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ROSTER form: List elements in { }
A = {1, 2, 3, 4, 5} B = {a, e, i, o, u}
SET-BUILDER form: Define property
A = {x : x is natural number, x ≤ 5}
B = {x : x is vowel in English alphabet}
TYPES OF SETS:
Empty/Null set: { } or ∅ (no elements)
Singleton: {5} (exactly one element)
Finite: {1,2,3,4,5} (countable elements)
Infinite: {1,2,3,...} (unending)
Universal set U: contains all sets under consideration
UNION (A ∪ B): All elements in A OR B or both
┌──────────────────────────┐
│ A │ B │
│ ┌────────┼────────┐ │
│ │ only A │ A∩B │ only B│
│ └────────┼────────┘ │
└──────────────────────────┘
A ∪ B = shaded completely
INTERSECTION (A ∩ B): Elements in BOTH A AND B
┌──────────────────────────┐
│ A │ B │
│ │████│ │
│ │████│ │
└──────────────────────────┘
A ∩ B = only middle shaded
COMPLEMENT (A'): Elements NOT in A (but in U)
┌──────────────────────────┐
│ ████│ A │███████ │
│ ████│ │███████ │
└──────────────────────────┘
A' = everything outside A
KEY FORMULAS:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
n(A ∪ B ∪ C) = n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(A∩C)+n(A∩B∩C)
FUNCTION f: A → B
A (Domain) B (Codomain)
┌──────┐ ┌──────────┐
│ 1 ──┼───────────┼→ a │
│ 2 ──┼──────────┐│ b │
│ 3 ──┼──────────┼┼→ c │
└──────┘ ↑│ d │
└┼──────────┘
Range = {a, c}
(images actually used)
For a function:
✓ Every element of A has exactly one image
✓ Two elements of A can map to same element of B
✗ One element of A CANNOT map to two elements of B
TYPES OF FUNCTIONS:
One-one (Injective): Different inputs → different outputs
f(a)=f(b) implies a=b
Onto (Surjective): Every element of B has at least one preimage
Range = Codomain
Bijective: Both one-one and onto (has inverse function)
STANDARD VALUES (must memorize):
Angle → │ 0° 30° 45° 60° 90°
─────────┼──────────────────────────────────────
sin θ │ 0 1/2 1/√2 √3/2 1
cos θ │ 1 √3/2 1/√2 1/2 0
tan θ │ 0 1/√3 1 √3 undefined
cosec θ │ ∞ 2 √2 2/√3 1
sec θ │ 1 2/√3 √2 2 ∞
cot θ │ ∞ √3 1 1/√3 0
MEMORY TRICK for sin:
sin 0°=√0/2=0, sin 30°=√1/2=1/2, sin 45°=√2/2,
sin 60°=√3/2, sin 90°=√4/2=1
(Numerators: 0,1,2,3,4 under √, denominator=2)
FUNDAMENTAL IDENTITIES (MUST KNOW):
sin²θ + cos²θ = 1 ← Most important!
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
COMPOUND ANGLE FORMULAS:
sin(A+B) = sinA cosB + cosA sinB
sin(A−B) = sinA cosB − cosA sinB
cos(A+B) = cosA cosB − sinA sinB
cos(A−B) = cosA cosB + sinA sinB
tan(A+B) = (tanA + tanB)/(1 − tanA tanB)
DOUBLE ANGLE:
sin 2A = 2 sinA cosA
cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A
tan 2A = 2tanA/(1 − tan²A)
ASTC Rule (All Students Take Calculus):
90°
│
II │ I
S(+) │ All(+) ← All positive
sin(+) │ sin, cos, tan
rest(-) │
─────────────────────── 0°/360°
│
III │ IV
T(+) │ C(+)
tan(+) │ cos(+)
rest(-) │ rest(-)
│
270°
Quadrant I (0°-90°): All positive
Quadrant II (90°-180°): Only sin positive
Quadrant III (180°-270°): Only tan positive
Quadrant IV (270°-360°): Only cos positive
COMPLEX NUMBER: z = a + bi
a = real part (Re(z))
b = imaginary part (Im(z))
i = √(-1) → i² = -1, i³ = -i, i⁴ = 1
ARGAND PLANE (Complex number diagram):
Imaginary (y-axis)
│
3i │ ● z = 2 + 3i
2i │
1i │
────┼────────── Real (x-axis)
-1i │ 1 2 3
-2i │
│
OPERATIONS:
Addition: (a+bi) + (c+di) = (a+c) + (b+d)i
Multiplication: (a+bi)(c+di) = ac + adi + bci + bdi²
= (ac-bd) + (ad+bc)i
Conjugate: If z = a+bi, then z̄ = a-bi
Modulus: |z| = √(a²+b²)
MULTIPLICATION PRINCIPLE:
If event A can occur in m ways AND
event B can occur in n ways,
then A and B together: m × n ways
Example: Shirts=3, Trousers=4
Total outfits = 3 × 4 = 12
ADDITION PRINCIPLE:
If A can occur in m ways OR B in n ways (mutually exclusive):
Total = m + n ways
FACTORIAL:
n! = n × (n-1) × (n-2) × ... × 2 × 1
5! = 5×4×3×2×1 = 120
0! = 1 (by definition)
PERMUTATION: Arrangement (ORDER matters)
nPr = n!/(n-r)!
3 people (A,B,C) in 2 seats:
AB, BA, AC, CA, BC, CB → 6 arrangements
3P2 = 3!/(3-2)! = 6/1 = 6 ✓
COMBINATION: Selection (ORDER doesn't matter)
nCr = n!/[r!(n-r)!]
Choose 2 from A,B,C:
{A,B}, {A,C}, {B,C} → 3 selections
3C2 = 3!/(2!×1!) = 6/2 = 3 ✓
KEY RELATION: nPr = nCr × r!
PROPERTIES:
nCr = nC(n-r) [symmetry]
nC0 = nCn = 1
nC1 = n
nCr + nC(r-1) = (n+1)Cr [Pascal's rule]
(a + b)ⁿ = Σ nCr × a^(n-r) × b^r (r from 0 to n)
GENERAL TERM: T(r+1) = nCr × a^(n-r) × b^r
Example: (x + 2)⁴
T1 = 4C0 × x⁴ × 2⁰ = x⁴
T2 = 4C1 × x³ × 2¹ = 8x³
T3 = 4C2 × x² × 2² = 24x²
T4 = 4C3 × x¹ × 2³ = 32x
T5 = 4C4 × x⁰ × 2⁴ = 16
(x+2)⁴ = x⁴ + 8x³ + 24x² + 32x + 16
PASCAL'S TRIANGLE:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Row n gives coefficients of (a+b)ⁿ
LIMIT: Value a function approaches as x approaches a value
lim f(x) = L means:
x→a
As x gets closer to a, f(x) gets closer to L
Example: f(x) = x²
x f(x) = x²
1.9 3.61
1.99 3.9601
1.999 3.996001
2.001 4.004001
2.01 4.0401
2.1 4.41
As x→2, f(x)→4 So: lim x² = 4
x→2
STANDARD LIMITS (MEMORIZE):
lim sin x/x = 1 (as x→0, x in radians)
x→0
lim (1-cos x)/x = 0
x→0
lim (xⁿ - aⁿ)/(x-a) = n×aⁿ⁻¹
x→a
DERIVATIVES (Rules):
d/dx (xⁿ) = nxⁿ⁻¹ Power rule
d/dx (constant) = 0
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec²x
d/dx (eˣ) = eˣ
d/dx (ln x) = 1/x
RULES:
Sum rule: d/dx[f+g] = f'+g'
Product rule: d/dx[fg] = f'g + fg'
Quotient rule: d/dx[f/g] = (f'g - fg')/g²
Chain rule: d/dx[f(g(x))] = f'(g(x)) × g'(x)
Class 11 Maths complete NCERT notes — sets, relations functions, trigonometry, complex numbers, quadratic equations, permutations combinations, binomial theorem, sequences series, coordinate geometry, calculus limits with solved examples.
58 pages · 2.2 MB · Updated 2026-03-11
A relation is any set of ordered pairs (x,y) from set A to set B. A function is a special relation where each element of A has exactly ONE image in B (no element of A maps to two different elements in B). Every function is a relation, but not every relation is a function.
Use the table: 0°,30°,45°,60°,90°. For sin: √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, 1/√2, √3/2, 1. For cos: reverse the sin values. For tan = sin/cos. Remember: 'Some People Have Curly Brown Hair Through Proper Brushing' for sin, cos, tan values.
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