CBSE Mathematics — Complete Formula Compendium

01

Real Numbers

1.1 Euclid's Division Lemma

a = bq + r, where 0 ≤ r < b → b = (a - r) / q → q = (a - r) / b → r = a - bq → a - r = bq → a mod b = r

1.2 HCF & LCM Relationship

HCF(a, b) × LCM(a, b) = a × b → LCM(a, b) = (a × b) / HCF(a, b) → HCF(a, b) = (a × b) / LCM(a, b) → a = HCF × (a/HCF) → b = HCF × (b/HCF) → LCM = HCF × (a/HCF) × (b/HCF) For three numbers: HCF(a,b,c) × LCM(a,b,c) ≠ a×b×c [NOT directly] LCM(a,b,c) = (a×b×c × HCF(a,b,c)) / [HCF(a,b) × HCF(b,c) × HCF(a,c)]

1.3 Fundamental Theorem of Arithmetic

Every composite number = product of primes (unique, order aside) n = p₁^a₁ × p₂^a₂ × p₃^a₃ × ... × pₖ^aₖ HCF = product of SMALLEST powers of common prime factors LCM = product of GREATEST powers of all prime factors

1.4 Rational & Irrational Numbers

Rational number p/q terminates iff q = 2^m × 5^n Number of decimal places = max(m, n) If q has prime factor other than 2 or 5 → non-terminating repeating √p is irrational if p is a prime number √(a/b) is irrational if a/b is not a perfect square of rationals

1.5 Divisibility Conditions

n² - 1 = (n-1)(n+1) n(n+1) = always even n(n+1)(n+2) = always divisible by 6 n(n+1)(2n+1)/6 = sum of squares of first n natural numbers

1.6 Key Number Theory Identities

(a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b² a² - b² = (a+b)(a-b) (a + b)³ = a³ + 3a²b + 3ab² + b³ (a - b)³ = a³ - 3a²b + 3ab² - b³ a³ + b³ = (a+b)(a² - ab + b²) a³ - b³ = (a-b)(a² + ab + b²) Twisted Forms: ab = [(a+b)² - a² - b²] / 2 ab = [(a+b)² - (a-b)²] / 4 a² + b² = (a+b)² - 2ab a² + b² = (a-b)² + 2ab (a+b)² - (a-b)² = 4ab (a+b)² + (a-b)² = 2(a²+b²)

02

Polynomials

2.1 Degree and Types

Linear: ax + b = 0 (degree 1) → 1 zero Quadratic: ax² + bx + c = 0 (degree 2) → 2 zeros Cubic: ax³+bx²+cx+d = 0 (degree 3) → 3 zeros

2.2 Quadratic Polynomial — Relationship Between Zeros and Coefficients

p(x) = ax² + bx + c, zeros: α, β Sum of zeros: α + β = -b/a Product of zeros: α × β = c/a → b = -a(α + β) → c = a(α × β) → a/b = -1/(α+β) → b/c = -(α+β)/(αβ) Twisted Derived Forms: α² + β² = (α+β)² - 2αβ = b²/a² - 2c/a = (b² - 2ac)/a² (α - β)² = (α+β)² - 4αβ = b²/a² - 4c/a = (b² - 4ac)/a² |α - β| = √[(α+β)² - 4αβ] = √(b²-4ac) / |a| α² - β² = (α+β)(α-β) 1/α + 1/β = (α+β)/(αβ) = -b/c 1/(αβ) = a/c α/β + β/α = (α²+β²)/(αβ) = (b²-2ac)/(ac) α² × β² = (αβ)² = c²/a² α³ + β³ = (α+β)³ - 3αβ(α+β) = (-b/a)³ - 3(c/a)(-b/a) = (-b³ + 3abc) / a³ α³ - β³ = (α-β)[(α+β)²-αβ] 1/α² + 1/β² = (α²+β²)/(αβ)² = (b²-2ac)/c² (α+1)(β+1) = αβ + α + β + 1 = c/a - b/a + 1 = (c - b + a)/a (α-1)(β-1) = αβ - (α+β) + 1 = c/a + b/a + 1 = (c + b + a)/a α²β + αβ² = αβ(α+β) = (c/a)(-b/a) = -bc/a²

2.3 Formation of Quadratic from Given Zeros

p(x) = x² - (α+β)x + αβ [when a=1] p(x) = k[x² - (α+β)x + αβ] [general, k ≠ 0] p(x) = k[x² - (sum of zeros)x + (product of zeros)]

2.4 Cubic Polynomial — Relationship with Zeros

p(x) = ax³ + bx² + cx + d, zeros: α, β, γ α + β + γ = -b/a αβ + βγ + γα = c/a αβγ = -d/a Twisted Forms: α² + β² + γ² = (α+β+γ)² - 2(αβ+βγ+γα) = b²/a² - 2c/a = (b²-2ac)/a² α³+β³+γ³-3αβγ = (α+β+γ)(α²+β²+γ²-αβ-βγ-γα) 1/α + 1/β + 1/γ = (αβ+βγ+γα)/(αβγ) = -c/d Formation: p(x) = x³ - (α+β+γ)x² + (αβ+βγ+γα)x - αβγ

2.5 Division Algorithm

Dividend = Divisor × Quotient + Remainder p(x) = g(x) × q(x) + r(x) where deg[r(x)] < deg[g(x)] → p(x) - r(x) = g(x) × q(x) [p-r is divisible by g] → q(x) = [p(x) - r(x)] / g(x)

2.6 Discriminant (Δ)

Δ = b² - 4ac Δ > 0 → two distinct real zeros Δ = 0 → two equal real zeros (coincident) Δ < 0 → no real zeros (complex) Zeros: α, β = [-b ± √(b²-4ac)] / 2a [Quadratic Formula] → α + β = -b/a (verify) → α - β = √Δ / a (difference of roots) → 2α = (-b/a) + √Δ/a = (-b + √Δ)/a → 2β = (-b/a) - √Δ/a = (-b - √Δ)/a

03

Pair of Linear Equations in Two Variables

3.1 Standard Form

a₁x + b₁y + c₁ = 0 a₂x + b₂y + c₂ = 0

3.2 Consistency Conditions

Unique solution (consistent): a₁/a₂ ≠ b₁/b₂ Infinitely many (consistent): a₁/a₂ = b₁/b₂ = c₁/c₂ No solution (inconsistent): a₁/a₂ = b₁/b₂ ≠ c₁/c₂

3.3 Cross-Multiplication Method

x y 1 ——————— = ——————— = ———————— b₁c₂-b₂c₁ c₁a₂-c₂a₁ a₁b₂-a₂b₁ → x = (b₁c₂ - b₂c₁) / (a₁b₂ - a₂b₁) → y = (c₁a₂ - c₂a₁) / (a₁b₂ - a₂b₁) Let D = a₁b₂ - a₂b₁ Dx = b₁c₂ - b₂c₁ [replace a-column with c values, negated] Dy = c₁a₂ - c₂a₁ x = Dx/D, y = Dy/D

3.4 Substitution Method (Steps)

From eq1: y = (c₁ - a₁x) / b₁ Substitute in eq2: a₂x + b₂[(c₁-a₁x)/b₁] + c₂ = 0 Solve for x, then back-substitute for y

3.5 Elimination Method

Multiply eq1 by a₂, eq2 by a₁: a₁a₂x + b₁a₂y = -c₁a₂ a₁a₂x + b₂a₁y = -c₂a₁ Subtract: y(b₁a₂ - b₂a₁) = c₂a₁ - c₁a₂ → y = (c₂a₁ - c₁a₂)/(b₁a₂ - b₂a₁)

3.6 Graphical Interpretation

Lines intersect → unique solution → (x,y) = point of intersection Lines parallel → no solution → slopes equal, y-intercepts differ Lines coincident → infinite solutions → same line Slope of a₁x + b₁y + c₁ = 0 is m = -a₁/b₁

04

Quadratic Equations

4.1 Standard Form & Quadratic Formula

ax² + bx + c = 0 (a ≠ 0) x = [-b ± √(b² - 4ac)] / 2a x₁ = (-b + √D) / 2a x₂ = (-b - √D) / 2a where D = b² - 4ac (Discriminant)

4.2 Nature of Roots

D > 0 → two distinct real roots D = 0 → two equal real roots: x = -b/2a D < 0 → no real roots (imaginary) D is perfect square → rational roots D > 0 but not perfect square → irrational roots

4.3 Sum and Product of Roots

x₁ + x₂ = -b/a x₁ × x₂ = c/a x₁ - x₂ = √D/a = √(b²-4ac)/a Equation from roots: x² - (x₁+x₂)x + x₁x₂ = 0

4.4 Completing the Square

ax² + bx + c = 0 x² + (b/a)x + c/a = 0 (x + b/2a)² = b²/4a² - c/a = (b²-4ac)/4a² x + b/2a = ± √(b²-4ac) / 2a x = -b/2a ± √(b²-4ac)/2a

4.5 Special Cases & Factoring Tricks

If sum of coefficients = 0: x=1 is always a root i.e., a + b + c = 0 → x=1 and x=c/a If a + c = b: x = -1 is always a root For x² - Sx + P = 0: roots are S±√(S²-4P)/2 Factoring: ax²+bx+c = a(x-x₁)(x-x₂) = a[x² - (x₁+x₂)x + x₁x₂]

4.6 Common Root Condition

If ax²+bx+c=0 and dx²+ex+f=0 share a common root r: ar²+br+c=0 and dr²+er+f=0 → r = (bf-ce)/(cd-af) = (ae-bd)/(bf-ce) [cross multiply] Condition: (bf-ce)(ae-bd) = (cd-af)²

05

Arithmetic Progressions (AP)

5.1 General Term (nth Term)

aₙ = a + (n-1)d where: a = first term d = common difference = aₙ - aₙ₋₁ n = number of terms → n = (aₙ - a)/d + 1 → d = (aₙ - a)/(n-1) → a = aₙ - (n-1)d → aₙ = l = last term Last term: l = a + (n-1)d Number of terms: n = (l-a)/d + 1

5.2 Sum of n Terms

Sₙ = n/2 [2a + (n-1)d] Sₙ = n/2 [a + l] (when last term l is known) Sₙ = n/2 [a + aₙ] → aₙ = Sₙ - Sₙ₋₁ (nth term from sum) → 2a + (n-1)d = 2Sₙ/n → d = 2(Sₙ - na) / [n(n-1)] → a = [2Sₙ - n(n-1)d] / 2n If Sₙ = An² + Bn: aₙ = A(2n-1) + B (nth term) a₁ = A + B d = 2A

5.3 Important Sum Formulas

Sum of first n natural numbers: Σn = n(n+1)/2 Sum of first n odd numbers: 1+3+5+...+(2n-1) = n² Sum of first n even numbers: 2+4+6+...+2n = n(n+1) Sum of squares of first n naturals: Σn² = n(n+1)(2n+1)/6 Sum of cubes of first n naturals: Σn³ = [n(n+1)/2]²

5.4 Arithmetic Mean

AM between a and b: A = (a+b)/2 → a, A, b are in AP ↔ A-a = b-A ↔ 2A = a+b n AMs between a and b: A₁ = a + d, A₂ = a + 2d, ..., Aₙ = a + nd where d = (b-a)/(n+1) Sum of n AMs between a and b = n(a+b)/2

5.5 Properties of AP

If a, b, c are in AP: 2b = a + c → b-a = c-b If each term multiplied by k → still AP with same ratio If constant added → still AP with same d If reversed → still AP with common difference -d aₘ + aₙ = aₚ + aₚ where m+n = p+q (same sum of indices → same sum of terms) Middle term of AP with odd n = Sₙ/n = average 3 numbers in AP: a-d, a, a+d [sum = 3a] 4 numbers in AP: a-3d, a-d, a+d, a+3d [sum = 4a] 5 numbers in AP: a-2d, a-d, a, a+d, a+2d [sum = 5a]

5.6 Twisted AP Identities

aₘ - aₙ = (m-n)d aₘ/aₙ = [a+(m-1)d] / [a+(n-1)d] If aₘ = n and aₙ = m: d = (n-m)/(m-n) = -1 aₘ₊ₙ = 0 If Sₘ/Sₙ = (am+b)/(cm+d) [ratio of sums given]: aₘ/aₙ = [a(2m-1)+b] / [c(2m-1)+d] [replace n by 2n-1] Sₘ = Sₙ (m≠n) → Sₘ₊ₙ = 0 → aₘ₊ₙ₊₁ = 0

06

Triangles

6.1 Basic Proportionality Theorem (Thales' Theorem)

If DE ∥ BC in △ABC, D on AB, E on AC: AD/DB = AE/EC Equivalent Forms: AD/AB = AE/AC (part/whole ratio) DB/AB = EC/AC AB/AD = AC/AE AB/DB = AC/EC DB/AD = EC/AE

6.2 Converse of BPT

If AD/DB = AE/EC, then DE ∥ BC

6.3 Angle Bisector Theorem

If AD bisects ∠A in △ABC: BD/DC = AB/AC → BD = (AB × BC)/(AB + AC) → DC = (AC × BC)/(AB + AC)

6.4 Criteria for Similarity

AAA / AA: two pairs of equal angles → similar SSS: all three sides proportional → similar SAS: two sides proportional and included angle equal → similar If △ABC ~ △DEF: AB/DE = BC/EF = CA/FD = k (scale factor) ∠A = ∠D, ∠B = ∠E, ∠C = ∠F

6.5 Properties of Similar Triangles

If △ABC ~ △DEF with ratio k: Ratio of perimeters = k = AB/DE Ratio of areas = k² = (AB/DE)² Ratio of altitudes = k Ratio of medians = k Ratio of angle bisectors = k Ratio of circumradii = k Ratio of inradii = k Area(△ABC)/Area(△DEF) = (AB/DE)² = (BC/EF)² = (CA/FD)²

6.6 Pythagoras Theorem & Converse

In right △ABC (right angle at C): AB² = BC² + AC² (Hypotenuse)² = (Base)² + (Perpendicular)² → BC² = AB² - AC² → AC² = AB² - BC² → BC = √(AB² - AC²) Converse: If AB² = BC² + AC², then ∠C = 90° Pythagorean Triplets (a,b,c) where a²+b²=c²: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41) (6,8,10), (10,24,26) [multiples of above] General: (m²-n², 2mn, m²+n²) for m>n>0

6.7 Important Results from Pythagoras

In △ABC, if altitude CD ⊥ AB: CD² = AD × DB [geometric mean relation] BC² = BD × BA [altitude on hypotenuse] AC² = AD × AB CD = (AC × BC)/AB Median formula (in △ABC, m_a = median from A): m_a² = (2b² + 2c² - a²)/4 Stewart's Theorem: b²m + c²n - a(mn + d²) = 0 [cevian d divides BC into m and n]

6.8 Congruence Criteria

SSS: three sides equal SAS: two sides and included angle equal ASA: two angles and included side equal AAS: two angles and non-included side equal RHS: right angle, hypotenuse, one side equal

07

Coordinate Geometry

7.1 Distance Formula

d = √[(x₂-x₁)² + (y₂-y₁)²] → d² = (x₂-x₁)² + (y₂-y₁)² → (x₂-x₁)² = d² - (y₂-y₁)² Distance from origin O(0,0) to P(x,y): d = √(x² + y²) Distance between P(x₁,y₁) and Q(x₂,y₂): PQ = √[(x₁-x₂)² + (y₁-y₂)²]

7.2 Section Formula

Point P dividing A(x₁,y₁) and B(x₂,y₂) in ratio m:n INTERNAL division: P = [(mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)] EXTERNAL division: P = [(mx₂ - nx₁)/(m-n), (my₂ - ny₁)/(m-n)] Finding ratio m:n given P(x,y) on AB: m/n = (x - x₁)/(x₂ - x) [using x-coordinate] m/n = (y - y₁)/(y₂ - y) [using y-coordinate]

7.3 Midpoint Formula

Midpoint M of A(x₁,y₁) and B(x₂,y₂): M = [(x₁+x₂)/2, (y₁+y₂)/2] → x₁ + x₂ = 2xₘ → x₁ = 2xₘ - x₂ [find other endpoint if midpoint known]

7.4 Centroid of Triangle

G = [(x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3] → x₁+x₂+x₃ = 3xG → If G = (0,0): x₁+x₂+x₃ = 0, y₁+y₂+y₃ = 0

7.5 Area of Triangle

7.6 Slope of a Line

m = (y₂-y₁)/(x₂-x₁) = tan θ → θ = arctan(m) (angle with positive x-axis) Parallel lines: m₁ = m₂ Perpendicular: m₁ × m₂ = -1 → m₂ = -1/m₁ Slope-intercept form: y = mx + c Point-slope form: y - y₁ = m(x - x₁) Two-point form: (y-y₁)/(y₂-y₁) = (x-x₁)/(x₂-x₁) Intercept form: x/a + y/b = 1

7.7 Area of Quadrilateral & Polygon

7.8 Key Distance Facts

Collinear: AB + BC = AC (if B is between A and C) Isosceles: two distances equal Equilateral: all three distances equal, each = same Right angle: (longest)² = (other)² + (other)² Rectangle: diagonals equal and bisect each other Rhombus: all sides equal, diagonals bisect perpendicularly Square: all sides equal + diagonals equal Parallelogram: diagonals bisect each other (midpoints same)

08–09

Trigonometry & Applications

8.1 Trigonometric Ratios (Right Triangle)

sin θ = Opposite/Hypotenuse = P/H cos θ = Adjacent/Hypotenuse = B/H tan θ = Opposite/Adjacent = P/B cosec θ = 1/sin θ = H/P sec θ = 1/cos θ = H/B cot θ = 1/tan θ = B/P = cos θ/sin θ tan θ = sin θ/cos θ cot θ = cos θ/sin θ

8.2 Reciprocal Relations

sin θ × cosec θ = 1 → cosec θ = 1/sin θ cos θ × sec θ = 1 → sec θ = 1/cos θ tan θ × cot θ = 1 → cot θ = 1/tan θ

8.3 Pythagorean Identities

sin²θ + cos²θ = 1 ← PRIMARY → sin²θ = 1 - cos²θ → cos²θ = 1 - sin²θ → sinθ = √(1-cos²θ) → cosθ = √(1-sin²θ) 1 + tan²θ = sec²θ ← SECONDARY → sec²θ - tan²θ = 1 → sec²θ - 1 = tan²θ → tan²θ = sec²θ - 1 → (secθ + tanθ)(secθ - tanθ) = 1 → secθ + tanθ = 1/(secθ - tanθ) 1 + cot²θ = cosec²θ ← TERTIARY → cosec²θ - cot²θ = 1 → cosec²θ - 1 = cot²θ → (cosecθ + cotθ)(cosecθ - cotθ) = 1 → cosecθ + cotθ = 1/(cosecθ - cotθ)

8.4 Standard Angle Values Table

Angle θ: 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 ∞ 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: √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, 1/√2, √3/2, 1

8.5 Complementary Angle Relations

sin(90°-θ) = cosθ → cosθ = sin(90°-θ) cos(90°-θ) = sinθ → sinθ = cos(90°-θ) tan(90°-θ) = cotθ → cotθ = tan(90°-θ) cot(90°-θ) = tanθ → tanθ = cot(90°-θ) sec(90°-θ) = cosecθ → cosecθ = sec(90°-θ) cosec(90°-θ) = secθ → secθ = cosec(90°-θ)

8.6 Useful Identity Proofs (Templates)

(sinθ + cosθ)² = 1 + 2sinθcosθ (sinθ - cosθ)² = 1 - 2sinθcosθ (sinθ + cosθ)² + (sinθ - cosθ)² = 2 sin⁴θ + cos⁴θ = 1 - 2sin²θcos²θ = 1 - 2sin²θ(1-sin²θ) sin⁶θ + cos⁶θ = 1 - 3sin²θcos²θ tanθ + cotθ = 1/(sinθcosθ) = secθcosecθ tanθ - cotθ = (sin²θ-cos²θ)/(sinθcosθ) = -cos2θ/(sinθcosθ) secθ + tanθ = (1+sinθ)/cosθ secθ - tanθ = (1-sinθ)/cosθ = 1/(secθ+tanθ) cosecθ + cotθ = (1+cosθ)/sinθ cosecθ - cotθ = (1-cosθ)/sinθ = 1/(cosecθ+cotθ)

8.7 Heights and Distances (Chapter 9)

Angle of Elevation: angle from horizontal UP to object Angle of Depression: angle from horizontal DOWN to object tan(angle of elevation) = Height / Horizontal Distance tan(angle of depression) = Height / Horizontal Distance Height h = d × tan θ Distance d = h / tan θ = h × cot θ If angle of elevation from A is α, from B (further) is β: (α > β since A is closer) Let AB = x, object height = h, base of object foot = C h/BC = tan α → BC = h/tanα = h cotα h/AC = tan β → AC = h cotβ AB = AC - BC = h(cotβ - cotα) → h = AB/(cotβ - cotα) = AB tanα tanβ/(tanα - tanβ) Two buildings problem: Height of taller - shorter = d × (tanα - tanβ) [careful about geometry]

10

Circles

10.1 Tangent Properties

Tangent ⊥ radius at point of contact Length of tangent from external point P to circle (center O, radius r): PQ = PT = √(PO² - r²) → PO² = PQ² + r² (Pythagoras) → r² = PO² - PQ² Tangents from same external point are equal: PA = PB (A,B are points of tangency)

10.2 Angle Properties of Tangent

∠OAP = ∠OBP = 90° (tangent-radius angle) In quadrilateral OAPB (P external, A,B tangent points): ∠APB + ∠AOB = 180° [opposite angles supplementary] ∠APO = ∠BPO [OP bisects angle between tangents] ∠AOT = ∠BOT [OP bisects angle between radii] OA = OB = r PA = PB OP = OP (common) → △OAP ≅ △OBP (RHS congruence)

10.3 Circle Theorems

Angle in semicircle = 90° Angles in same segment are equal Angle at center = 2 × angle at circumference (same arc) Opposite angles of cyclic quadrilateral sum to 180° Exterior angle of cyclic quadrilateral = interior opposite angle Equal chords subtend equal angles at center Perpendicular from center to chord bisects the chord

10.4 Chord-Tangent Angle

Angle between tangent and chord = angle in alternate segment (Tangent-chord angle = inscribed angle on opposite side)

11

Areas Related to Circles

11.1 Basic Circle Formulas

Area of circle = πr² Circumference = 2πr = πd Diameter d = 2r Area = π(d/2)² = πd²/4 r = √(Area/π) r = Circumference/(2π)

11.2 Sector Formulas

Area of sector = (θ/360°) × πr² [θ in degrees] = (1/2)r²θ [θ in radians] = (1/2) × l × r [l = arc length] Arc length = (θ/360°) × 2πr = (θ/180°) × πr = rθ [θ in radians] Perimeter of sector = 2r + l = 2r + (θ/360°)×2πr = 2r + (πrθ/180°) → r = 2 × Area / l [from A = ½lr] → θ = (l/r) radians = (l/r) × (180°/π) degrees

11.3 Segment Formulas

Area of minor segment = Area of sector - Area of triangle = (θ/360°)πr² - (1/2)r² sin θ = r²[(πθ/360°) - (sinθ/2)] = (r²/2)[θ_rad - sinθ] [θ in radians] Area of major segment = πr² - Area of minor segment Chord length = 2r sin(θ/2) Height of segment = r(1 - cosθ/2) = r - r cos(θ/2)

11.4 Combination Formulas

Area of ring (annulus) = π(R²-r²) = π(R+r)(R-r) where R=outer radius, r=inner radius Area of semi-circle = πr²/2 Perimeter of semi-circle = πr + 2r = r(π+2) Area of quadrant = πr²/4 Perimeter of quadrant = πr/2 + 2r = r(π/2 + 2) Shaded region problems: Area of figure = Area₁ ± Area₂ ± Area₃...

12

Surface Areas and Volumes

12.1 Cuboid

LSA (Lateral Surface Area) = 2h(l+b) TSA (Total Surface Area) = 2(lb + bh + hl) Volume = l × b × h Diagonal = √(l²+b²+h²) Face diagonal (on lb face) = √(l²+b²) → l = V/(bh) → LSA = TSA - 2lb [remove top and bottom]

12.2 Cube (side = a)

LSA = 4a² TSA = 6a² Volume = a³ Diagonal = a√3 Face diagonal = a√2 → a = (V)^(1/3) → a = √(TSA/6)

12.3 Cylinder (radius r, height h)

CSA (Curved SA) = 2πrh TSA = 2πr(r+h) = 2πr² + 2πrh Volume = πr²h → r = CSA/(2πh) → h = CSA/(2πr) → h = V/(πr²) → r = √(V/πh) → r+h = TSA/(2πr) [if r known]

12.4 Cone (radius r, height h, slant l)

Slant height l = √(r²+h²) CSA = πrl = πr√(r²+h²) TSA = πrl + πr² = πr(l+r) Volume = (1/3)πr²h → h = √(l²-r²) → r = √(l²-h²) → l = √(r²+h²) → r = CSA/(πl) → h = 3V/(πr²) → r = √(3V/(πh))

12.5 Sphere (radius r)

SA (Surface Area) = 4πr² Volume = (4/3)πr³ → r = √(SA/4π) → r = (3V/4π)^(1/3) → V = (SA)^(3/2) / (6√π) [from eliminating r] → SA³ = 36π V² [classic relation]

12.6 Hemisphere (radius r)

CSA = 2πr² TSA = 3πr² [CSA + base circle πr²] Volume = (2/3)πr³ → r = √(CSA/2π) → r = √(TSA/3π) → r = (3V/2π)^(1/3)

12.7 Frustum of Cone (R=bigger radius, r=smaller, h=height, l=slant)

Slant height l = √[h² + (R-r)²] CSA = π(R+r)l TSA = π(R+r)l + πR² + πr² = π[(R+r)l + R² + r²] Volume = (πh/3)(R² + r² + Rr) → h = 3V / [π(R²+r²+Rr)] → l = √[h²+(R-r)²] When r=0 → cone: V = πR²h/3 ✓ When r=R → cylinder: V = πR²h ✓

12.8 Combination of Solids

Total Volume = V₁ + V₂ + ... Total SA = SA of outer surface only (remove hidden parts) Hemisphere on cylinder: TSA = CSA of cylinder + CSA of hemisphere + base circle = 2πrh + 2πr² + πr² = πr(2h + 3r) Volume = πr²h + (2/3)πr³ = πr²(h + 2r/3) Cone on cylinder: TSA = base of cylinder + CSA cylinder + CSA cone = πr² + 2πrh + πrl Volume = πr²H + (1/3)πr²h_cone

12.9 Volume Conversion Formulas

If solid A melted and recast into solid B: Volume of A = Volume of B (if no wastage) n × Volume of B = Volume of A (n pieces of B from A) If melted cube of side a → n spheres of radius r: a³ = n × (4/3)πr³ → n = 3a³/(4πr³) → r = a × (3/4πn)^(1/3)

13

Statistics

13.1 Mean (Ungrouped)

x̄ = (x₁+x₂+...+xₙ)/n = Σxᵢ/n = Σfᵢxᵢ/Σfᵢ [frequency dist] → Σxᵢ = n × x̄ → n = Σxᵢ/x̄

13.2 Mean (Grouped Data — Direct Method)

x̄ = Σfᵢxᵢ / Σfᵢ = Σfᵢxᵢ / N where xᵢ = class mark = (lower limit + upper limit)/2 fᵢ = frequency of ith class N = Σfᵢ = total frequency

13.3 Assumed Mean (Shortcut) Method

x̄ = A + (Σfᵢdᵢ/N) where A = assumed mean (usually central class mark) dᵢ = xᵢ - A (deviation from assumed mean) N = Σfᵢ

13.4 Step Deviation Method

x̄ = A + [(Σfᵢuᵢ/N) × h] where uᵢ = (xᵢ - A)/h h = class width (size) A = assumed mean → dᵢ = uᵢ × h → xᵢ = A + uᵢh

13.5 Median (Grouped Data)

Median = L + [(N/2 - cf)/f] × h where L = lower limit of median class N = Σfᵢ (total frequency) cf = cumulative frequency of class before median class f = frequency of median class h = class width Steps: 1. Find N/2 2. Find class with cf ≥ N/2 → median class 3. Apply formula Twisted: → L = Median - [(N/2 - cf)/f] × h → cf = N/2 - f(Median - L)/h

13.6 Mode (Grouped Data)

Mode = L + [f₁-f₀ / (2f₁-f₀-f₂)] × h where L = lower limit of modal class f₁ = frequency of modal class (highest frequency) f₀ = frequency of class before modal class f₂ = frequency of class after modal class h = class width Twisted: → L = Mode - [(f₁-f₀)/(2f₁-f₀-f₂)] × h → If f₀ = f₂: Mode = L + h/2 [symmetric case]

13.7 Empirical Relationship

Mode = 3 × Median - 2 × Mean → Mean = (3 × Median - Mode)/2 → Median = (Mode + 2 × Mean)/3 → Mean - Mode = 3(Mean - Median) → Mean - Median = (1/3)(Mean - Mode) [This holds approximately for moderately skewed distributions]

13.8 Ogive and Cumulative Frequency

Less-than ogive: plot (upper limit, cf) More-than ogive: plot (lower limit, N - cf) Median = x-coordinate where two ogives intersect

14

Probability

14.1 Classical (Theoretical) Probability

P(E) = n(E)/n(S) = Number of favorable outcomes / Total outcomes 0 ≤ P(E) ≤ 1 P(certain event) = 1 P(impossible event) = 0

14.2 Complementary Probability

P(Ē) = 1 - P(E) (E-bar = complement of E) P(E) + P(Ē) = 1 → P(E) = 1 - P(Ē) → P(Ē) = 1 - P(E) P(not E) = 1 - P(E) P(at least one) = 1 - P(none)

14.3 Standard Sample Spaces

Coin toss: S = {H, T}, n(S) = 2 Two coins: S = {HH,HT,TH,TT}, n(S) = 4 Three coins: n(S) = 8 n coins: n(S) = 2ⁿ One die: n(S) = 6 Two dice: n(S) = 36 Cards (full deck): n(S) = 52 - Hearts: 13, Diamonds: 13, Clubs: 13, Spades: 13 - Red cards: 26, Black cards: 26 - Face cards (J,Q,K): 12 (3 per suit) - Aces: 4, Kings: 4, Queens: 4, Jacks: 4 - Non-face number cards: 36

14.4 Two Dice Probability Reference

Sum = 2: {(1,1)} → P = 1/36 Sum = 3: {(1,2),(2,1)} → P = 2/36 = 1/18 Sum = 4: 3 ways → P = 3/36 = 1/12 Sum = 5: 4 ways → P = 4/36 = 1/9 Sum = 6: 5 ways → P = 5/36 Sum = 7: 6 ways (MAXIMUM) → P = 6/36 = 1/6 Sum = 8: 5 ways → P = 5/36 Sum = 9: 4 ways → P = 4/36 Sum = 10: 3 ways → P = 3/36 = 1/12 Sum = 11: 2 ways → P = 2/36 = 1/18 Sum = 12: {(6,6)} → P = 1/36 Doublets: (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) → P = 6/36 = 1/6

14.5 Addition Rule

P(A∪B) = P(A) + P(B) - P(A∩B) If A and B are mutually exclusive: P(A∩B) = 0 P(A∪B) = P(A) + P(B) P(A∩B) = P(A) + P(B) - P(A∪B)

14.6 Odds

Odds in favor of E = P(E) : P(Ē) = n(E) : n(Ē) Odds against E = P(Ē) : P(E) = n(Ē) : n(E) If odds in favor = m:n: P(E) = m/(m+n) P(Ē) = n/(m+n)

B1

Relations and Functions

1.1 Types of Relations

Empty Relation: R = ∅ Universal Relation: R = A × A Reflexive: (a,a) ∈ R for all a ∈ A Symmetric: (a,b) ∈ R → (b,a) ∈ R Transitive: (a,b)∈R and (b,c)∈R → (a,c)∈R Equivalence: Reflexive + Symmetric + Transitive

1.2 Types of Functions

One-one (injective): f(a)=f(b) ⟹ a=b Onto (surjective): Range = Codomain Bijective: One-one + Onto Number of functions from A to B: |B|^|A| = n(B)^n(A) Number of one-one functions: ⁿ⁽ᴮ⁾Pₙ₍ₐ₎ = P(n(B), n(A)) [if n(B)≥n(A)] Number of onto functions (|A|=m, |B|=n): Σ(-1)^k C(n,k)(n-k)^m [inclusion-exclusion]

1.3 Composition of Functions

(g∘f)(x) = g(f(x)) (f∘g)(x) = f(g(x)) (g∘f) ≠ (f∘g) in general [not commutative] (h∘g)∘f = h∘(g∘f) [associative] If f and g both one-one → g∘f one-one If f and g both onto → g∘f onto If g∘f one-one → f is one-one If g∘f onto → g is onto

1.4 Inverse Function

If f: A→B bijective, then f⁻¹: B→A exists f⁻¹(f(x)) = x f(f⁻¹(y)) = y f∘f⁻¹ = f⁻¹∘f = I (identity) To find f⁻¹: let y = f(x), solve for x in terms of y

1.5 Binary Operations

Closure: a*b ∈ A for all a,b ∈ A Commutative: a*b = b*a Associative: (a*b)*c = a*(b*c) Identity element e: a*e = e*a = a Inverse of a: a*a⁻¹ = e (if identity exists) Number of binary operations on set with n elements = n^(n²) Number of commutative binary operations = n^(n(n+1)/2)

B2

Inverse Trigonometric Functions

2.1 Domain and Range

Function Domain Range (Principal) sin⁻¹x [-1, 1] [-π/2, π/2] cos⁻¹x [-1, 1] [0, π] tan⁻¹x (-∞, ∞) = ℝ (-π/2, π/2) cosec⁻¹x (-∞,-1]∪[1,∞) [-π/2,π/2]\{0} sec⁻¹x (-∞,-1]∪[1,∞) [0,π]\{π/2} cot⁻¹x (-∞, ∞) = ℝ (0, π)

2.2 Basic Identities

sin⁻¹x + cos⁻¹x = π/2 (|x| ≤ 1) tan⁻¹x + cot⁻¹x = π/2 (x ∈ ℝ) sec⁻¹x + cosec⁻¹x = π/2 (|x| ≥ 1) → cos⁻¹x = π/2 - sin⁻¹x → sin⁻¹x = π/2 - cos⁻¹x → cot⁻¹x = π/2 - tan⁻¹x → tan⁻¹x = π/2 - cot⁻¹x

2.3 Negative Argument

sin⁻¹(-x) = -sin⁻¹x (odd function) tan⁻¹(-x) = -tan⁻¹x (odd function) cosec⁻¹(-x) = -cosec⁻¹x (odd function) cos⁻¹(-x) = π - cos⁻¹x (not odd!) sec⁻¹(-x) = π - sec⁻¹x cot⁻¹(-x) = π - cot⁻¹x

2.4 Reciprocal Relations

sin⁻¹(1/x) = cosec⁻¹x (x ≥ 1 or x ≤ -1) cos⁻¹(1/x) = sec⁻¹x (x ≥ 1 or x ≤ -1) tan⁻¹(1/x) = cot⁻¹x (x > 0) tan⁻¹(1/x) = cot⁻¹x - π (x < 0) → cosec⁻¹x = sin⁻¹(1/x) → sec⁻¹x = cos⁻¹(1/x) → cot⁻¹x = tan⁻¹(1/x) for x>0, cot⁻¹x = π + tan⁻¹(1/x) for x<0

2.5 Double Angle Formulas

2sin⁻¹x = sin⁻¹(2x√(1-x²)) (|x| ≤ 1/√2) 2cos⁻¹x = cos⁻¹(2x²-1) (0 ≤ x ≤ 1) 2tan⁻¹x = tan⁻¹(2x/(1-x²)) (|x| < 1) 2tan⁻¹x = sin⁻¹(2x/(1+x²)) (|x| ≤ 1) 2tan⁻¹x = cos⁻¹((1-x²)/(1+x²)) (x ≥ 0) 3tan⁻¹x = tan⁻¹(3x-x³)/(1-3x²) (|x|<1/√3)

2.6 Sum and Difference Formulas

sin⁻¹x ± sin⁻¹y = sin⁻¹[x√(1-y²) ± y√(1-x²)] cos⁻¹x ± cos⁻¹y = cos⁻¹[xy ∓ √(1-x²)√(1-y²)] tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1-xy)] (xy < 1) = π + tan⁻¹[(x+y)/(1-xy)] (xy > 1, x>0, y>0) = -π + tan⁻¹[(x+y)/(1-xy)] (xy > 1, x<0, y<0) tan⁻¹x - tan⁻¹y = tan⁻¹[(x-y)/(1+xy)] (xy > -1) = π + tan⁻¹[(x-y)/(1+xy)] (x>0, xy<-1) = -π + tan⁻¹[(x-y)/(1+xy)] (x<0, xy<-1) Derived: tan⁻¹(1/2) + tan⁻¹(1/3) = π/4 tan⁻¹1 + tan⁻¹2 + tan⁻¹3 = π

2.7 Conversion Formulas

sin⁻¹x = cos⁻¹(√(1-x²)) (x ≥ 0) = tan⁻¹(x/√(1-x²)) (|x|<1) cos⁻¹x = sin⁻¹(√(1-x²)) (x ≥ 0) = tan⁻¹(√(1-x²)/x) tan⁻¹x = sin⁻¹(x/√(1+x²)) = cos⁻¹(1/√(1+x²)) For θ = sin⁻¹x: sinθ=x, cosθ=√(1-x²), tanθ=x/√(1-x²) For θ = cos⁻¹x: cosθ=x, sinθ=√(1-x²), tanθ=√(1-x²)/x For θ = tan⁻¹x: tanθ=x, sinθ=x/√(1+x²), cosθ=1/√(1+x²)

B3

Matrices

3.1 Matrix Notation

A = [aᵢⱼ]ₘₓₙ (m rows, n columns) Order = m × n, Total elements = mn aᵢⱼ = element in ith row, jth column

3.2 Types of Matrices

Row matrix: 1 × n Column matrix: m × 1 Square matrix: m = n Diagonal matrix: aᵢⱼ = 0 if i ≠ j Scalar matrix: Diagonal with all diagonal elements equal Identity matrix: I, diagonal with all 1s Zero matrix: O, all elements 0 Upper triangular: aᵢⱼ = 0 if i > j Lower triangular: aᵢⱼ = 0 if i < j

3.3 Transpose

If A = [aᵢⱼ]ₘₓₙ, then Aᵀ = [aⱼᵢ]ₙₓₘ Properties: (Aᵀ)ᵀ = A (A+B)ᵀ = Aᵀ + Bᵀ (kA)ᵀ = kAᵀ (AB)ᵀ = BᵀAᵀ ← REVERSED ORDER! Symmetric matrix: A = Aᵀ → aᵢⱼ = aⱼᵢ Skew-symmetric: A = -Aᵀ → aᵢⱼ = -aⱼᵢ, diagonal = 0 Any square matrix A: A = ½(A+Aᵀ) + ½(A-Aᵀ) [symmetric + skew-symmetric]

3.4 Matrix Operations

Addition: (A+B)ᵢⱼ = aᵢⱼ + bᵢⱼ [same order] Scalar: (kA)ᵢⱼ = kaᵢⱼ Multiplication: (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ [A: m×n, B: n×p → AB: m×p] AB ≠ BA in general [not commutative] A(BC) = (AB)C [associative] A(B+C) = AB+AC [distributive] AO = OA = O AI = IA = A

3.5 Elementary Row/Column Operations

Rᵢ ↔ Rⱼ [swap rows i and j] Rᵢ → kRᵢ [multiply row i by k ≠ 0] Rᵢ → Rᵢ + kRⱼ [add k times row j to row i]

3.6 Invertible Matrix

AA⁻¹ = A⁻¹A = I A⁻¹ = adjA / |A| (if |A| ≠ 0) (A⁻¹)⁻¹ = A (AB)⁻¹ = B⁻¹A⁻¹ ← REVERSED! (Aᵀ)⁻¹ = (A⁻¹)ᵀ (kA)⁻¹ = (1/k)A⁻¹ |A⁻¹| = 1/|A|

3.7 2×2 Matrix — Quick Formulas

A = [a b; c d] |A| = ad - bc A⁻¹ = (1/(ad-bc)) × [d -b; -c a] adj A = [d -b; -c a] (transpose of cofactor matrix) For 2×2: adj A is obtained by: swapping diagonal elements negating off-diagonal elements

B4

Determinants

4.1 Determinant of 2×2

|A| = |a b| = ad - bc |c d|

4.2 Determinant of 3×3 (Expansion along R₁)

|a₁ b₁ c₁| |a₂ b₂ c₂| = a₁(b₂c₃-b₃c₂) - b₁(a₂c₃-a₃c₂) + c₁(a₂b₃-a₃b₂) |a₃ b₃ c₃| = a₁M₁₁ - b₁M₁₂ + c₁M₁₃ (Mᵢⱼ = minor of element aᵢⱼ) Cofactor Cᵢⱼ = (-1)^(i+j) Mᵢⱼ Sign pattern for cofactors: + - + - + - + - +

4.3 Properties of Determinants

|Aᵀ| = |A| |kA|ₙₓₙ = kⁿ|A| |AB| = |A||B| |A⁻¹| = 1/|A| |A²| = |A|² |Aⁿ| = |A|ⁿ |adj A| = |A|^(n-1) [for n×n matrix] If any row/column = 0 → |A| = 0 If two rows/columns identical → |A| = 0 If one row is k×another row → |A| = 0 Swapping two rows/columns → |A| changes sign

4.4 Adjoint and Inverse

adj A = transpose of cofactor matrix A(adj A) = (adj A)A = |A| × I A⁻¹ = adj A / |A| (|A| ≠ 0) |adj A| = |A|^(n-1) adj(AB) = (adj B)(adj A) adj(adj A) = |A|^(n-2) × A [n×n] adj(Aᵀ) = (adj A)ᵀ adj(kA) = k^(n-1) adj A

4.5 Cramer's Rule

For AX = B: x = D₁/D, y = D₂/D, z = D₃/D D = |A| (coefficient determinant) D₁ = replace column 1 of A by B D₂ = replace column 2 of A by B D₃ = replace column 3 of A by B If D ≠ 0 → unique solution If D = 0 and D₁ = D₂ = D₃ = 0 → infinitely many / no solution If D = 0 and any Dᵢ ≠ 0 → no solution (inconsistent)

4.6 Area Using Determinants

Area of △ with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃): Area = ½ |x₁ y₁ 1| |x₂ y₂ 1| |x₃ y₃ 1| Collinear if Area = 0: |x₁ y₁ 1| |x₂ y₂ 1| = 0 |x₃ y₃ 1| Equation of line through (x₁,y₁) and (x₂,y₂): |x y 1| |x₁ y₁ 1| = 0 |x₂ y₂ 1| → (y-y₁)(x₂-x₁) = (x-x₁)(y₂-y₁)

B5

Continuity and Differentiability

5.1 Continuity Condition

f is continuous at x=a if: 1. f(a) is defined 2. lim[x→a] f(x) exists 3. lim[x→a] f(x) = f(a) LHL = lim[x→a⁻] f(x) = lim[h→0] f(a-h) RHL = lim[x→a⁺] f(x) = lim[h→0] f(a+h) Continuous ↔ LHL = RHL = f(a)

5.2 Standard Derivatives

d/dx (constant) = 0 d/dx (x) = 1 d/dx (xⁿ) = nxⁿ⁻¹ d/dx (√x) = 1/(2√x) d/dx (1/x) = -1/x² d/dx (1/xⁿ) = -n/x^(n+1) Exponential & Log: d/dx (eˣ) = eˣ d/dx (aˣ) = aˣ ln a d/dx (ln x) = 1/x d/dx (log_a x) = 1/(x ln a) Trigonometric: d/dx (sin x) = cos x d/dx (cos x) = -sin x d/dx (tan x) = sec²x d/dx (cot x) = -cosec²x d/dx (sec x) = sec x tan x d/dx (cosec x) = -cosec x cot x Inverse Trig: d/dx (sin⁻¹x) = 1/√(1-x²) d/dx (cos⁻¹x) = -1/√(1-x²) d/dx (tan⁻¹x) = 1/(1+x²) d/dx (cot⁻¹x) = -1/(1+x²) d/dx (sec⁻¹x) = 1/(|x|√(x²-1)) d/dx (cosec⁻¹x) = -1/(|x|√(x²-1))

5.3 Rules of Differentiation

Sum/Difference: (u ± v)' = u' ± v' Scalar: (cu)' = cu' Product: (uv)' = u'v + uv' ← Product Rule (Leibniz) Quotient: (u/v)' = (u'v - uv')/v² ← Quotient Rule (v≠0) Chain: d/dx[f(g(x))] = f'(g(x))·g'(x) ← Chain Rule Extended Product Rule: (uvw)' = u'vw + uv'w + uvw' Extended Chain Rule: d/dx[f(g(h(x)))] = f'(g(h(x)))·g'(h(x))·h'(x)

5.4 Implicit Differentiation

For F(x,y) = 0: dy/dx = -(∂F/∂x)/(∂F/∂y) [partial derivatives method] Or differentiate both sides w.r.t. x, treating y as function of x Example: x² + y² = r² → 2x + 2y(dy/dx) = 0 → dy/dx = -x/y

5.5 Parametric Differentiation

x = f(t), y = g(t) dy/dx = (dy/dt)/(dx/dt) = g'(t)/f'(t) d²y/dx² = (d/dt[dy/dx]) / (dx/dt)

5.6 Logarithmic Differentiation

For y = [f(x)]^g(x): ln y = g(x) ln[f(x)] (1/y)(dy/dx) = g'(x)ln[f(x)] + g(x)·f'(x)/f(x) dy/dx = y × [g'(x)ln f(x) + g(x)f'(x)/f(x)] For y = (f₁·f₂·...)/(g₁·g₂·...): ln y = ln f₁ + ln f₂ + ... - ln g₁ - ln g₂ - ... (1/y)y' = f₁'/f₁ + f₂'/f₂ + ... - g₁'/g₁ - ...

5.7 Higher Order Derivatives

y = f(x) y' = f'(x) = dy/dx [first derivative] y'' = f''(x) = d²y/dx² [second derivative] y''' = f'''(x) = d³y/dx³ [third derivative] yⁿ = f⁽ⁿ⁾(x) = dⁿy/dxⁿ [nth derivative] Leibniz Formula (nth derivative of product): (uv)ⁿ = Σₖ₌₀ⁿ C(n,k) u⁽ᵏ⁾ v⁽ⁿ⁻ᵏ⁾

5.8 Rolle's Theorem

If f: [a,b]→ℝ is: 1. Continuous on [a,b] 2. Differentiable on (a,b) 3. f(a) = f(b) Then ∃ c ∈ (a,b) such that f'(c) = 0

5.9 Mean Value Theorem (Lagrange's MVT)

If f: [a,b]→ℝ is: 1. Continuous on [a,b] 2. Differentiable on (a,b) Then ∃ c ∈ (a,b) such that: f'(c) = [f(b)-f(a)] / (b-a) → f(b) - f(a) = f'(c)(b-a) → c = value(s) satisfying the above [Geometric meaning: slope of chord = slope of tangent at c]

B6

Application of Derivatives

6.1 Rate of Change

Rate of change of y w.r.t. x = dy/dx Rate of change of y w.r.t. t = dy/dt If y = f(x) and x = g(t): dy/dt = (dy/dx)(dx/dt) Rate of change of area of circle w.r.t. radius: dA/dr = 2πr Rate of change of volume of sphere w.r.t. radius: dV/dr = 4πr² Rate of change of volume of cube w.r.t. side: dV/da = 3a²

6.2 Tangent and Normal

Equation of tangent at (x₁,y₁): y - y₁ = m(x - x₁), where m = (dy/dx) at (x₁,y₁) Equation of normal at (x₁,y₁): y - y₁ = -1/m (x - x₁), where m = (dy/dx) at (x₁,y₁) Slope of tangent: m_t = f'(x₁) Slope of normal: m_n = -1/f'(x₁) m_t × m_n = -1 (tangent ⊥ normal) Length of tangent = y√(1+m²)/m Length of normal = y√(1+m²) Length of subtangent = y/m = y/(dy/dx) Length of subnormal = y × m = y(dy/dx) Tangent parallel to x-axis: dy/dx = 0 Tangent parallel to y-axis: dy/dx → ∞ (dx/dy = 0) Tangent passes through origin: y/x = dy/dx (i.e., y=mx, m=dy/dx)

6.3 Increasing and Decreasing Functions

f is strictly increasing on (a,b) if f'(x) > 0 ∀ x∈(a,b) f is strictly decreasing on (a,b) if f'(x) < 0 ∀ x∈(a,b) f is constant on (a,b) if f'(x) = 0 ∀ x∈(a,b) At a critical point: f'(c) = 0 Monotonically increasing: f(x₁) < f(x₂) whenever x₁ < x₂ Monotonically decreasing: f(x₁) > f(x₂) whenever x₁ < x₂

6.4 Maxima and Minima

First Derivative Test: f'(c) = 0 and f' changes + to - at c → local maximum f' changes - to + at c → local minimum f' doesn't change sign → neither (inflection point) Second Derivative Test: f'(c) = 0 and f''(c) < 0 → local maximum at x=c; f(c) = local max value f''(c) > 0 → local minimum at x=c; f(c) = local min value f''(c) = 0 → test inconclusive (use first derivative test) Global/Absolute maximum/minimum on [a,b]: Compare f(a), f(b), and all f(c) where f'(c)=0

6.5 Optimization Formulas

Rectangle of maximum area for given perimeter P: Square: side = P/4, Area = P²/16 Rectangle of minimum perimeter for given area A: Square: side = √A, Perimeter = 4√A Cylinder of maximum volume: r = h Cone of maximum volume inscribed in sphere of radius R: h = 4R/3, r = 2R√2/3 Rectangle inscribed in circle of radius R: Maximum area square: side = R√2, Area = 2R² Cylinder inscribed in sphere of radius R: Maximum volume: h = 2R/√3, r = R√(2/3) V_max = 4πR³/(3√3) Wire of length L bent to maximize area: If circle: r = L/2π, A = L²/4π If square: side = L/4, A = L²/16 Circle has more area than square for same perimeter Box (open top) from square sheet of side a, corners cut x: V = x(a-2x)² dV/dx = 0 → x = a/6 for max volume V_max = 2a³/27

6.6 Approximations (Linear Approximation)

Δy ≈ dy = f'(x)·Δx [for small Δx] f(x + Δx) ≈ f(x) + f'(x)·Δx Relative error: dy/y Percentage error: (dy/y)×100 If y = xⁿ: Δy/y ≈ n(Δx/x) [percentage error multiplied by n]

B7

Integrals

7.1 Standard Integration Formulas

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1) ∫ 1/x dx = ln|x| + C ∫ eˣ dx = eˣ + C ∫ aˣ dx = aˣ/ln a + C ∫ 1 dx = x + C Trigonometric: ∫ sin x dx = -cos x + C ∫ cos x dx = sin x + C ∫ tan x dx = ln|sec x| + C = -ln|cos x| + C ∫ cot x dx = ln|sin x| + C ∫ sec x dx = ln|sec x + tan x| + C ∫ cosec x dx = ln|cosec x - cot x| + C ∫ sec²x dx = tan x + C ∫ cosec²x dx = -cot x + C ∫ sec x tan x dx = sec x + C ∫ cosec x cot x dx = -cosec x + C Special: ∫ 1/√(1-x²) dx = sin⁻¹x + C = -cos⁻¹x + C ∫ -1/√(1-x²) dx = cos⁻¹x + C ∫ 1/(1+x²) dx = tan⁻¹x + C = -cot⁻¹x + C ∫ 1/(x√(x²-1)) dx = sec⁻¹x + C = -cosec⁻¹x + C

7.2 Standard Forms (Key Templates)

∫ 1/(x²+a²) dx = (1/a)tan⁻¹(x/a) + C ∫ 1/(x²-a²) dx = (1/2a)ln|(x-a)/(x+a)| + C (|x|>a) ∫ 1/(a²-x²) dx = (1/2a)ln|(a+x)/(a-x)| + C (|x|<a) ∫ 1/√(x²+a²) dx = ln|x + √(x²+a²)| + C ∫ 1/√(x²-a²) dx = ln|x + √(x²-a²)| + C ∫ 1/√(a²-x²) dx = sin⁻¹(x/a) + C ∫ √(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 ∫ √(x²-a²) dx = (x/2)√(x²-a²) - (a²/2)ln|x+√(x²-a²)| + C

7.3 (a) Substitution Method (u-substitution)

∫ f(g(x))·g'(x) dx = ∫ f(u) du [let u = g(x)] Common substitutions: √(a²-x²) → x = a sinθ √(a²+x²) → x = a tanθ √(x²-a²) → x = a secθ (a-x)/(a+x) or similar → x = a cos2θ

7.3 (b) Integration by Parts (IBP)

∫ u dv = uv - ∫ v du or ∫ u·v dx = u(∫v dx) - ∫[u'(∫v dx)] dx ILATE priority rule (choose u in this order): I — Inverse trig (sin⁻¹x, tan⁻¹x...) L — Logarithmic (ln x, log x) A — Algebraic (xⁿ, polynomials) T — Trigonometric (sin x, cos x...) E — Exponential (eˣ, aˣ) Standard IBP results: ∫ xeˣ dx = eˣ(x-1) + C ∫ x²eˣ dx = eˣ(x²-2x+2) + C ∫ xⁿeˣ dx = eˣ[xⁿ - nxⁿ⁻¹ + n(n-1)xⁿ⁻² - ...] + C ∫ x sin x dx = -x cos x + sin x + C ∫ x cos x dx = x sin x + cos x + C ∫ x² sin x dx = -x²cos x + 2x sin x + 2cos x + C ∫ ln x dx = x ln x - x + C ∫ x ln x dx = x²/2 ln x - x²/4 + C ∫ sin⁻¹x dx = x sin⁻¹x + √(1-x²) + C ∫ cos⁻¹x dx = x cos⁻¹x - √(1-x²) + C ∫ tan⁻¹x dx = x tan⁻¹x - (1/2)ln(1+x²) + C Special formula: ∫ eˣ[f(x) + f'(x)] dx = eˣ f(x) + C

7.3 (c) Partial Fractions

Proper fraction: degree(numerator) < degree(denominator) Case 1: Distinct linear factors P(x)/[(ax+b)(cx+d)] = A/(ax+b) + B/(cx+d) Case 2: Repeated linear factors P(x)/(ax+b)² = A/(ax+b) + B/(ax+b)² P(x)/(ax+b)³ = A/(ax+b) + B/(ax+b)² + C/(ax+b)³ Case 3: Irreducible quadratic P(x)/[(ax+b)(x²+bx+c)] = A/(ax+b) + (Bx+C)/(x²+bx+c) Case 4: Improper → divide first If degree(P) ≥ degree(Q): P/Q = quotient + remainder/Q

7.3 (d) Integration of Rational Trig Functions

∫ 1/(a + b sinx) dx: let t = tan(x/2) sinx = 2t/(1+t²), cosx = (1-t²)/(1+t²), dx = 2dt/(1+t²) ∫ 1/(a sinx + b cosx) dx = (1/√(a²+b²)) ln|tan(x/2 + α)| + C where tan α = a/b R sin(x+α) form: a sinx + b cosx = R sin(x+α) R = √(a²+b²), tan α = b/a R cos(x-α) form: a cosx + b sinx = R cos(x-α) R = √(a²+b²), tan α = b/a

7.4 Definite Integral Properties

∫ₐᵇ f(x) dx = F(b) - F(a) [where F'(x)=f(x)] P1: ∫ₐᵇ f(x) dx = -∫ᵦₐ f(x) dx P2: ∫ₐᵃ f(x) dx = 0 P3: ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫ᶜᵇ f(x) dx P4: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(a+b-x) dx ← KING Property P5: ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a-x) dx ← Special King P6: ∫₀²ᵃ f(x) dx = 2∫₀ᵃ f(x) dx [if f(2a-x) = f(x)] = 0 [if f(2a-x) = -f(x)] P7: ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx [if f even: f(-x)=f(x)] = 0 [if f odd: f(-x)=-f(x)] P8: ∫₀ⁿᵀ f(x) dx = n∫₀ᵀ f(x) dx [if f periodic with period T]

7.5 Definite Integral as Limit of Sum

∫ₐᵇ f(x) dx = lim[n→∞] h Σₖ₌₀ⁿ⁻¹ f(a+kh) where h = (b-a)/n Or: ∫₀¹ f(x) dx = lim[n→∞] (1/n) Σₖ₌₁ⁿ f(k/n)

7.6 Important Definite Integrals

∫₀^(π/2) sin x dx = ∫₀^(π/2) cos x dx = 1 ∫₀^π sin x dx = 2, ∫₀^π cos x dx = 0 ∫₀^(π/2) sin²x dx = ∫₀^(π/2) cos²x dx = π/4 ∫₀^(π/2) sinⁿx dx: n=2: π/4 n=3: 2/3 n=4: 3π/16 ∫₀^(π/2) ln(sinx) dx = ∫₀^(π/2) ln(cosx) dx = -(π/2)ln2 ∫₀^π x f(sinx) dx = (π/2)∫₀^π f(sinx) dx [using King property]

7.7 Walli's Formula

∫₀^(π/2) sinⁿx dx = ∫₀^(π/2) cosⁿx dx = [(n-1)(n-3)...3·1]/[n(n-2)...4·2] × π/2 (n even) [(n-1)(n-3)...4·2]/[n(n-2)...3·1] (n odd)

B8

Application of Integrals

8.1 Area Under a Curve

Area between y=f(x) and x-axis, from x=a to x=b: A = ∫ₐᵇ |f(x)| dx If f(x) ≥ 0: A = ∫ₐᵇ f(x) dx If f(x) ≤ 0: A = -∫ₐᵇ f(x) dx = ∫ₐᵇ |f(x)| dx Area between x=g(y) and y-axis, from y=c to y=d: A = ∫ᶜᵈ |g(y)| dy

8.2 Area Between Two Curves

Area between y=f(x) and y=g(x), a to b [f(x) ≥ g(x)]: A = ∫ₐᵇ [f(x) - g(x)] dx Area between x=f(y) and x=g(y) [f(y) ≥ g(y)]: A = ∫ᶜᵈ [f(y) - g(y)] dy Finding intersection points: solve f(x) = g(x)

8.3 Standard Areas

Circle x²+y²=r²: Area = πr² (full) Area = πr²/2 (semicircle) Ellipse x²/a²+y²/b²=1: Area = πab Area of parabola y=x² and y=x: Intersect at x=0,1 A = ∫₀¹(x-x²)dx = [x²/2 - x³/3]₀¹ = 1/2-1/3 = 1/6 Area of ellipse arc with x-axis: ∫₀ᵃ b√(1-x²/a²)dx = πab/4 [first quadrant]

B9

Differential Equations

9.1 Order, Degree, and Type

Order = highest derivative present Degree = power of highest order derivative (when polynomial in derivatives) Degree undefined if sin(y'), eʸ' etc. Linear DE: highest power of y and its derivatives is 1 Non-linear: otherwise

9.2 Variable Separable Method

dy/dx = f(x)·g(y) → dy/g(y) = f(x)dx → ∫ dy/g(y) = ∫ f(x)dx + C Particular solution: use initial condition to find C

9.3 Homogeneous Differential Equation

dy/dx = f(x,y)/g(x,y) where f,g are homogeneous of same degree Substitution: y = vx → dy/dx = v + x(dv/dx) v + x(dv/dx) = F(v) x(dv/dx) = F(v) - v ∫ dv/[F(v)-v] = ∫ dx/x + C [After integrating, back-substitute v = y/x]

9.4 Linear Differential Equation (First Order)

dy/dx + P(x)y = Q(x) Integrating Factor (IF) = e^(∫P dx) Solution: y × IF = ∫[Q × IF] dx + C → ye^(∫Pdx) = ∫[Q·e^(∫Pdx)] dx + C For dx/dy + P(y)x = Q(y) [x as dependent]: IF = e^(∫P(y)dy) x × IF = ∫[Q(y)·IF] dy + C

9.5 Bernoulli's Equation

dy/dx + P(x)y = Q(x)yⁿ (n ≠ 0, 1) Substitution: z = y^(1-n) dz/dx = (1-n)y^(-n)(dy/dx) Becomes linear: dz/dx + (1-n)P(x)z = (1-n)Q(x)

9.6 General Solutions

∫ dy/y = ln|y| ∫ dy/y² = -1/y ∫ dy/(1+y²) = tan⁻¹y For xdy + ydx = d(xy) [exact differential] For (xdy-ydx)/x² = d(y/x) For (ydx-xdy)/y² = d(x/y) For (xdx+ydy) = (1/2)d(x²+y²)

9.7 Important Differential Equations and Solutions

dy/dx = ky: y = Ce^(kx) [growth/decay] d²y/dx² + n²y = 0: y = A cos(nx) + B sin(nx) d²y/dx² - n²y = 0: y = Ae^(nx) + Be^(-nx) d²y/dx² = f(x): integrate twice Population: dP/dt = kP → P = P₀e^(kt) Radioactive: dN/dt = -λN → N = N₀e^(-λt) Newton's cooling: dT/dt = -k(T-T₀) → T-T₀ = (T₁-T₀)e^(-kt)

B10

Vector Algebra

10.1 Vector Notation

Position vector of A(x,y,z): a⃗ = xî + yĵ + zk̂ Magnitude: |a⃗| = √(x²+y²+z²) Unit vector: â = a⃗/|a⃗| Zero vector: 0⃗ = 0î + 0ĵ + 0k̂ Direction cosines: l = x/|a⃗|, m = y/|a⃗|, n = z/|a⃗| l² + m² + n² = 1 Direction ratios: proportional to (x,y,z)

10.2 Vector Operations

Addition: a⃗ + b⃗ = (a₁+b₁)î + (a₂+b₂)ĵ + (a₃+b₃)k̂ Subtraction: a⃗ - b⃗ = (a₁-b₁)î + (a₂-b₂)ĵ + (a₃-b₃)k̂ Scalar multiple: ka⃗ = ka₁î + ka₂ĵ + ka₃k̂ |a⃗ + b⃗|² = |a⃗|² + 2a⃗·b⃗ + |b⃗|² |a⃗ - b⃗|² = |a⃗|² - 2a⃗·b⃗ + |b⃗|² |a⃗ + b⃗|² + |a⃗ - b⃗|² = 2(|a⃗|² + |b⃗|²) [Parallelogram law] |a⃗ + b⃗|² - |a⃗ - b⃗|² = 4(a⃗·b⃗)

10.3 Dot Product (Scalar Product)

a⃗·b⃗ = |a⃗||b⃗|cosθ = a₁b₁ + a₂b₂ + a₃b₃ → cosθ = a⃗·b⃗ / (|a⃗||b⃗|) → θ = cos⁻¹[a⃗·b⃗ / (|a⃗||b⃗|)] Perpendicular ↔ a⃗·b⃗ = 0 (a⃗ ≠ 0⃗, b⃗ ≠ 0⃗) Parallel ↔ a⃗×b⃗ = 0⃗ î·î = ĵ·ĵ = k̂·k̂ = 1 î·ĵ = ĵ·k̂ = k̂·î = 0 a⃗·a⃗ = |a⃗|² a⃗·b⃗ = b⃗·a⃗ [commutative] Projection of a⃗ on b⃗ = (a⃗·b⃗)/|b⃗| Component of a⃗ along b⃗ = (a⃗·b⃗)/|b⃗| [scalar] Vector projection = [(a⃗·b⃗)/|b⃗|²] b⃗

10.4 Cross Product (Vector Product)

a⃗×b⃗ = |a⃗||b⃗|sinθ n̂ |a⃗×b⃗| = |a⃗||b⃗|sinθ → sinθ = |a⃗×b⃗| / (|a⃗||b⃗|) a⃗×b⃗ = |î ĵ k̂ | |a₁ a₂ a₃| |b₁ b₂ b₃| = î(a₂b₃-a₃b₂) - ĵ(a₁b₃-a₃b₁) + k̂(a₁b₂-a₂b₁) î×î = ĵ×ĵ = k̂×k̂ = 0⃗ î×ĵ = k̂, ĵ×k̂ = î, k̂×î = ĵ [cyclic] ĵ×î = -k̂, k̂×ĵ = -î, î×k̂ = -ĵ [anti-cyclic] a⃗×b⃗ = -(b⃗×a⃗) [anti-commutative] Area of parallelogram = |a⃗×b⃗| Area of triangle = ½|a⃗×b⃗| Area of △ABC = ½|AB⃗×AC⃗|

10.5 Scalar Triple Product

[a⃗ b⃗ c⃗] = a⃗·(b⃗×c⃗) = scalar = |a₁ a₂ a₃| |b₁ b₂ b₃| |c₁ c₂ c₃| Volume of parallelepiped = |[a⃗ b⃗ c⃗]| Volume of tetrahedron = (1/6)|[a⃗ b⃗ c⃗]| Coplanar vectors ↔ [a⃗ b⃗ c⃗] = 0 [a⃗ b⃗ c⃗] = [b⃗ c⃗ a⃗] = [c⃗ a⃗ b⃗] [cyclic] [a⃗ b⃗ c⃗] = -[b⃗ a⃗ c⃗] [swap any two → sign changes]

10.6 Vector Triple Product

a⃗×(b⃗×c⃗) = (a⃗·c⃗)b⃗ - (a⃗·b⃗)c⃗ [BAC-CAB rule] (a⃗×b⃗)×c⃗ = (a⃗·c⃗)b⃗ - (b⃗·c⃗)a⃗

10.7 Section Formula in Vectors

If P divides AB in ratio m:n internally: p⃗ = (mb⃗ + na⃗)/(m+n) Midpoint: p⃗ = (a⃗ + b⃗)/2 Externally: p⃗ = (mb⃗ - na⃗)/(m-n)

B11

Three Dimensional Geometry

11.1 Direction Cosines and Ratios

For line with direction ratios (a,b,c): l = a/√(a²+b²+c²) m = b/√(a²+b²+c²) n = c/√(a²+b²+c²) l²+m²+n² = 1 always cos α = l, cos β = m, cos γ = n (α, β, γ = angles with x,y,z axes) cos²α + cos²β + cos²γ = 1 sin²α + sin²β + sin²γ = 2 [derived]

11.2 Angle Between Two Lines

Lines with DRs (a₁,b₁,c₁) and (a₂,b₂,c₂): cos θ = |a₁a₂+b₁b₂+c₁c₂| / [√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)] Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂ Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0 With DCs (l₁,m₁,n₁) and (l₂,m₂,n₂): cos θ = |l₁l₂ + m₁m₂ + n₁n₂|

11.3 Equation of a Line

Vector Form: r⃗ = a⃗ + λb⃗ [passing through point a⃗, direction b⃗] r⃗ = a⃗ + λ(b⃗-a⃗) [passing through a⃗ and b⃗] Cartesian/Symmetric Form: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c = λ x = x₁+aλ, y = y₁+bλ, z = z₁+cλ

11.4 Distance Between Two Lines

Skew lines (r⃗=a⃗₁+λb⃗₁ and r⃗=a⃗₂+μb⃗₂): d = |(a⃗₂-a⃗₁)·(b⃗₁×b⃗₂)| / |b⃗₁×b⃗₂| Parallel lines (b⃗₁ ∥ b⃗₂=b⃗): d = |(a⃗₂-a⃗₁)×b⃗| / |b⃗| Intersecting lines: d = 0 Coplanar lines: (a⃗₂-a⃗₁)·(b⃗₁×b⃗₂) = 0

11.5 Equation of a Plane

Vector Form: r⃗·n̂ = d [n̂ = unit normal, d = distance from origin] r⃗·n⃗ = D [n⃗ = normal vector (not unit)] Cartesian Form: ax + by + cz = d [a,b,c = normal direction] ax + by + cz + d = 0 Normal form: lx + my + nz = p Intercept Form: x/α + y/β + z/γ = 1 [α,β,γ = x,y,z intercepts] Plane through 3 points A(x₁,y₁,z₁), B, C: |x-x₁ y-y₁ z-z₁| |x₂-x₁ y₂-y₁ z₂-z₁| = 0 |x₃-x₁ y₃-y₁ z₃-z₁|

11.6 Angle Between Two Planes

Planes a₁x+b₁y+c₁z+d₁=0 and a₂x+b₂y+c₂z+d₂=0: cos θ = |a₁a₂+b₁b₂+c₁c₂| / [√(a₁²+b₁²+c₁²)·√(a₂²+b₂²+c₂²)] Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂ Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0

11.7 Distance Formulas in 3D

Distance between P(x₁,y₁,z₁) and Q(x₂,y₂,z₂): d = √[(x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²] Distance from point P(x₁,y₁,z₁) to plane ax+by+cz+d=0: d = |ax₁+by₁+cz₁+d| / √(a²+b²+c²) Distance between parallel planes ax+by+cz=d₁ and ax+by+cz=d₂: d = |d₁-d₂| / √(a²+b²+c²) Foot of perpendicular from P(x₁,y₁,z₁) to plane ax+by+cz+d=0: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c = -(ax₁+by₁+cz₁+d)/(a²+b²+c²)

11.8 Angle Between Line and Plane

Line: r⃗ = a⃗ + λb⃗, Plane: r⃗·n⃗ = d sin φ = |b⃗·n⃗| / (|b⃗||n⃗|) [φ = angle with plane] cos θ = |b⃗·n⃗| / (|b⃗||n⃗|) [θ = angle with normal] Line ∥ plane ↔ b⃗·n⃗ = 0 Line ⊥ plane ↔ b⃗ ∥ n⃗ ↔ b⃗×n⃗ = 0⃗ Line in plane ↔ b⃗·n⃗=0 AND a⃗·n⃗=d

11.9 Family of Planes

Plane through intersection of P₁=0 and P₂=0: P₁ + λP₂ = 0 (for varying λ) (a₁x+b₁y+c₁z+d₁) + λ(a₂x+b₂y+c₂z+d₂) = 0

B12

Linear Programming

12.1 Standard Formulation

Objective Function: Z = ax + by (maximize or minimize) Subject to constraints: a₁x+b₁y ≤/≥/= c₁ etc. Non-negativity: x ≥ 0, y ≥ 0 Feasible Region: set of all points satisfying all constraints Optimal Solution: max/min value of Z at a corner point Corner Point Method: 1. Graph all constraints 2. Find feasible region 3. Find corner points (vertices) 4. Evaluate Z at each corner point 5. Max/min Z is the answer

12.2 Fundamental Theorem

If optimal solution exists → it is at a corner point of feasible region If Z = c (constant) is parallel to a boundary → Z takes same value along that boundary (infinite solutions) Unbounded feasible region → Z may or may not have optimum

12.3 Key Results

For maximize Z = ax+by: → Look for corner with largest x if a>b, etc. For minimize Z = ax+by: → Look for corner nearest to origin (usually) If feasible region is empty → no solution (infeasible problem) If feasible region is bounded → both max and min exist If unbounded → only one of max/min may exist

B13

Probability (Class 12)

13.1 Conditional Probability

P(A|B) = P(A∩B)/P(B) [probability of A given B, P(B)≠0] → P(A∩B) = P(B)·P(A|B) = P(A)·P(B|A) → P(B|A) = P(A∩B)/P(A) → P(A|B) · P(B) = P(B|A) · P(A) Properties: 0 ≤ P(A|B) ≤ 1 P(S|B) = 1 P(A'|B) = 1 - P(A|B) P(A∪C|B) = P(A|B) + P(C|B) - P(A∩C|B)

13.2 Multiplication Theorem

13.3 Total Probability Theorem

If B₁, B₂, ..., Bₙ are mutually exclusive and exhaustive events: P(A) = Σᵢ P(Bᵢ)·P(A|Bᵢ) = P(B₁)P(A|B₁) + P(B₂)P(A|B₂) + ... + P(Bₙ)P(A|Bₙ)

13.4 Bayes' Theorem

13.5 Random Variables and Probability Distribution

If X is a random variable taking values x₁, x₂, ..., xₙ with probabilities p₁, p₂, ..., pₙ: Conditions: pᵢ ≥ 0 for all i, Σpᵢ = 1 Mean (Expected Value): E(X) = μ = Σxᵢpᵢ = x₁p₁ + x₂p₂ + ... + xₙpₙ Variance: Var(X) = σ² = E(X²) - [E(X)]² = Σxᵢ²pᵢ - (Σxᵢpᵢ)² = E[(X-μ)²] = Σ(xᵢ-μ)²pᵢ Standard Deviation: σ = √Var(X) = √[E(X²) - (E(X))²] Properties: E(aX+b) = aE(X) + b Var(aX+b) = a²Var(X) Var(aX) = a²Var(X) Var(X+b) = Var(X)

13.6 Binomial Distribution

X ~ B(n, p) [n = trials, p = success prob, q = 1-p] P(X = r) = C(n,r) · pʳ · qⁿ⁻ʳ (r = 0,1,2,...,n) where C(n,r) = n! / [r!(n-r)!] = ⁿCᵣ Mean: E(X) = np Variance: Var(X) = npq = np(1-p) SD: σ = √(npq) Mode: (n+1)p [if integer, two modes: (n+1)p and (n+1)p-1] ⌊(n+1)p⌋ [if not integer] P(X=0) = qⁿ P(X=n) = pⁿ P(X≥1) = 1 - P(X=0) = 1 - qⁿ Recurrence: P(X=r+1)/P(X=r) = [(n-r)/(r+1)] × (p/q) For large n, small p (np=λ fixed) → Poisson approximation: P(X=r) ≈ e^(-λ)λʳ/r!

13.7 Important Probability Results

P(A only) = P(A∩B') = P(A) - P(A∩B) P(B only) = P(A'∩B) = P(B) - P(A∩B) P(exactly one of A,B) = P(A) + P(B) - 2P(A∩B) P(neither A nor B) = 1 - P(A∪B) = 1 - P(A) - P(B) + P(A∩B) For three events: P(A∪B∪C) = P(A)+P(B)+P(C)-P(A∩B)-P(B∩C)-P(C∩A)+P(A∩B∩C) If A and B independent: A and B' independent A' and B independent A' and B' independent P(A∩B) = P(A)P(B) [independent] P(A∩B) = 0 [mutually exclusive] Independent ≠ mutually exclusive (unless P(A)=0 or P(B)=0)

C1

Advanced Algebra Identities

Algebraic Expansions

(a+b+c)² = a²+b²+c²+2ab+2bc+2ca (a-b-c)² = a²+b²+c²-2ab+2bc-2ca (a+b-c)² = a²+b²+c²+2ab-2bc-2ca a²+b²+c² = (a+b+c)² - 2(ab+bc+ca) ab+bc+ca = [(a+b+c)²-(a²+b²+c²)]/2 (a+b+c)³ = a³+b³+c³+3(a+b)(b+c)(c+a) a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca) = ½(a+b+c)[(a-b)²+(b-c)²+(c-a)²] If a+b+c=0: a³+b³+c³ = 3abc (a+b)⁴ = a⁴+4a³b+6a²b²+4ab³+b⁴ (a-b)⁴ = a⁴-4a³b+6a²b²-4ab³+b⁴ a⁴-b⁴ = (a+b)(a-b)(a²+b²) a⁴+b⁴ = (a²+b²)²-2a²b² Binomial: (a+b)ⁿ = Σₖ₌₀ⁿ C(n,k)aⁿ⁻ᵏbᵏ General term: T(r+1) = C(n,r)aⁿ⁻ʳbʳ Middle term: T((n/2)+1) for even n Sophie Germain Identity: a⁴+4b⁴ = (a²+2b²+2ab)(a²+2b²-2ab)

C2

Complete Trigonometry Identities

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) tan(A-B) = (tanA - tanB)/(1 + tanA tanB) sin(A+B)·sin(A-B) = sin²A - sin²B = cos²B - cos²A cos(A+B)·cos(A-B) = cos²A - sin²B = cos²B - sin²A

Double Angle Formulas

sin 2A = 2 sinA cosA = 2tanA/(1+tan²A) cos 2A = cos²A - sin²A = 1-2sin²A = 2cos²A-1 = (1-tan²A)/(1+tan²A) tan 2A = 2tanA/(1-tan²A) sin²A = (1-cos2A)/2 cos²A = (1+cos2A)/2 tan²A = (1-cos2A)/(1+cos2A) sin A = 2sin(A/2)cos(A/2) cos A = 1-2sin²(A/2) = 2cos²(A/2)-1

Triple Angle Formulas

sin 3A = 3sinA - 4sin³A cos 3A = 4cos³A - 3cosA tan 3A = (3tanA - tan³A)/(1-3tan²A) sin³A = (3sinA - sin3A)/4 cos³A = (3cosA + cos3A)/4

Product to Sum Formulas

2sinA cosB = sin(A+B) + sin(A-B) 2cosA sinB = sin(A+B) - sin(A-B) 2cosA cosB = cos(A+B) + cos(A-B) 2sinA sinB = cos(A-B) - cos(A+B)

Sum to Product Formulas

sinC + sinD = 2sin((C+D)/2)cos((C-D)/2) sinC - sinD = 2cos((C+D)/2)sin((C-D)/2) cosC + cosD = 2cos((C+D)/2)cos((C-D)/2) cosC - cosD = -2sin((C+D)/2)sin((C-D)/2)

Important Trig Values

sin 15° = (√6-√2)/4 cos 15° = (√6+√2)/4 sin 18° = (√5-1)/4 cos 36° = (√5+1)/4 sin 36° = √(10-2√5)/4 cos 18° = √(10+2√5)/4 sin 22.5° = √((√2-1)/(2√2)) tan 15° = 2-√3 tan 75° = 2+√3

C3

Mensuration Advanced Formulas

Regular polygon (n sides, side a): Perimeter = na Interior angle = (n-2)×180°/n Exterior angle = 360°/n Area = (na²/4)cot(π/n) Sum of interior angles = (n-2)×180° Heron's Formula (triangle with sides a,b,c): s = (a+b+c)/2 [semi-perimeter] Area = √[s(s-a)(s-b)(s-c)] Area using angles (triangle): Area = (1/2)ab sinC = (1/2)bc sinA = (1/2)ca sinB Circumradius R of triangle: R = abc/(4 × Area) = a/(2sinA) Inradius r of triangle: r = Area/s = (s-a)tan(A/2) Cosine Rule: a² = b²+c²-2bc cosA b² = a²+c²-2ac cosB c² = a²+b²-2ab cosC → cosA = (b²+c²-a²)/(2bc) Sine Rule: a/sinA = b/sinB = c/sinC = 2R

C4

Coordinate Geometry Advanced (Class 12 Level)

Standard curves: Circle: x²+y²+2gx+2fy+c=0 Center: (-g,-f), Radius: √(g²+f²-c) Parabola: y²=4ax (opens right, vertex origin) Directrix: x=-a, Focus: (a,0), Axis: y=0 y²=-4ax (opens left) x²=4ay (opens up), x²=-4ay (opens down) Ellipse: x²/a²+y²/b²=1 (a>b>0) c²=a²-b² (c=focal distance) e=c/a<1 (eccentricity) Foci: (±c,0), Vertices: (±a,0) Directrices: x=±a/e=±a²/c Hyperbola: x²/a²-y²/b²=1 b²=c²-a², e=c/a>1 Asymptotes: y=±(b/a)x Conjugate: y²/a²-x²/b²=1 Tangent to circle x²+y²=r² at (x₁,y₁): xx₁+yy₁=r² Tangent to parabola y²=4ax at (x₁,y₁): yy₁=2a(x+x₁) Tangent to ellipse x²/a²+y²/b²=1 at (x₁,y₁): xx₁/a²+yy₁/b²=1 Tangent of slope m to parabola y²=4ax: y=mx+a/m Tangent of slope m to ellipse: y=mx±√(a²m²+b²)

C5

Limits — Key Formulas

Standard Limits: lim[x→0] sinx/x = 1 lim[x→0] tanx/x = 1 lim[x→0] (1-cosx)/x = 0 lim[x→0] (1-cosx)/x² = 1/2 lim[x→0] sin⁻¹x/x = 1 lim[x→0] tan⁻¹x/x = 1 lim[x→0] (eˣ-1)/x = 1 lim[x→0] (aˣ-1)/x = ln a lim[x→0] (ln(1+x))/x = 1 lim[x→∞] (1+1/x)ˣ = e lim[x→0] (1+x)^(1/x) = e lim[x→a] (xⁿ-aⁿ)/(x-a) = naⁿ⁻¹ L'Hôpital's Rule (0/0 or ∞/∞ form): lim f(x)/g(x) = lim f'(x)/g'(x) Special: lim[x→∞] xⁿ/eˣ = 0 (exponential grows faster) lim[x→∞] ln x/xⁿ = 0 (power grows faster than log)

C6

Sequences and Series (Additional)

Geometric Progression (GP)

aₙ = ar^(n-1) [nth term] r = common ratio = aₙ₊₁/aₙ Sum of n terms: Sₙ = a(rⁿ-1)/(r-1) (r ≠ 1, r > 1 preferred) Sₙ = a(1-rⁿ)/(1-r) (r ≠ 1, r < 1 preferred) Sₙ = na (r = 1) Sum to infinity (|r|<1): S∞ = a/(1-r) Product of first n terms of GP: P = aⁿ × r^(n(n-1)/2) GM between a and b: G = √(ab) AM ≥ GM for positive reals: (a+b)/2 ≥ √(ab) HM between a and b: H = 2ab/(a+b) AM ≥ GM ≥ HM GM² = AM × HM 3 terms in GP: a/r, a, ar [product = a³] 4 terms in GP: a/r³, a/r, ar, ar³ [product = a⁴]

Harmonic Progression (HP)

If a,b,c in HP: 1/a, 1/b, 1/c in AP → 2/b = 1/a + 1/c → b = 2ac/(a+c) nth term of HP: 1/(A + (n-1)D)

AGP (Arithmetico-Geometric Progression)

a, (a+d)r, (a+2d)r², ... S∞ = a/(1-r) + dr/(1-r)² [|r|<1]

C7

Combinatorics (for Probability)

Permutation: ⁿPᵣ = n!/(n-r)! [order matters] Combination: ⁿCᵣ = n!/[r!(n-r)!] [order doesn't matter] ⁿCᵣ = ⁿCₙ₋ᵣ ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ [Pascal's identity] ⁿCₒ = ⁿCₙ = 1 ⁿC₁ = n ⁿC₂ = n(n-1)/2 ⁿCᵣ = (n/r)ⁿ⁻¹Cᵣ₋₁ n! = n × (n-1)! 0! = 1 Circular permutation: (n-1)! Necklace/bracelet: (n-1)!/2 Multinomial: n!/(n₁!n₂!...nₖ!) Stars and bars: distributing n identical into k distinct: C(n+k-1, k-1)

C8

Complex Numbers

z = a + ib [a = real part, b = imaginary part] i = √(-1), i² = -1, i³ = -i, i⁴ = 1 Modulus: |z| = √(a²+b²) Argument: arg(z) = θ = tan⁻¹(b/a) [in correct quadrant] Conjugate: z̄ = a - ib z + z̄ = 2a = 2Re(z) z - z̄ = 2ib = 2i·Im(z) z·z̄ = |z|² = a²+b² (z̄₁+z₂) = z̄₁+z̄₂ (z₁z₂) = z̄₁·z̄₂ Polar form: z = r(cosθ + i sinθ) = reⁱθ r = |z|, θ = arg(z) De Moivre's theorem: (cosθ+i sinθ)ⁿ = cos(nθ)+i sin(nθ) |z₁z₂| = |z₁||z₂| arg(z₁z₂) = arg(z₁)+arg(z₂) Triangle inequality: |z₁+z₂| ≤ |z₁|+|z₂| |z₁-z₂| ≥ ||z₁|-|z₂|| nth roots of unity: zⁿ = 1 zₖ = cos(2πk/n)+i sin(2πk/n) for k=0,1,...,n-1 Sum of nth roots = 0 Product of nth roots = (-1)^(n-1) Cube roots of unity: 1, ω, ω² ω = (-1+i√3)/2 = e^(2πi/3) 1+ω+ω² = 0 ω³ = 1 ω² = (-1-i√3)/2

C9

Complete Derivative Table (Quick Reference)

f(x) f'(x) ───────────────────────────────────────── c (constant) 0 x 1 xⁿ nxⁿ⁻¹ √x 1/(2√x) 1/x -1/x² 1/xⁿ -n/x^(n+1) eˣ eˣ eᵃˣ aeᵃˣ aˣ aˣ ln a ln x 1/x log_a x 1/(x ln a) sin x cos x cos x -sin x tan x sec²x cot x -cosec²x sec x sec x tan x cosec x -cosec x cot x sin(ax+b) a cos(ax+b) cos(ax+b) -a sin(ax+b) tan(ax+b) a sec²(ax+b) sin⁻¹x 1/√(1-x²) cos⁻¹x -1/√(1-x²) tan⁻¹x 1/(1+x²) cot⁻¹x -1/(1+x²) sec⁻¹x 1/(|x|√(x²-1)) cosec⁻¹x -1/(|x|√(x²-1)) sin⁻¹(x/a) 1/√(a²-x²) cos⁻¹(x/a) -1/√(a²-x²) tan⁻¹(x/a) a/(a²+x²) |x| x/|x| = sgn(x) (x≠0) xˣ xˣ(1+ln x) f(g(x)) f'(g(x))·g'(x)

C10

Complete Integral Table (Quick Reference)

f(x) ∫f(x)dx ──────────────────────────────────────────────────────────── xⁿ (n≠-1) xⁿ⁺¹/(n+1) + C 1/x ln|x| + C eˣ eˣ + C eᵃˣ eᵃˣ/a + C aˣ aˣ/ln a + C sin x -cos x + C cos x sin x + C tan x ln|sec x| + C cot x ln|sin x| + C sec x ln|sec x + tan x| + C cosec x ln|cosec x - cot x| + C sec²x tan x + C cosec²x -cot x + C sec x tan x sec x + C cosec x cot x -cosec x + C 1/√(1-x²) sin⁻¹x + C -1/√(1-x²) cos⁻¹x + C 1/(1+x²) tan⁻¹x + C -1/(1+x²) cot⁻¹x + C 1/(x²+a²) (1/a)tan⁻¹(x/a) + C 1/(x²-a²) (1/2a)ln|(x-a)/(x+a)| + C 1/(a²-x²) (1/2a)ln|(a+x)/(a-x)| + C 1/√(a²-x²) sin⁻¹(x/a) + C 1/√(x²+a²) ln|x+√(x²+a²)| + C 1/√(x²-a²) ln|x+√(x²-a²)| + C √(a²-x²) (x/2)√(a²-x²)+(a²/2)sin⁻¹(x/a)+C √(x²+a²) (x/2)√(x²+a²)+(a²/2)ln|x+√(x²+a²)|+C √(x²-a²) (x/2)√(x²-a²)-(a²/2)ln|x+√(x²-a²)|+C eˣ[f(x)+f'(x)] eˣf(x) + C x eˣ eˣ(x-1) + C x² eˣ eˣ(x²-2x+2) + C ln x x ln x - x + C x ln x (x²/2)ln x - x²/4 + C sin⁻¹x x sin⁻¹x + √(1-x²) + C tan⁻¹x x tan⁻¹x - (1/2)ln(1+x²) + C cos⁻¹x x cos⁻¹x - √(1-x²) + C sin²x x/2 - sin(2x)/4 + C cos²x x/2 + sin(2x)/4 + C tan²x tan x - x + C cot²x -cot x - x + C sin³x -cos x + cos³x/3 + C cos³x sin x - sin³x/3 + C

C11

Matrices — Additional Important Results

Trace(A) = sum of diagonal elements Trace(AB) = Trace(BA) Rank of matrix = number of non-zero rows in row echelon form For A (m×n): rank(A) ≤ min(m,n) A is invertible ↔ rank(A) = n (full rank) A is singular ↔ |A| = 0 ↔ rank(A) < n Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation. For A 2×2 with char. eq λ²-(trA)λ+|A|=0: A² - (trA)A + |A|I = 0 → A⁻¹ = (A - (trA)I) / |A| [for 2×2 non-singular] For 2×2 A = [a b; c d]: Characteristic equation: λ² - (a+d)λ + (ad-bc) = 0 λ₁+λ₂ = a+d = Trace(A) λ₁·λ₂ = ad-bc = |A| Eigenvalues of diagonal matrix = diagonal elements Eigenvalues of triangular matrix = diagonal elements |A| = product of eigenvalues Trace = sum of eigenvalues If A is orthogonal: AAᵀ = AᵀA = I → A⁻¹ = Aᵀ If A is idempotent: A² = A If A is nilpotent: Aⁿ = O for some n If A is involutory: A² = I → A⁻¹ = A

C12

Special Technique Formulas (Exam Tricks)

Rationalization Tricks

1/(a+b√c) = (a-b√c)/(a²-b²c) 1/(√a+√b) = (√a-√b)/(a-b) Nested radicals: √(a+√b) = √((a+√(a²-b))/2) + √((a-√(a²-b))/2)

Telescoping Sums

Σ[f(k)-f(k-1)] = f(n) - f(0) [collapses to endpoints] 1/(k(k+1)) = 1/k - 1/(k+1) 1/(k(k+1)(k+2)) = (1/2)[1/(k(k+1)) - 1/((k+1)(k+2))] Σ 1/(k(k+1)) from k=1 to n = 1 - 1/(n+1) = n/(n+1)

Inequality Formulas

AM-GM: (a+b)/2 ≥ √(ab) [equality when a=b] (a₁+a₂+...+aₙ)/n ≥ (a₁a₂...aₙ)^(1/n) Cauchy-Schwarz: (a₁b₁+a₂b₂+...+aₙbₙ)² ≤ (a₁²+...+aₙ²)(b₁²+...+bₙ²) Triangle inequality: |a+b| ≤ |a| + |b| ||a|-|b|| ≤ |a-b| For positive x: x + 1/x ≥ 2 [min value 2 at x=1] x² + 1/x² ≥ 2

Useful Number Patterns

1+2+3+...+n = n(n+1)/2 1+3+5+...+(2n-1) = n² 2+4+6+...+2n = n(n+1) 1²+2²+...+n² = n(n+1)(2n+1)/6 1³+2³+...+n³ = [n(n+1)/2]² 1×2+2×3+...+n(n+1) = n(n+1)(n+2)/3 1×1!+2×2!+...+n×n! = (n+1)!-1

Important Constants and Values

π ≈ 3.14159265... e ≈ 2.71828182... √2 ≈ 1.41421356... √3 ≈ 1.73205080... √5 ≈ 2.23606797... 1/√2 = √2/2 ≈ 0.7071 √3/2 ≈ 0.8660 ln 2 ≈ 0.6931 ln 10 ≈ 2.3026 log₁₀ e ≈ 0.4343 Radian to degree: 1 rad = 180°/π ≈ 57.296° Degree to radian: 1° = π/180 rad ≈ 0.01745 rad 30° = π/6 rad 45° = π/4 rad 60° = π/3 rad 90° = π/2 rad 120° = 2π/3 rad 135° = 3π/4 rad 150° = 5π/6 rad 180° = π rad 270° = 3π/2 rad 360° = 2π rad

Differential Calculus — Twisted Forms

If y = sin(ax+b): y' = a cos(ax+b), y'' = -a² sin(ax+b), yⁿ = aⁿ sin(ax+b+nπ/2) If y = cos(ax+b): yⁿ = aⁿ cos(ax+b+nπ/2) If y = eᵃˣ: yⁿ = aⁿeᵃˣ If y = ln x: y' = 1/x, y'' = -1/x², yⁿ = (-1)^(n-1)(n-1)!/xⁿ Partial derivatives (for Lagrange multipliers): ∂/∂x[f(x,y)] — treat y as constant Jacobian: J = ∂(u,v)/∂(x,y) = |∂u/∂x ∂u/∂y| |∂v/∂x ∂v/∂y| Euler's theorem for homogeneous f of degree n: x(∂f/∂x) + y(∂f/∂y) = nf Total differential: df = (∂f/∂x)dx + (∂f/∂y)dy

3D Geometry — Extra Formulas

Sphere with center (h,k,l) and radius r: (x-h)²+(y-k)²+(z-l)² = r² General sphere: x²+y²+z²+2ux+2vy+2wz+d=0 Center: (-u,-v,-w), Radius: √(u²+v²+w²-d) Image of point P(α,β,γ) in plane ax+by+cz+d=0: (x-α)/a = (y-β)/b = (z-γ)/c = -2(aα+bβ+cγ+d)/(a²+b²+c²) Centroid of tetrahedron (4 vertices): G = [(x₁+x₂+x₃+x₄)/4, (y₁+y₂+y₃+y₄)/4, (z₁+z₂+z₃+z₄)/4] Volume of tetrahedron with one vertex at origin, others A,B,C: V = (1/6)|[a⃗ b⃗ c⃗]| Distance from line to line (skew): d = |[(b⃗₁×b⃗₂)·(a⃗₂-a⃗₁)]| / |b⃗₁×b⃗₂|

Vector — Additional Identities

|a⃗×b⃗|² = |a⃗|²|b⃗|² - (a⃗·b⃗)² [Lagrange's identity] (a⃗×b⃗)·(c⃗×d⃗) = (a⃗·c⃗)(b⃗·d⃗) - (a⃗·d⃗)(b⃗·c⃗) (a⃗×b⃗)×(c⃗×d⃗) = [a⃗ b⃗ d⃗]c⃗ - [a⃗ b⃗ c⃗]d⃗ = [a⃗ c⃗ d⃗]b⃗ - [b⃗ c⃗ d⃗]a⃗ Area of parallelogram with diagonals d⃗₁, d⃗₂: Area = ½|d⃗₁×d⃗₂| Normal to plane through a⃗, b⃗, c⃗: n⃗ = (b⃗-a⃗)×(c⃗-a⃗) Angle bisector of a⃗ and b⃗ (unit vectors â,b̂): bisector direction = â + b̂ [internal] = â - b̂ [external]

Exam Strategy — Key Identity Shortcuts

a² + b² + c² - ab - bc - ca = ½[(a-b)² + (b-c)² + (c-a)²] ≥ 0 always (a+b+c)² = a²+b²+c² + 2(ab+bc+ca) If a+b+c=0: a²+b²+c² = -2(ab+bc+ca) ab+bc+ca = -(a²+b²+c²)/2 a³+b³+c³ = 3abc Determinant tricks: R₁→R₁+R₂+R₃ (sum of rows): factor out the sum C₁→C₁-C₂, C₂→C₂-C₃: create zeros If rows in AP: middle row = average of outer rows → use R₂→2R₂-R₁-R₃ Common factor: |ka kb kc| = k × |a b c| |d e f | |d e f| |g h i | |g h i|

CBSE Board Frequent Formula Combos

Volume related conversion: If a sphere of radius R is melted into n small spheres of radius r: (4/3)πR³ = n × (4/3)πr³ → n = R³/r³ = (R/r)³ Cone from circle (sector folded): If a sector of radius L and angle θ is folded into a cone: Slant height of cone = L (radius of sector) 2πr = (θ/360°) × 2πL → r = θL/360° → h = √(L²-r²) → V = (1/3)πr²h Ratio of volumes of cone:cylinder:sphere (same r, same h=2r): V_cone : V_cyl : V_sph = 1 : 3 : 2 [FAMOUS RATIO] Sphere inscribed in cylinder: r_sphere = r_cyl = h_cyl/2 SA_sphere/CSA_cyl = 1 [Equal surface areas!] Largest cone inscribed in cylinder: same base and height V_cone = (1/3)V_cyl Largest sphere inscribed in cube of side a: r = a/2 Largest sphere inscribed in cone of radius R, height h: r = Rh/(√(R²+h²)+R) = Rh/(l+R)

C13

Final Master Summary — The Golden Rules

1. a² - b² = (a+b)(a-b) — always factorize first! 2. D = b²-4ac — discriminant rules everything 3. sin²+cos²=1 — most common identity 4. AM≥GM≥HM — inequalities 5. P(A)+P(Ā)=1 — complement rule 6. aₙ=a+(n-1)d, Sₙ=n/2[2a+(n-1)d] — AP core 7. LCM×HCF = a×b — always 8. Area = ½|base×height| — universally 9. Chain rule: d/dx[f(g(x))]=f'(g(x))g'(x) 10. ∫eˣ[f+f']=eˣf+C — IBP shortcut Key Values to Remember: sin30=1/2, cos60=1/2, tan45=1, sin90=1, cos0=1 √2≈1.414, √3≈1.732, π≈3.14159, e≈2.71828 Key Theorems: - Pythagoras: H²=B²+P² - Thales (BPT): AD/DB=AE/EC - MVT: f'(c)=(f(b)-f(a))/(b-a) - Bayes: P(H|E)=P(H)P(E|H)/P(E) - Euclid: a=bq+r - FTA: Every composite number has unique prime factorization

Complete MathematicsFormula Compendium

Complete Mathematics
Formula Compendium