Chapter Mind Maps | Ishu (My Wife)

📦 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

Chapter Mind Maps 🗺️