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Chapter 1
Sets
⭕ Practice on Venn Diagram Builder →
| Formula | Name | Note |
|---|---|---|
| n(A∪B) = n(A) + n(B) – n(A∩B) | Union of two sets | Inclusion-exclusion principle |
| n(A∪B∪C) = n(A)+n(B)+n(C) –n(A∩B)–n(B∩C)–n(A∩C)+n(A∩B∩C) | Union of three sets | Extended inclusion-exclusion |
| A' = U – A | Complement of A | All elements not in A |
| (A∪B)' = A'∩B' | De Morgan's Law 1 | Complement of union |
| (A∩B)' = A'∪B' | De Morgan's Law 2 | Complement of intersection |
| n(A×B) = n(A)·n(B) | Cartesian Product | Number of ordered pairs |
Chapter 3
Trigonometric Functions
📐 Practice on Trigonometry Visualizer →
| Formula | Name | Note |
|---|---|---|
| sin²θ + cos²θ = 1 | Pythagorean Identity 1 | Most important identity |
| 1 + tan²θ = sec²θ | Pythagorean Identity 2 | Divide identity 1 by cos²θ |
| 1 + cot²θ = cosec²θ | Pythagorean Identity 3 | Divide identity 1 by sin²θ |
| sin(A+B) = sinA·cosB + cosA·sinB | Addition Formula | For sin(A–B) change + to – |
| cos(A+B) = cosA·cosB – sinA·sinB | Addition Formula | For cos(A–B) change – to + |
| tan(A+B) = (tanA+tanB)/(1–tanA·tanB) | tan Addition | Signs flip for subtraction |
| sin2A = 2sinA·cosA | Double Angle (sin) | Put B=A in addition formula |
| cos2A = cos²A – sin²A = 1–2sin²A = 2cos²A–1 | Double Angle (cos) | 3 equivalent forms |
| sinC + sinD = 2sin((C+D)/2)·cos((C–D)/2) | Sum to Product | Very important for solving equations |
| a/sinA = b/sinB = c/sinC | Sine Rule | For any triangle |
| c² = a² + b² – 2ab·cosC | Cosine Rule | For any triangle |
Chapter 5
Complex Numbers & Quadratic Equations
🌀 Practice on Complex Number Explorer →
| Formula | Name | Note |
|---|---|---|
| i = √–1, i² = –1, i³ = –i, i⁴ = 1 | Powers of i | Pattern repeats every 4 |
| z = a + ib | Standard Form | a = real part, b = imaginary part |
| |z| = √(a² + b²) | Modulus | Distance from origin in Argand plane |
| z̄ = a – ib | Conjugate | z·z̄ = |z|² |
| z = r(cosθ + i·sinθ) | Polar Form | r = |z|, θ = arg(z) |
| x = (–b ± √(b²–4ac)) / 2a | Quadratic Formula | Discriminant D = b²–4ac |
Chapter 7
Permutations & Combinations
🔢 Practice on P&C Calculator →
| Formula | Name | Note |
|---|---|---|
| n! = n×(n–1)×...×2×1 | Factorial | 0! = 1 |
| ⁿPr = n!/(n–r)! | Permutation | Order matters, r items from n |
| ⁿCr = n!/[r!(n–r)!] | Combination | Order doesn't matter |
| ⁿCr = ⁿC(n–r) | Symmetry Property | ⁿC0 = ⁿCn = 1 |
| ⁿCr + ⁿC(r–1) = ⁿ⁺¹Cr | Pascal's Identity | Pascal's triangle relation |
Chapter 8
Binomial Theorem
📊 Practice on Binomial Expander →
| Formula | Name | Note |
|---|---|---|
| (a+b)ⁿ = Σ ⁿCr · aⁿ⁻ʳ · bʳ (r=0 to n) | Binomial Theorem | n must be a positive integer |
| T(r+1) = ⁿCr · aⁿ⁻ʳ · bʳ | General Term | (r+1)th term of expansion |
| Total terms = n + 1 | Number of Terms | In expansion of (a+b)ⁿ |
| Sum of coefficients = 2ⁿ | Coefficient Sum | Put a = b = 1 |
Chapter 9
Sequences & Series
📈 Practice on AP/GP Visualizer →
| Formula | Name | Note |
|---|---|---|
| aₙ = a + (n–1)d | AP: nth term | a = first term, d = common difference |
| Sₙ = n/2·[2a + (n–1)d] | AP: Sum of n terms | Also: Sₙ = n/2·(a + l), l = last term |
| aₙ = a·rⁿ⁻¹ | GP: nth term | r = common ratio |
| Sₙ = a(rⁿ–1)/(r–1), r≠1 | GP: Sum of n terms | If r=1, Sₙ = na |
| S∞ = a/(1–r), |r|<1 | GP: Infinite sum | Only valid if |r| < 1 |
| Σn = n(n+1)/2 | Sum of first n naturals | 1+2+3+...+n |
| Σn² = n(n+1)(2n+1)/6 | Sum of squares | 1²+2²+...+n² |
| Σn³ = [n(n+1)/2]² | Sum of cubes | 1³+2³+...+n³ |
Chapter 10
Straight Lines
📉 Practice on Coordinate Plotter →
| Formula | Name | Note |
|---|---|---|
| m = (y₂–y₁)/(x₂–x₁) | Slope | m = tanθ, θ = inclination angle |
| y – y₁ = m(x – x₁) | Point-Slope Form | Line through (x₁,y₁) with slope m |
| y = mx + c | Slope-Intercept Form | c = y-intercept |
| x/a + y/b = 1 | Intercept Form | a = x-intercept, b = y-intercept |
| d = |ax₁+by₁+c| / √(a²+b²) | Distance from Point to Line | Line: ax + by + c = 0 |
| d = |c₁–c₂| / √(a²+b²) | Distance between Parallel Lines | ax+by+c₁=0 and ax+by+c₂=0 |
Chapter 11
Conic Sections
⭕ Practice on Conic Sections Visualizer →
| Shape | Standard Equation | Key Parameters |
|---|---|---|
| Circle | x² + y² = r² | Centre (0,0), radius r |
| Circle (general) | (x–h)² + (y–k)² = r² | Centre (h,k) |
| Parabola | y² = 4ax | Focus (a,0), directrix x = –a |
| Ellipse | x²/a² + y²/b² = 1, a>b | c² = a²–b², e = c/a < 1 |
| Hyperbola | x²/a² – y²/b² = 1 | c² = a²+b², e = c/a > 1 |
Chapter 13
Limits & Derivatives
∞ Practice on Limits Explorer →
| Formula | Name | Note |
|---|---|---|
| lim(xⁿ–aⁿ)/(x–a) = naⁿ⁻¹ as x→a | Standard Limit 1 | Very commonly tested |
| lim(sinx/x) = 1 as x→0 | Standard Limit 2 | x must be in radians |
| lim(tanx/x) = 1 as x→0 | Standard Limit 3 | x in radians |
| d/dx(xⁿ) = nxⁿ⁻¹ | Power Rule | Most used derivative rule |
| d/dx(sinx) = cosx | Derivative of sin | d/dx(cosx) = –sinx |
| d/dx(UV) = U·V' + V·U' | Product Rule | Also called Leibniz rule |
| d/dx(U/V) = (VU' – UV')/V² | Quotient Rule | V ≠ 0 |
Chapter 15
Statistics
📊 Practice on Statistics Lab →
| Formula | Name | Note |
|---|---|---|
| x̄ = Σxᵢ/n | Mean (ungrouped) | Sum of all values / count |
| x̄ = Σfᵢxᵢ/Σfᵢ | Mean (grouped) | fᵢ = frequency |
| σ² = Σ(xᵢ–x̄)²/n | Variance | Mean of squared deviations |
| σ = √[Σ(xᵢ–x̄)²/n] | Standard Deviation | σ = √variance |
| CV = (σ/x̄) × 100 | Coefficient of Variation | For comparing variability |