📐 Key Concepts:
• Types of Relations: Reflexive (aRa ∀a), Symmetric (aRb ⇒ bRa), Transitive (aRb ∧ bRc ⇒ aRc)
• Equivalence Relation = Reflexive + Symmetric + Transitive
• Types of Functions: One-one (Injective): f(a)=f(b) ⇒ a=b | Onto (Surjective): Range = Codomain | Bijective = One-one + Onto
• Composition: (fog)(x) = f(g(x)) — apply g first, then f
• Inverse: f⁻¹ exists only if f is bijective

📐 Domain & Range Table:
• sin⁻¹x: Domain [-1,1], Range [-π/2, π/2]
• cos⁻¹x: Domain [-1,1], Range [0, π]
• tan⁻¹x: Domain ℝ, Range (-π/2, π/2)
• cot⁻¹x: Domain ℝ, Range (0, π)
• sec⁻¹x: Domain (-∞,-1]∪[1,∞), Range [0,π]-{π/2}
• cosec⁻¹x: Domain (-∞,-1]∪[1,∞), Range [-π/2,π/2]-{0}

📐 Key Formulas:
• Types: Row (1×n), Column (m×1), Square (n×n), Diagonal, Scalar, Identity (I), Zero, Symmetric (A=Aᵀ), Skew-Symmetric (A=-Aᵀ)
• Operations: A+B (same order), kA (scalar), AB (columns of A = rows of B)
• Transpose: (AB)ᵀ = BᵀAᵀ | (A+B)ᵀ = Aᵀ+Bᵀ | (Aᵀ)ᵀ = A
• Inverse: A⁻¹ = (1/|A|) × adj(A) | AA⁻¹ = I
• Every square matrix: A = ½(A+Aᵀ) + ½(A-Aᵀ) [Symmetric + Skew-Symmetric]
• Elementary Operations: Rᵢ ↔ Rⱼ, Rᵢ → kRᵢ, Rᵢ → Rᵢ + kRⱼ

📐 Key Formulas:
• 2×2: |A| = ad - bc for [a,b; c,d]
• 3×3: Expand along R1 = a₁₁(M₁₁) - a₁₂(M₁₂) + a₁₃(M₁₃)
• Properties: |Aᵀ| = |A| | Rows/Columns swap → sign change | Two identical rows → |A|=0 | Row × k → |A|×k | |kA| = kⁿ|A| (n = matrix का order, i.e. 3×3 तो n=3)
• Area of Triangle: ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
• Cramer's Rule: x = D₁/D, y = D₂/D, z = D₃/D
• Adjoint: adj(A) = transpose of cofactor matrix
• A(adj A) = |A|·I | |adj A| = |A|ⁿ⁻¹

📐 Differentiation Formulas:
• d/dx(xⁿ) = nxⁿ⁻¹ | d/dx(eˣ) = eˣ | d/dx(aˣ) = aˣ·ln a | d/dx(ln x) = 1/x
• d/dx(sin x) = cos x | d/dx(cos x) = -sin x | d/dx(tan x) = sec²x
• d/dx(sec x) = sec x·tan x | d/dx(cosec x) = -cosec x·cot x | d/dx(cot x) = -cosec²x
• d/dx(sin⁻¹x) = 1/√(1-x²) | d/dx(cos⁻¹x) = -1/√(1-x²) | d/dx(tan⁻¹x) = 1/(1+x²)
• Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x)
• Product Rule: (uv)' = u'v + uv' | Quotient: (u/v)' = (u'v - uv')/v²
• Logarithmic: y = [f(x)]^g(x) → log y = g(x)·log f(x) → differentiate both sides
• Parametric: dy/dx = (dy/dt)/(dx/dt)
• Rolle's Theorem: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) ⇒ ∃c: f'(c)=0
• LMVT: f'(c) = (f(b)-f(a))/(b-a)

📐 Key Applications:
• Rate of Change: dy/dt = (dy/dx)·(dx/dt)
• Tangent at (x₁,y₁): y - y₁ = f'(x₁)(x - x₁) | Normal: y - y₁ = -(1/f'(x₁))(x - x₁)
• Increasing: f'(x) > 0 | Decreasing: f'(x) < 0
• Maxima/Minima (First Derivative Test): f'(x) changes + to - = maxima | - to + = minima
• Second Derivative Test: f'(c)=0 and f''(c)<0 = maxima | f''(c)>0 = minima
• Absolute Max/Min: Compare f(a), f(b), f(critical points)

📐 MEGA Integration Formula Sheet:
Basic:
• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C | ∫1/x dx = ln|x| + C | ∫eˣ dx = eˣ + C | ∫aˣ dx = aˣ/ln a + C
Trigonometric:
• ∫sin x dx = -cos x | ∫cos x dx = sin x | ∫tan x dx = -ln|cos x| = ln|sec x|
• ∫cot x dx = ln|sin x| | ∫sec x dx = ln|sec x + tan x| | ∫cosec x dx = ln|cosec x - cot x|
• ∫sec²x dx = tan x | ∫cosec²x dx = -cot x | ∫sec x·tan x dx = sec x
Special Forms:
• ∫dx/(x²+a²) = (1/a)tan⁻¹(x/a) | ∫dx/√(a²-x²) = sin⁻¹(x/a)
• ∫dx/(x²-a²) = (1/2a)ln|(x-a)/(x+a)| | ∫dx/(a²-x²) = (1/2a)ln|(a+x)/(a-x)|
• ∫dx/√(x²+a²) = ln|x+√(x²+a²)| | ∫dx/√(x²-a²) = ln|x+√(x²-a²)|
• ∫√(a²-x²) dx = (x/2)√(a²-x²) + (a²/2)sin⁻¹(x/a) + C
• ∫√(x²+a²) dx = (x/2)√(x²+a²) + (a²/2)ln|x+√(x²+a²)| + C

📐 Area Under Curves:
• Area = ∫ₐᵇ f(x) dx (between curve and x-axis)
• Between two curves: ∫ₐᵇ [f(x) - g(x)] dx where f(x) ≥ g(x)
• Circle: x² + y² = r² → y = √(r²-x²) → Area = πr²
• Ellipse: x²/a² + y²/b² = 1 → Area = πab
• Parabola: y² = 4ax → Area between 0 to a = (4a)(2a)/3 × 2

📐 Types & Methods:
• Order: highest derivative | Degree: power of highest order derivative
• Variable Separable: f(x)dx = g(y)dy → integrate both sides
• Homogeneous: Put y = vx → dy/dx = v + x(dv/dx) → separate v and x
• Linear DE: dy/dx + P(x)·y = Q(x) → IF = e^∫P dx → y × IF = ∫Q × IF dx + C
• Linear DE (dx/dy): dx/dy + P(y)·x = Q(y) → similar method

📐 Key Formulas:
• Magnitude: |a⃗| = √(a₁² + a₂² + a₃²)
• Unit Vector: â = a⃗/|a⃗|
• Dot Product: a⃗·b⃗ = a₁b₁ + a₂b₂ + a₃b₃ = |a⃗||b⃗|cos θ
• Cross Product: a⃗ × b⃗ = |i⃗ j⃗ k⃗; a₁ a₂ a₃; b₁ b₂ b₃| | |a⃗ × b⃗| = |a⃗||b⃗|sin θ
• Area of ∆: ½|a⃗ × b⃗| | Area of ∥gram: |a⃗ × b⃗|
• Projection: proj of a⃗ on b⃗ = (a⃗·b⃗)/|b⃗|
• Scalar Triple Product: [a⃗ b⃗ c⃗] = a⃗·(b⃗ × c⃗) = Volume of parallelepiped
• Coplanar vectors: [a⃗ b⃗ c⃗] = 0

📐 Key Formulas:
Direction Cosines: l² + m² + n² = 1 | l = a/√(a²+b²+c²)

Equation of Line:
• Vector: r⃗ = a⃗ + λb⃗ | Cartesian: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c

Angle between Lines: cos θ = |a₁a₂+b₁b₂+c₁c₂| / (√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²))
• Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂ | Perpendicular: a₁a₂+b₁b₂+c₁c₂ = 0

Shortest Distance between Skew Lines:
• d = |(a⃗₂-a⃗₁)·(b⃗₁×b⃗₂)| / |b⃗₁×b⃗₂|
Parallel Lines (b⃗₁×b⃗₂ = 0): d = |(a⃗₂-a⃗₁) × b⃗| / |b⃗|
⚠️ पहले check करो: lines skew हैं या parallel? b⃗₁×b⃗₂ = 0 → parallel formula use करो!

Equation of Plane:
• General: ax + by + cz = d | Vector: r⃗·n⃗ = d
• Through 3 points: determinant form
• Distance of point from plane: d = |ax₁+by₁+cz₁-d| / √(a²+b²+c²)
• Angle between planes: cos θ = |a₁a₂+b₁b₂+c₁c₂| / (√·√)
• Angle between line & plane: sin θ = |(a⃗·n⃗)| / (|a⃗|·|n⃗|)

📐 Steps:
1. Define variables (Let x = ..., y = ...)
2. Write objective function: Maximize/Minimize Z = ax + by
3. Write constraints as inequalities (≤ or ≥)
4. Plot constraints on graph → shade feasible region
5. Find corner points of feasible region
6. Evaluate Z at each corner → highest = max, lowest = min

📐 Key Formulas:
• Conditional: P(A|B) = P(A∩B)/P(B)
• Multiplication: P(A∩B) = P(A)·P(B|A) = P(B)·P(A|B)
• Independent: P(A∩B) = P(A)·P(B)
• Bayes' Theorem: P(Eᵢ|A) = P(Eᵢ)·P(A|Eᵢ) / Σ P(Eⱼ)·P(A|Eⱼ) ⭐
• Total Probability: P(A) = Σ P(Eᵢ)·P(A|Eᵢ)

Random Variable & Distribution:
• Mean (E(X)) = Σ xᵢ·P(xᵢ) | Variance = E(X²) - [E(X)]²
• Binomial: P(X=r) = ⁿCᵣ · pʳ · qⁿ⁻ʳ where q = 1-p
• Mean = np | Variance = npq