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🗺️ Visual Chapter Maps

Chapter Mind Maps 🗺️

See how every concept in a chapter connects! Ishu (My Wife) — one glance at this map and the whole chapter will be in your head! 🎯

📦 Relations & Functions (Ch 1)
🔗 Relations
  • Relation R on set A ⊆ A×A
  • Reflexive: (a,a) ∈ R ∀a
  • Symmetric: (a,b)→(b,a)
  • Transitive: (a,b),(b,c)→(a,c)
  • Equivalence = All 3!
  • Empty: not reflexive
  • Universal: is equivalence
📈 Functions
  • One-One: f(a)=f(b)⟹a=b
  • Onto: Range = Codomain
  • Bijective = One-One + Onto
  • Inverse exists ↔ Bijective
  • Composition: gof(x)=g(f(x))
  • gof ≠ fog in general!
⚙️ Binary Operations
  • Commutative: a*b = b*a
  • Associative: (a*b)*c = a*(b*c)
  • Identity: a*e = e*a = a
  • Inverse: a*b = b*a = e
📊 Board Focus
  • ⭐ Prove Equivalence (3M)
  • ⭐ One-One & Onto check (3M)
  • ⭐ Find inverse of bijection (2M)
  • Case studies on composition
🔄 Inverse Trigonometric Functions (Ch 2)
📐 Principal Ranges
  • sin⁻¹: [-π/2, π/2] · Domain [-1,1]
  • cos⁻¹: [0, π] · Domain [-1,1]
  • tan⁻¹: (-π/2, π/2) · Domain ℝ
  • cot⁻¹: (0, π)
  • sec⁻¹: [0,π]-{π/2}
  • cosec⁻¹: [-π/2,π/2]-{0}
⭐ Key Identities
  • sin⁻¹x + cos⁻¹x = π/2
  • tan⁻¹x + cot⁻¹x = π/2
  • cos⁻¹(-x) = π - cos⁻¹x
  • sin⁻¹(-x) = -sin⁻¹x (odd)
$2\tan^{-1}x = \sin^{-1}\frac{2x}{1+x^2}$
➕ Addition Formulas
$\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}$ (xy<1)
  • If xy>1 and x,y>0: add π
  • If xy>1 and x,y<0: subtract π
📊 Board Focus
  • ⭐ Simplify ITF expressions
  • ⭐ Prove identity questions
  • ⭐ Find principal values
  • Watch: cos⁻¹(-x) is NOT negative!
📉 Continuity & Differentiability (Ch 5)
🔵 Continuity
  • 3 conditions: LHL=RHL=f(c)
  • Polynomial/rational/trig: continuous
  • Sum, product, composite: continuous
  • Discontinuity: jump, infinite, removable
📐 Differentiability
  • Diff at c: LHD = RHD
  • Diff ⟹ Continuous (not vice versa)
  • |x| is continuous but NOT diff at 0
  • Rolle's: f'(c)=0 exists
  • LMVT: f'(c)=(f(b)-f(a))/(b-a)
⚙️ Key Derivatives
  • d/dx(sin⁻¹x) = 1/√(1-x²)
  • d/dx(tan⁻¹x) = 1/(1+x²)
  • d/dx(eˣ) = eˣ
  • d/dx(ln x) = 1/x
  • Chain Rule: dy/dx = (dy/du)(du/dx)
  • Log diff for xˣ type
📊 Board Focus
  • ⭐ Find k for continuity (3M)
  • ⭐ Prove diff ⟹ continuous
  • ⭐ Log differentiation
  • ⭐ Implicit/parametric diff
∫ Integrals (Ch 7)
📝 Methods
  • Substitution: u = g(x)
  • By Parts (ILATE rule)
  • Partial Fractions
  • Special Formulas (a²±x² types)
✨ Key Properties
  • ∫₀ᵃ f = ∫₀ᵃ f(a-x)dx
  • ∫₋ₐᵃ even function = 2∫₀ᵃ
  • ∫₋ₐᵃ odd function = 0
  • ∫ₐᵇ = ∫ₐᵇ f(a+b-x)dx
⭐ Special Formulas
$\int e^x[f(x)+f'(x)]dx = e^xf(x)+C$
$\int\frac{1}{x^2+a^2}dx = \frac{1}{a}\tan^{-1}\frac{x}{a}+C$
📊 Board Focus
  • ⭐ Always write +C
  • ⭐ Definite integral properties
  • ⭐ Integration by parts (5M)
  • ⭐ Change limits on substitution
🎲 Probability (Ch 13)
🔀 Conditional Probability
$P(A|B)=\frac{P(A\cap B)}{P(B)}$
  • Multiplication: P(A∩B)=P(A)·P(B|A)
  • Independent: P(A∩B)=P(A)·P(B)
🏛️ Bayes' Theorem
$P(E_i|A)=\frac{P(E_i)P(A|E_i)}{\sum P(E_j)P(A|E_j)}$
  • Denominator = Total Probability P(A)
  • Always compute P(A) first!
🎰 Random Variables
  • E(X) = Σx·P(x)
  • Var(X) = E(X²) - [E(X)]²
  • Binomial: P(X=r) = ⁿCᵣpʳqⁿ⁻ʳ
  • Mean=np, Var=npq
📊 Board Focus
  • ⭐ Bayes' Theorem (5M) always!
  • ⭐ Binomial distribution
  • ⭐ Mean & variance of R.V.
  • High scoring chapter — don't skip!
🔢 Matrices (Ch 3)
📋 Types
  • Row/Column/Square/Zero
  • Identity (I): aᵢᵢ=1, rest 0
  • Symmetric: A=Aᵀ
  • Skew-Symmetric: A=-Aᵀ, diag=0
⚙️ Operations
  • (AB)ᵀ = BᵀAᵀ (order reverses!)
  • (AB)⁻¹ = B⁻¹A⁻¹
  • A = ½(A+Aᵀ) + ½(A-Aᵀ)
📊 Board Focus
  • ⭐ Sym+Skew decomposition
  • ⭐ Find A⁻¹ using adj
  • ⭐ Transpose properties
📊 Determinants (Ch 4)
🔑 Key Properties
  • |kA|=kⁿ|A| for n×n
  • |AB|=|A||B|
  • |Aᵀ|=|A|
  • Two same rows → det=0
📐 Area & Collinearity
  • Area = ½|det| (always positive!)
  • Collinear ↔ det = 0
$A^{-1}=\frac{1}{|A|}\text{adj}A$
📊 Board Focus
  • ⭐ Matrix inverse (5M)
  • ⭐ Solve AX=B system
  • ⭐ Cofactor sign: (-1)^(i+j)
📈 Application of Derivatives (Ch 6)
📍 Rate of Change
  • Rate: dy/dt = (dy/dx)·(dx/dt)
  • Tangent slope = f'(x₁)
  • Normal slope = -1/f'(x₁)
🔺 Max & Min
  • f'(x)=0 → critical points
  • f''(c)<0 → Maximum
  • f''(c)>0 → Minimum
  • Absolute: check endpoints too!
📊 Board Focus
  • ⭐ Optimization problems (6M)
  • ⭐ Increasing/decreasing intervals
  • ⭐ Tangent & normal equations
📐 Application of Integrals (Ch 8)
🎯 Core Formula
$A = \int_a^b|f(x)-g(x)|dx$
  • Find intersection points first!
  • Area always positive (take |·|)
  • Sketch graph always
📊 Standard Curves
  • Circle x²+y²=r² → Area=πr²
  • Ellipse → Area=πab
  • Parabola y=x² → Area=b³/3
📊 Board Focus
  • ⭐ Area between two curves (5M)
  • ⭐ Parabola + line area
  • ⭐ Circle/ellipse area
📋 Differential Equations (Ch 9)
🔢 Basics
  • Order = highest derivative
  • Degree = power of highest
  • General: has C. Particular: C found
⚙️ Methods
  • Variable Separable: f(x)dx=g(y)dy
  • Homogeneous: substitute y=vx
  • Linear: dy/dx+Py=Q → I.F.=e^∫P
📊 Board Focus
  • ⭐ Linear DE (5M) — always comes
  • ⭐ Homogeneous DE
  • ⭐ Formation of DE
→ Vector Algebra (Ch 10)
🔵 Dot Product
$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta$
  • ⊥ ↔ dot product = 0
  • Projection: a·b/|b|
✖️ Cross Product
$|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin\theta$
  • ∥ ↔ cross product = 0
  • a×b = -(b×a)
  • Area △ = ½|a×b|
📊 Board Focus
  • ⭐ Find angle between vectors
  • ⭐ Area using cross product
  • ⭐ Triple product (coplanar)
📦 3D Geometry (Ch 11)
📏 Lines
  • Vector: r=a+λb
  • Cartesian: (x-x₁)/a=(y-y₁)/b=(z-z₁)/c
  • Angle: cos θ=|b₁·b₂|/|b₁||b₂|
📐 Planes
  • Normal vector: (a,b,c)
  • Eq: ax+by+cz+d=0
  • Pt to plane: |ax₁+by₁+cz₁+d|/√(a²+b²+c²)
📊 Board Focus
  • ⭐ Skew line SD formula
  • ⭐ Plane from 3 points
  • ⭐ Angle between line & plane
📊 Linear Programming (Ch 12)
📋 Steps
  • 1. Define variables x,y
  • 2. Write objective Z=ax+by
  • 3. Write all constraints
  • 4. Graph feasible region
  • 5. Find ALL corner points
  • 6. Evaluate Z at each corner
  • 7. Conclude max/min
📊 Board Focus
  • ⭐ Make Z table (mandatory!)
  • ⭐ Shade correct side
  • ⭐ State conclusion clearly
  • ⭐ Mention Corner Pt Theorem