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ONE-ONE (Injective):
Different inputs → Different outputs
f(a) = f(b) implies a = b
A: 1 ──→ A :B
A: 2 ──→ B
A: 3 ──→ C ← Each arrow to different element = One-one
ONTO (Surjective):
Every element of B has at least one preimage
Range = Codomain
A: 1 ──┐
├──→ A :B
A: 2 ──┘
A: 3 ──→ B :B ← Every B element has arrow coming = Onto
BIJECTIVE = ONE-ONE + ONTO
Has perfect pairing, has inverse function
If f: A→B and g: B→C, then gof: A→C
A ──f──→ B ──g──→ C
gof
A ─────────────→ C
(gof)(x) = g(f(x))
Example: f(x) = x+1, g(x) = x²
(gof)(x) = g(f(x)) = g(x+1) = (x+1)²
(fog)(x) = f(g(x)) = f(x²) = x²+1
Note: gof ≠ fog generally (not commutative)
Function Domain Range (Principal value branch)
─────────────────────────────────────────────────────────────
sin⁻¹(x) [-1, 1] [-π/2, π/2]
cos⁻¹(x) [-1, 1] [0, π]
tan⁻¹(x) ℝ (all reals) (-π/2, π/2)
cot⁻¹(x) ℝ (0, π)
sec⁻¹(x) |x| ≥ 1 [0,π] − {π/2}
cosec⁻¹(x) |x| ≥ 1 [-π/2,π/2] − {0}
IMPORTANT FORMULAS:
sin⁻¹(x) + cos⁻¹(x) = π/2
tan⁻¹(x) + cot⁻¹(x) = π/2
sin⁻¹(-x) = -sin⁻¹(x) (odd function)
cos⁻¹(-x) = π - cos⁻¹(x) (not odd!)
tan⁻¹(-x) = -tan⁻¹(x) (odd function)
tan⁻¹(1/x) = π/2 - tan⁻¹(x) if x > 0
MATRIX: Rectangular array of numbers
Order = rows × columns
Addition: Same order only
[1 2] + [5 6] = [6 8]
[3 4] [7 8] [10 12]
Multiplication: A(m×n) × B(n×p) = C(m×p)
Rows of A × Columns of B
[1 2][5] = [1×5+2×7] = [19]
[3 4][7] [3×5+4×7] [43]
TRANSPOSE: A^T — rows become columns
A = [1 2 3] A^T = [1 4]
[4 5 6] [2 5]
[3 6]
Symmetric Matrix: A^T = A
Skew-Symmetric: A^T = -A
2×2 DETERMINANT:
|a b| = ad - bc
|c d|
3×3 DETERMINANT (expansion along row 1):
|a₁ b₁ c₁|
|a₂ b₂ c₂| = a₁|b₂ c₂| - b₁|a₂ c₂| + c₁|a₂ b₂|
|a₃ b₃ c₃| |b₃ c₃| |a₃ c₃| |a₃ b₃|
AREA OF TRIANGLE (vertices (x₁,y₁),(x₂,y₂),(x₃,y₃)):
Area = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
= ½|x₁ y₁ 1|
|x₂ y₂ 1|
|x₃ y₃ 1|
INVERSE MATRIX:
A⁻¹ = (1/|A|) × adj(A)
A⁻¹ exists only if |A| ≠ 0 (non-singular)
CONTINUITY at x=a:
f is continuous if lim f(x) = f(a)
x→a
3 conditions:
1. f(a) exists (defined)
2. lim f(x) exists (left limit = right limit)
x→a
3. Both are equal
DIFFERENTIABILITY:
f'(a) = lim [f(a+h) - f(a)] / h
h→0
Differentiable → Continuous (always)
Continuous → NOT necessarily differentiable
Example: |x| is continuous at x=0 but not differentiable
CHAIN RULE:
If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x)
Example: y = sin(x²)
dy/dx = cos(x²) × 2x = 2x cos(x²)
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 (sin⁻¹x) = 1/√(1-x²)
d/dx (cos⁻¹x) = -1/√(1-x²)
d/dx (tan⁻¹x) = 1/(1+x²)
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C [n ≠ -1]
∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C
∫ 1/√(1-x²) dx = sin⁻¹x + C
∫ 1/(1+x²) dx = tan⁻¹x + C
DEFINITE INTEGRAL:
∫ₐᵇ f(x)dx = F(b) - F(a) [where F' = f]
Represents AREA under curve from a to b
PROPERTIES:
∫ₐᵇ f(x)dx = -∫ᵦᵃ f(x)dx
∫ₐᵃ f(x)dx = 0
∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx [if f is even]
∫₋ₐᵃ f(x)dx = 0 [if f is odd]
VECTOR has magnitude AND direction
Position vector: OP = (x,y,z) from origin O to P(x,y,z)
Magnitude: |a| = √(a₁² + a₂² + a₃²)
Unit vector: â = a/|a| (magnitude = 1)
DOT PRODUCT (Scalar product):
a·b = |a||b|cosθ = a₁b₁ + a₂b₂ + a₃b₃
Use: find angle between vectors
cosθ = a·b / (|a||b|)
If a·b = 0, vectors are perpendicular (θ=90°)
CROSS PRODUCT (Vector product):
a × b = |i j k |
|a₁ a₂ a₃|
|b₁ b₂ b₃|
|a × b| = |a||b|sinθ
If a × b = 0, vectors are parallel
Direction: perpendicular to both a and b (right-hand rule)
CONDITIONAL PROBABILITY:
P(A|B) = P(A∩B) / P(B) [probability of A given B occurred]
MULTIPLICATION RULE:
P(A∩B) = P(A) × P(B|A) = P(B) × P(A|B)
INDEPENDENT EVENTS:
P(A∩B) = P(A) × P(B) [A and B don't affect each other]
BAYES' THEOREM:
P(Aᵢ|B) = P(Aᵢ) × P(B|Aᵢ) / Σ P(Aⱼ) × P(B|Aⱼ)
Example: Box 1: 3 red, 4 blue. Box 2: 5 red, 2 blue.
Box selected randomly. Red ball drawn. What's P(Box 2)?
P(Box1) = P(Box2) = 1/2
P(Red|Box1) = 3/7, P(Red|Box2) = 5/7
P(Box2|Red) = P(Box2)×P(Red|Box2) / [P(Box1)×P(Red|Box1) + P(Box2)×P(Red|Box2)]
= (1/2)(5/7) / [(1/2)(3/7) + (1/2)(5/7)]
= (5/14) / (8/14) = 5/8
BINOMIAL DISTRIBUTION:
n independent trials, each with probability p of success
P(X=r) = nCr × pʳ × (1-p)ⁿ⁻ʳ
Mean = np
Variance = np(1-p)
Standard deviation = √(np(1-p))
Example: Coin tossed 6 times. P(exactly 4 heads)?
n=6, r=4, p=0.5
P(X=4) = 6C4 × (0.5)⁴ × (0.5)² = 15 × 0.0625 × 0.25 = 0.234
Class 12 Maths complete NCERT notes — relations functions, inverse trigonometry, matrices, determinants, continuity differentiability, integrals, differential equations, vectors, 3D geometry, linear programming, probability with solved examples.
68 pages · 2.5 MB · Updated 2026-03-11
Part 1: If F(x) = ∫ₐˣ f(t)dt, then F'(x) = f(x) — derivative of integral is the integrand. Part 2: ∫ₐᵇ f(x)dx = F(b) - F(a) where F is antiderivative of f — definite integral equals change in antiderivative. This links differentiation and integration as inverse operations.
Bayes' theorem: P(A|B) = P(B|A)×P(A) / P(B). It calculates conditional probability — probability of A given B has occurred, using prior probability of A and likelihood of B given A. Used in spam filters, medical diagnosis, Bayesian statistics.
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