

AA Similarity

If two angles of one △ equal two angles of another △, they are similar

Area Ratio Theorem

ar(△ABC) / ar(△DEF) = (AB/DE)² = (BC/EF)²

Ratio of areas = square of ratio of corresponding sides

Pythagoras Theorem

AC² = AB² + BC²

In a right-angled triangle

Converse of Pythagoras

If AC² = AB² + BC², then ∠B = 90°

Distance Formula

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Section Formula

P = ((m·x₂ + n·x₁)/(m+n), (m·y₂ + n·y₁)/(m+n))

Midpoint Formula

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Area of Triangle

A = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

Collinearity Condition

Area of triangle = 0

sin θ

sin θ = Opposite / Hypotenuse

cos θ

cos θ = Adjacent / Hypotenuse

tan θ

tan θ = sin θ / cos θ = Opposite / Adjacent

Identity 1

sin²θ + cos²θ = 1

Identity 2

1 + tan²θ = sec²θ

Identity 3

1 + cot²θ = cosec²θ

Standard Values

sin: 0, 1/2, 1/√2, √3/2, 1 | cos: 1, √3/2, 1/√2, 1/2, 0

For θ = 0°, 30°, 45°, 60°, 90°

Complementary Angles

sin(90°-θ) = cosθ, tan(90°-θ) = cotθ, sec(90°-θ) = cosecθ

Angle of Elevation

tan θ = Height / Horizontal distance

Looking upward

Angle of Depression

tan θ = Height difference / Horizontal distance

Looking downward

Height from angle + distance

h = d × tan θ

Distance from angle + height

d = h / tan θ = h × cot θ

Tangent ⊥ Radius

Tangent at any point is perpendicular to the radius at contact

Equal Tangent Lengths

PA = PB (tangents from external point P)

Angle Between Tangents

∠APB + ∠AOB = 180°

Number of Tangents

Inside: 0 | On circle: 1 | Outside: 2

Area of Circle

A = πr²

Circumference

C = 2πr

Area of Sector

A = (θ/360°) × πr²

Length of Arc

l = (θ/360°) × 2πr

Area of Segment

Area of sector - Area of △ = (θ/360°)πr² - ½r²sinθ

Area of Ring

A = π(R² - r²)

R = outer, r = inner

Cylinder CSA / TSA / Volume

CSA = 2πrh | TSA = 2πr(r+h) | V = πr²h

Cone CSA / TSA / Volume

CSA = πrl | TSA = πr(r+l) | V = (1/3)πr²h

l = √(r²+h²)

Sphere SA / Volume

SA = 4πr² | V = (4/3)πr³

Hemisphere CSA / TSA / Volume

CSA = 2πr² | TSA = 3πr² | V = (2/3)πr³

Frustum Volume

V = (πh/3)(r₁² + r₂² + r₁r₂)

Mean (Direct)

Mean = Σfᵢxᵢ / Σfᵢ

Mean (Assumed Mean)

Mean = a + (Σfᵢdᵢ / Σfᵢ)

dᵢ = xᵢ - a

Mean (Step Deviation)

Mean = a + h(Σfᵢuᵢ / Σfᵢ)

uᵢ = (xᵢ - a)/h

Median

Median = l + [(n/2 - cf) / f] × h

Mode

Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h

Empirical Relationship

3 Median = Mode + 2 Mean

Probability

P(E) = Favourable outcomes / Total outcomes

Complementary Events

P(E) + P(Ē) = 1

Range

0 ≤ P(E) ≤ 1

Dice

1 die = 6 outcomes | 2 dice = 36 outcomes

Cards

52 total | 4 suits × 13 | Face cards = 12 | Aces = 4

Coins

1 coin = 2 | 2 coins = 4 | n coins = 2ⁿ