1. Number Systems
Basic Number Properties
- Even + Even = Even
- Odd + Odd = Even
- Even + Odd = Odd
- Even × Even = Even
- Odd × Odd = Odd
- Even × Odd = Even
- Even ÷ Even = Even or Odd
- Odd ÷ Odd = Odd
- Even ÷ Odd = Even
Divisibility Rules (Complete Set)
- Divisible by 1: All numbers
- Divisible by 2: Last digit is 0, 2, 4, 6, 8
- Divisible by 3: Sum of digits divisible by 3
- Divisible by 4: Last two digits divisible by 4
- Divisible by 5: Last digit is 0 or 5
- Divisible by 6: Divisible by both 2 and 3
- Divisible by 7: Remove last digit, double it, subtract from remaining number. If result divisible by 7, original is too
- Divisible by 8: Last three digits divisible by 8
- Divisible by 9: Sum of digits divisible by 9
- Divisible by 10: Last digit is 0
- Divisible by 11: Alternating sum of digits divisible by 11
- Divisible by 12: Divisible by both 3 and 4
- Divisible by 13: Add 4 times the last digit to remaining number
- Divisible by 15: Divisible by both 3 and 5
- Divisible by 16: Last four digits divisible by 16
- Divisible by 17: Subtract 5 times the last digit from remaining number
- Divisible by 18: Divisible by both 2 and 9
- Divisible by 19: Add 2 times the last digit to remaining number
- Divisible by 25: Last two digits are 00, 25, 50, or 75
HCF and LCM Formulas
HCF(a,b) × LCM(a,b) = a × b
HCF(a,b,c) = HCF[HCF(a,b), c]
LCM(a,b,c) = LCM[LCM(a,b), c]
For coprime numbers: HCF = 1, LCM = Product
HCF by Euclidean Algorithm: HCF(a,b) = HCF(b, a mod b)
Factors and Multiples
Number of Factors
If n = a^p × b^q × c^r, then:
- Total factors = (p+1)(q+1)(r+1)
- Even factors = p(q+1)(r+1) [if a=2]
- Odd factors = (q+1)(r+1) [if a=2]
- Prime factors = p + q + r
- Composite factors = Total factors - Prime factors - 1
Sum of Factors
Sum of factors = [(a^(p+1) - 1)/(a-1)] × [(b^(q+1) - 1)/(b-1)] × [(c^(r+1) - 1)/(c-1)]
Product of Factors
Product of factors = n^(d(n)/2) where d(n) = number of factors
Perfect Numbers
- Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400...
- Perfect Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...
- Perfect Fourth Powers: 1, 16, 81, 256, 625, 1296...
- Perfect Fifth Powers: 1, 32, 243, 1024, 3125...
Sum Formulas
Sum of first n natural numbers: Σn = n(n+1)/2
Sum of first n even numbers: Σ2n = n(n+1)
Sum of first n odd numbers: Σ(2n-1) = n²
Sum of squares: Σn² = n(n+1)(2n+1)/6
Sum of cubes: Σn³ = [n(n+1)/2]²
Sum of fourth powers: Σn⁴ = n(n+1)(2n+1)(3n²+3n-1)/30
Sum of fifth powers: Σn⁵ = n²(n+1)²(2n²+2n-1)/12
Unit Digit Patterns
- Powers of 2: 2, 4, 8, 6, 2, 4, 8, 6... (cycle of 4)
- Powers of 3: 3, 9, 7, 1, 3, 9, 7, 1... (cycle of 4)
- Powers of 4: 4, 6, 4, 6... (cycle of 2)
- Powers of 5: Always 5
- Powers of 6: Always 6
- Powers of 7: 7, 9, 3, 1, 7, 9, 3, 1... (cycle of 4)
- Powers of 8: 8, 4, 2, 6, 8, 4, 2, 6... (cycle of 4)
- Powers of 9: 9, 1, 9, 1... (cycle of 2)
Remainder Theorems
(a + b) mod n = [(a mod n) + (b mod n)] mod n
(a - b) mod n = [(a mod n) - (b mod n)] mod n
(a × b) mod n = [(a mod n) × (b mod n)] mod n
a^n mod m = [(a mod m)^n] mod m
Chinese Remainder Theorem
If gcd(m₁, m₂) = 1, then system x ≡ a₁ (mod m₁), x ≡ a₂ (mod m₂) has unique solution modulo m₁m₂
Fermat's Little Theorem
If p is prime and gcd(a,p) = 1, then a^(p-1) ≡ 1 (mod p)
Wilson's Theorem
If p is prime, then (p-1)! ≡ -1 (mod p)
2. Algebra
Basic Identities (Complete Set)
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b)(a - b) = a² - b²
(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)
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
(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)
If a + b + c = 0, then a³ + b³ + c³ = 3abc
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
(a - b)⁴ = a⁴ - 4a³b + 6a²b² - 4ab³ + b⁴
a⁴ - b⁴ = (a² + b²)(a + b)(a - b)
a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² - ab + b²)
Advanced Identities
(a + b + c + d)² = Σa² + 2Σ(ab)
a⁵ + b⁵ = (a + b)(a⁴ - a³b + a²b² - ab³ + b⁴)
a⁵ - b⁵ = (a - b)(a⁴ + a³b + a²b² + ab³ + b⁴)
If a + b + c + d = 0, then a³ + b³ + c³ + d³ = 3(ab + bc + cd + da + ac + bd)(a + b + c + d)/2
Binomial Theorem
(a + b)ⁿ = Σ(r=0 to n) ⁿCᵣ aⁿ⁻ʳ bʳ
General term (r+1)th = ⁿCᵣ aⁿ⁻ʳ bʳ
Middle term (when n is even) = (n/2 + 1)th term
Middle terms (when n is odd) = (n+1)/2 th and (n+3)/2 th terms
Greatest coefficient = ⁿCₙ/₂ (when n is even) or ⁿC₍ₙ₋₁₎/₂ = ⁿC₍ₙ₊₁₎/₂ (when n is odd)
Special Cases
(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ...
(1 - x)ⁿ = 1 - nx + n(n-1)x²/2! - n(n-1)(n-2)x³/3! + ...
Quadratic Equations
Standard Form: ax² + bx + c = 0
Discriminant: Δ = b² - 4ac
Roots: x = [-b ± √(b² - 4ac)]/2a
Sum of roots: α + β = -b/a
Product of roots: αβ = c/a
Difference of roots: |α - β| = √Δ/|a| = √(b² - 4ac)/|a|
Nature of Roots
- Δ > 0: Real and distinct roots
- Δ = 0: Real and equal roots
- Δ < 0: Complex/imaginary roots
- Δ > 0 and perfect square: Rational roots
- Δ > 0 and not perfect square: Irrational roots
Formation of Quadratic Equation
Given roots α, β: x² - (α + β)x + αβ = 0
Common roots: If ax² + bx + c = 0 and dx² + ex + f = 0 have common root, then (bf - ce)² = (cd - af)(ae - bd)
Quadratic Inequalities
ax² + bx + c > 0:
- If a > 0 and Δ < 0: Always true
- If a > 0 and Δ > 0: x < α or x > β
- If a < 0 and Δ > 0: α < x < β
Cubic Equations
Standard Form: ax³ + bx² + cx + d = 0
Sum of roots: α + β + γ = -b/a
Sum of products taken two at a time: αβ + βγ + γα = c/a
Product of roots: αβγ = -d/a
Sequences and Series
Arithmetic Progression (AP)
nth term: aₙ = a + (n-1)d
Sum of n terms: Sₙ = n/2[2a + (n-1)d] = n/2[first term + last term]
Sum of n terms: Sₙ = n/2[a₁ + aₙ]
Arithmetic Mean: AM = (a + b)/2
Three numbers in AP: a-d, a, a+d
Four numbers in AP: a-3d, a-d, a+d, a+3d
Property: aₙ = (aₙ₋₁ + aₙ₊₁)/2
Geometric Progression (GP)
nth term: aₙ = ar^(n-1)
Sum of n terms: Sₙ = a(r^n - 1)/(r - 1) for r ≠ 1
Sum of n terms: Sₙ = na for r = 1
Sum to infinity: S∞ = a/(1-r) for |r| < 1
Geometric Mean: GM = √(ab)
Three numbers in GP: a/r, a, ar
Four numbers in GP: a/r³, a/r, ar, ar³
Property: aₙ² = aₙ₋₁ × aₙ₊₁
Harmonic Progression (HP)
If a₁, a₂, a₃, ... are in HP, then 1/a₁, 1/a₂, 1/a₃, ... are in AP
nth term: 1/aₙ = 1/a + (n-1)/d
Harmonic Mean: HM = 2ab/(a+b)
Three numbers in HP: 1/(a-d), 1/a, 1/(a+d)
Relationship between Means
AM ≥ GM ≥ HM
AM × HM = GM²
For two numbers a, b: AM = (a+b)/2, GM = √(ab), HM = 2ab/(a+b)
Arithmetic-Geometric Progression (AGP)
General term: aₙ = [a + (n-1)d] × r^(n-1)
Sum of n terms: Complex formula involving both AP and GP sums
Logarithms
Basic Properties
log_a(xy) = log_a x + log_a y
log_a(x/y) = log_a x - log_a y
log_a(x^n) = n log_a x
log_a x = log_b x / log_b a (Change of base)
log_a x = 1/log_x a
a^(log_a x) = x
log_a a = 1
log_a 1 = 0
Advanced Properties
log_a b × log_b c × log_c a = 1
log_a b + log_b a = log_a b + 1/log_a b ≥ 2
log_(a^n) x = (1/n) log_a x
log_a (x^(1/n)) = (1/n) log_a x
Natural and Common Logarithms
ln x = log_e x (Natural logarithm)
log x = log₁₀ x (Common logarithm)
ln(e^x) = x
e^(ln x) = x
log(10^x) = x
10^(log x) = x
3. Arithmetic
Percentage
Basic Formulas
Percentage = (Part/Whole) × 100
Part = (Percentage × Whole)/100
Whole = (Part × 100)/Percentage
Percentage increase = [(New Value - Original Value)/Original Value] × 100
Percentage decrease = [(Original Value - New Value)/Original Value] × 100
Successive Percentage Changes
Two changes of a% and b%: Net change = a + b + (ab/100)%
Three changes of a%, b%, c%: Net change = a + b + c + (ab + bc + ca)/100 + (abc/10000)%
n equal changes of r%: Final value = Initial × (1 ± r/100)ⁿ
Population Growth
P = P₀(1 + r/100)ⁿ where P₀ = initial population, r = rate, n = time
For depreciation: P = P₀(1 - r/100)ⁿ
Profit and Loss
Basic Formulas
Profit = SP - CP
Loss = CP - SP
Profit% = (Profit/CP) × 100
Loss% = (Loss/CP) × 100
SP = CP(100 + Profit%)/100
SP = CP(100 - Loss%)/100
CP = SP × 100/(100 + Profit%)
CP = SP × 100/(100 - Loss%)
Marked Price and Discount
Discount = MP - SP
Discount% = (Discount/MP) × 100
SP = MP(100 - Discount%)/100
MP = SP × 100/(100 - Discount%)
Advanced Concepts
If article sold at x% profit instead of y% loss: CP = (SP₁ - SP₂)/(x + y) × 100
Break-even point: SP = CP (no profit, no loss)
If two articles sold at same SP, one at x% profit, other at x% loss: Net loss = (x/10)²%
Simple Interest (SI)
Basic Formulas
SI = PRT/100
Amount = P + SI = P(1 + RT/100)
P = SI × 100/(RT)
R = SI × 100/(PT)
T = SI × 100/(PR)
Advanced Formulas
If sum becomes n times in T years: R = 100(n-1)/T %
If sum doubles in T years: R = 100/T %
If sum triples in T years: R = 200/T %
Compound Interest (CI)
Basic Formulas
Amount = P(1 + R/100)ᵀ
CI = P(1 + R/100)ᵀ - P
CI = P[(1 + R/100)ᵀ - 1]
Different Compounding Periods
Annually: A = P(1 + R/100)ᵀ
Semi-annually: A = P(1 + R/200)²ᵀ
Quarterly: A = P(1 + R/400)⁴ᵀ
Monthly: A = P(1 + R/1200)¹²ᵀ
Daily: A = P(1 + R/36500)³⁶⁵ᵀ
Continuously: A = Pe^(RT/100)
Special Cases
When rates are different for different years: A = P(1 + R₁/100)(1 + R₂/100)(1 + R₃/100)...
Population growth/decay: P = P₀(1 ± r/100)ⁿ
Difference between CI and SI
For 2 years: CI - SI = PR²/(100)²
For 3 years: CI - SI = PR²(300 + R)/(100)³
Time and Work
Basic Formulas
Work done = Rate × Time
If A can finish work in 'a' days: A's rate = 1/a work per day
Combined rate: 1/A + 1/B = 1/T (where T = time to complete together)
Time for A and B together: T = AB/(A + B)
Advanced Concepts
Efficiency: If A is n times efficient as B, then A:B = n:1 (efficiency ratio)
Time ratio: If efficiency ratio is m:n, then time ratio is n:m
If A, B, C can finish work in a, b, c days respectively: Combined time = abc/(bc + ac + ab)
Pipes and Cisterns
Filling pipe rate: 1/time to fill
Emptying pipe rate: -1/time to empty
Net rate: Sum of all rates
If inlet pipe fills in A hours and outlet empties in B hours: Time to fill = AB/(B-A) hours (when B > A)
Time, Speed and Distance
Basic Formulas
Speed = Distance/Time
Distance = Speed × Time
Time = Distance/Speed
Average Speed = Total Distance/Total Time
Relative Speed
Same direction: Relative speed = |S₁ - S₂|
Opposite direction: Relative speed = S₁ + S₂
Time to meet (opposite direction): T = Distance/(S₁ + S₂)
Time for faster to catch slower (same direction): T = Distance/(S₁ - S₂)
Unit Conversions
km/hr to m/s: Multiply by 5/18
m/s to km/hr: Multiply by 18/5
1 km/hr = 5/18 m/s
1 m/s = 18/5 km/hr
Boats and Streams
Basic Formulas
Downstream speed = u + v (u = boat speed, v = stream speed)
Upstream speed = u - v
Boat speed in still water = (Downstream + Upstream)/2
Stream speed = (Downstream - Upstream)/2
Time downstream = Distance/(u + v)
Time upstream = Distance/(u - v)
Advanced Formulas
If time downstream is t₁ and upstream is t₂:
u = d(t₁ + t₂)/(2t₁t₂)
v = d(t₂ - t₁)/(2t₁t₂)
Races and Games
Basic Formulas
If A beats B by n meters in a race of d meters: When A covers d meters, B covers (d-n) meters
Speed ratio: A:B = d:(d-n)
Time ratio: A:B = (d-n):d
If A gives B a start of n meters: B starts n meters ahead
Trains
Basic Formulas
Time to cross a stationary object = Length of train/Speed of train
Time to cross a platform = (Length of train + Length of platform)/Speed
Time to cross a man/pole = Length of train/Speed
Two Trains
Time to cross each other (opposite direction) = (L₁ + L₂)/(S₁ + S₂)
Time to cross each other (same direction) = (L₁ + L₂)/(S₁ - S₂)
Time for faster train to completely overtake slower = (L₁ + L₂)/(S₁ - S₂)
Mixtures and Alligations
Basic Formulas
Mean price = (q₁p₁ + q₂p₂ + ... + qₙpₙ)/(q₁ + q₂ + ... + qₙ)
Alligation rule: (Cheaper quantity):(Dearer quantity) = (Mean - Cheaper price):(Dearer price - Mean)
Replacement Formula
After n operations, quantity of original liquid = Initial quantity × (1 - r)ⁿ where r = fraction of mixture replaced each time
Ratio and Proportion
Basic Properties
a:b = c:d ⟺ ad = bc (Cross multiplication)
a:b = c:d ⟺ a/b = c/d
If a/b = c/d = e/f = k, then each ratio = (a+c+e)/(b+d+f)
Componendo and Dividendo
Componendo: If a/b = c/d, then (a+b)/b = (c+d)/d
Dividendo: If a/b = c/d, then (a-b)/b = (c-d)/d
Componendo-Dividendo: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d)
Continued Proportion
a, b, c are in continued proportion if a/b = b/c
Then b² = ac (b is mean proportional)
Third proportional to a, b is c where b/c = a/b
Fourth proportional to a, b, c is d where a/b = c/d
Variation
Direct Proportion: y = kx (y ∝ x)
Inverse Proportion: xy = k (y ∝ 1/x)
Joint Variation: z = kxy (z ∝ xy)
Combined Variation: z = kx/y (z ∝ x/y)
4. Geometry
Basic Angle Properties
- Vertically opposite angles are equal
- Linear pair of angles sum to 180°
- Angles around a point sum to 360°
- Alternate interior angles are equal (parallel lines)
- Corresponding angles are equal (parallel lines)
- Co-interior angles sum to 180° (parallel lines)
Triangles
Basic Properties
Angle sum property: ∠A + ∠B + ∠C = 180°
Exterior angle = Sum of two non-adjacent interior angles
Sum of any two sides > Third side (Triangle inequality)
Difference of any two sides < Third side
Area Formulas
Area = ½ × base × height
Area = ½ab sin C (where a, b are sides and C is included angle)
Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Area = abc/(4R) where R = circumradius
Area = rs where r = inradius, s = semiperimeter
Special Triangles
Equilateral Triangle (all sides = a)
- All angles = 60°
- Area = (√3/4)a²
- Height = (√3/2)a
- Circumradius R = a/√3
- Inradius r = a/(2√3)
Right Triangle
Pythagoras theorem: a² + b² = c² (c = hypotenuse)
Area = ½ × base × height
Circumradius R = c/2 (hypotenuse/2)
Inradius r = (a + b - c)/2
Isosceles Triangle
- Two sides equal, two angles equal
- If equal sides = a, base = b: Area = (b/4)√(4a² - b²)
Special Right Triangles
30-60-90 Triangle
- Sides in ratio 1 : √3 : 2
- If shortest side = a: Sides are a, a√3, 2a
45-45-90 Triangle
- Sides in ratio 1 : 1 : √2
- If equal sides = a: Sides are a, a, a√2
Congruence Rules
- SSS: Three sides equal
- SAS: Two sides and included angle equal
- ASA: Two angles and included side equal
- AAS: Two angles and one side equal
- RHS: Right angle, hypotenuse, and one side equal
Similarity Rules
- AAA or AA: All angles equal (or two angles equal)
- SSS: All sides proportional
- SAS: Two sides proportional and included angle equal
Properties of Similar Triangles
Ratio of areas = (Ratio of corresponding sides)²
Ratio of perimeters = Ratio of corresponding sides
Ratio of altitudes = Ratio of corresponding sides
Ratio of medians = Ratio of corresponding sides
Centers of Triangle
Centroid (G)
- Intersection of medians
- Divides each median in ratio 2:1
- Distance from vertex = (2/3) × median length
Circumcenter (O)
- Intersection of perpendicular bisectors
- Equidistant from all vertices
- Circumradius R = abc/(4×Area)
Incenter (I)
- Intersection of angle bisectors
- Center of inscribed circle
- Inradius r = Area/s where s = semiperimeter
Orthocenter (H)
- Intersection of altitudes
- For right triangle: Orthocenter = vertex of right angle
- For obtuse triangle: Orthocenter lies outside
Stewart's Theorem
If cevian of length d divides side a into segments m and n:
b²m + c²n = a(d² + mn)
Quadrilaterals
General Properties
Sum of interior angles = 360°
Sum of exterior angles = 360°
Rectangle
- Opposite sides equal and parallel
- All angles = 90°
- Diagonals equal and bisect each other
- Area = length × breadth
- Perimeter = 2(length + breadth)
- Diagonal = √(l² + b²)
Square
- All sides equal, all angles = 90°
- Diagonals equal, perpendicular, bisect each other
- Area = side²
- Perimeter = 4 × side
- Diagonal = side × √2
Parallelogram
- Opposite sides equal and parallel
- Opposite angles equal
- Diagonals bisect each other
- Area = base × height
- Area = ab sin θ (where θ is angle between sides)
- Perimeter = 2(a + b)
Rhombus
- All sides equal
- Opposite angles equal
- Diagonals perpendicular and bisect each other
- Area = side² × sin θ
- Area = ½ × d₁ × d₂ (d₁, d₂ are diagonals)
- Perimeter = 4 × side
Trapezium
- One pair of opposite sides parallel
- Area = ½(sum of parallel sides) × height
- Area = ½(a + b) × h
- For isosceles trapezium: Diagonals are equal
Kite
- Two pairs of adjacent sides equal
- Diagonals perpendicular
- Area = ½ × d₁ × d₂
Circles
Basic Properties
Circumference = 2πr = πd
Area = πr²
Diameter = 2 × radius
Arc and Sector
Arc length = (θ/360°) × 2πr = (θ/180°) × πr (θ in degrees)
Arc length = rθ (θ in radians)
Sector area = (θ/360°) × πr² (θ in degrees)
Sector area = ½r²θ (θ in radians)
Chord Properties
Chord length = 2r sin(θ/2) (θ = central angle)
Distance from center to chord = r cos(θ/2)
Equal chords are equidistant from center
Perpendicular from center bisects chord
Segment
Minor segment area = Sector area - Triangle area
Major segment area = Circle area - Minor segment area
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°
- Tangent perpendicular to radius at point of contact
- Two tangents from external point are equal
- Angle between tangent and chord = angle in alternate segment
Power of Point
For point P outside circle with secants: PA × PB = PC × PD
For point P with tangent PT and secant PAB: PT² = PA × PB
Power of point = |OP² - r²| where O is center
Common Tangents
Two external tangents length = √(d² - (r₁ - r₂)²)
Two internal tangents length = √(d² - (r₁ + r₂)²) where d = distance between centers
Polygons
Regular Polygon (n sides)
Each interior angle = (n-2) × 180°/n
Each exterior angle = 360°/n
Sum of interior angles = (n-2) × 180°
Sum of exterior angles = 360°
Number of diagonals = n(n-3)/2
Specific Regular Polygons
Pentagon (n = 5)
- Each interior angle = 108°
- Each exterior angle = 72°
Hexagon (n = 6)
- Each interior angle = 120°
- Each exterior angle = 60°
- Area = (3√3/2) × s² (s = side)
Octagon (n = 8)
- Each interior angle = 135°
- Each exterior angle = 45°
5. Coordinate Geometry
Distance Formula
Distance between (x₁,y₁) and (x₂,y₂) = √[(x₂-x₁)² + (y₂-y₁)²]
Section Formula
Internal Division
Point dividing line segment in ratio m:n internally:
x = (mx₂ + nx₁)/(m+n)
y = (my₂ + ny₁)/(m+n)
External Division
Point dividing line segment in ratio m:n externally:
x = (mx₂ - nx₁)/(m-n)
y = (my₂ - ny₁)/(m-n)
Midpoint Formula
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Centroid Formula
Centroid of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃):
G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
Area of Triangle
Area = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|
For collinear points: Area = 0
Straight Lines
Slope of Line
Slope = (y₂-y₁)/(x₂-x₁) = tan θ
Slope of line making angle θ with x-axis = tan θ
Equation of Line
Point-Slope Form
y - y₁ = m(x - x₁)
Slope-Intercept Form
y = mx + c (m = slope, c = y-intercept)
Two-Point Form
(y-y₁)/(y₂-y₁) = (x-x₁)/(x₂-x₁)
Intercept Form
x/a + y/b = 1 (a = x-intercept, b = y-intercept)
Normal Form
x cos α + y sin α = p (p = perpendicular distance from origin)
General Form
Ax + By + C = 0
Slope = -A/B
x-intercept = -C/A
y-intercept = -C/B
Properties of Lines
Parallel Lines
Slopes are equal: m₁ = m₂
Distance between parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0:
Distance = |C₁ - C₂|/√(A² + B²)
Perpendicular Lines
Product of slopes = -1: m₁ × m₂ = -1
If one line has slope m, perpendicular line has slope -1/m
Angle Between Lines
tan θ = |(m₁-m₂)/(1+m₁m₂)|
For lines A₁x + B₁y + C₁ = 0 and A₂x + B₂y + C₂ = 0:
tan θ = |(A₁B₂ - A₂B₁)/(A₁A₂ + B₁B₂)|
Distance from Point to Line
Distance from (x₀,y₀) to Ax + By + C = 0:
d = |Ax₀ + By₀ + C|/√(A² + B²)
Circles in Coordinate Geometry
Standard Form
Center (h,k), radius r: (x-h)² + (y-k)² = r²
Center at origin: x² + y² = r²
General Form
x² + y² + 2gx + 2fy + c = 0
Center = (-g, -f)
Radius = √(g² + f² - c)
Condition for real circle: g² + f² - c > 0
Equation of Tangent
At point (x₁,y₁) on circle x² + y² = r²: xx₁ + yy₁ = r²
At point (x₁,y₁) on circle (x-h)² + (y-k)² = r²: (x-h)(x₁-h) + (y-k)(y₁-k) = r²
Length of Tangent
From external point (x₁,y₁) to circle x² + y² = r²:
Length = √(x₁² + y₁² - r²)
Parabola
Standard Forms
y² = 4ax (opens rightward)
- Vertex: (0,0)
- Focus: (a,0)
- Directrix: x = -a
- Axis: y = 0
(y-k)² = 4a(x-h) (vertex at (h,k))
x² = 4ay (opens upward)
x² = -4ay (opens downward)
(x-h)² = 4a(y-k) (vertex at (h,k))
Ellipse
Standard Form
x²/a² + y²/b² = 1 (center at origin)
- Semi-major axis = a (if a > b)
- Semi-minor axis = b
- Eccentricity e = √(1 - b²/a²) (if a > b)
- Foci: (±ae, 0) if a > b
- Directrices: x = ±a/e
Hyperbola
Standard Form
x²/a² - y²/b² = 1
- Eccentricity e = √(1 + b²/a²)
- Foci: (±ae, 0)
- Directrices: x = ±a/e
- Asymptotes: y = ±(b/a)x
6. Trigonometry
Basic Trigonometric Ratios
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent = sin θ/cos θ
cosec θ = 1/sin θ = hypotenuse/opposite
sec θ = 1/cos θ = hypotenuse/adjacent
cot θ = 1/tan θ = cos θ/sin θ = adjacent/opposite
Fundamental Identities
Pythagorean Identities
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
Reciprocal Identities
sin θ × cosec θ = 1
cos θ × sec θ = 1
tan θ × cot θ = 1
Quotient Identities
tan θ = sin θ/cos θ
cot θ = cos θ/sin θ
Standard Angles and Values
| Angle | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1/2 | -1/√2 | -√3/2 | -1 | 0 | 1 |
| tan | 0 | 1/√3 | 1 | √3 | ∞ | -√3 | -1 | -1/√3 | 0 | ∞ | 0 |
Signs in Different Quadrants
- First Quadrant (0° to 90°): All positive
- Second Quadrant (90° to 180°): Only sin positive
- Third Quadrant (180° to 270°): Only tan positive
- Fourth Quadrant (270° to 360°): Only cos positive
- Mnemonic: All Students Take Chemistry
Compound Angle Formulas
Addition Formulas
sin(A + B) = sin A cos B + cos A sin B
sin(A - B) = sin A cos B - cos A sin B
cos(A + B) = cos A cos B - sin A sin B
cos(A - B) = cos A cos B + sin A sin B
tan(A + B) = (tan A + tan B)/(1 - tan A tan B)
tan(A - B) = (tan A - tan B)/(1 + tan A tan B)
Multiple Angle Formulas
Double Angle Formulas
sin 2A = 2 sin A cos A
cos 2A = cos²A - sin²A = 2cos²A - 1 = 1 - 2sin²A
tan 2A = 2tan A/(1 - tan²A)
Triple Angle Formulas
sin 3A = 3sin A - 4sin³A
cos 3A = 4cos³A - 3cos A
tan 3A = (3tan A - tan³A)/(1 - 3tan²A)
Half Angle Formulas
sin²(A/2) = (1 - cos A)/2
cos²(A/2) = (1 + cos A)/2
tan²(A/2) = (1 - cos A)/(1 + cos A)
tan(A/2) = sin A/(1 + cos A) = (1 - cos A)/sin A
Product-to-Sum Formulas
sin A sin B = ½[cos(A-B) - cos(A+B)]
cos A cos B = ½[cos(A-B) + cos(A+B)]
sin A cos B = ½[sin(A+B) + sin(A-B)]
cos A sin B = ½[sin(A+B) - sin(A-B)]
Sum-to-Product Formulas
sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)
sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2)
cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)
cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2)
Transformation Formulas
a sin θ + b cos θ = √(a² + b²) sin(θ + α) where tan α = b/a
a sin θ + b cos θ = √(a² + b²) cos(θ - β) where tan β = a/b
Conditional Identities
If A + B + C = π (180°)
sin(A + B) = sin C
cos(A + B) = -cos C
tan(A + B) = -tan C
sin(A/2) sin(B/2) sin(C/2) = sin(A/2 + B/2 + C/2) = cos((A+B+C)/2) = cos(π/2) = 0
cos(A/2) cos(B/2) cos(C/2) = sin((A+B+C)/2) = sin(π/2) = 1
If A + B + C = π/2 (90°)
sin(A + B) = cos C
cos(A + B) = sin C
tan(A + B) = cot C
Inverse Trigonometric Functions
Domains and Ranges
- sin⁻¹x: Domain [-1,1], Range [-π/2, π/2]
- cos⁻¹x: Domain [-1,1], Range [0, π]
- tan⁻¹x: Domain (-∞,∞), Range (-π/2, π/2)
Properties
sin⁻¹x + cos⁻¹x = π/2
tan⁻¹x + cot⁻¹x = π/2
sin⁻¹(-x) = -sin⁻¹x
cos⁻¹(-x) = π - cos⁻¹x
tan⁻¹(-x) = -tan⁻¹x
Addition Formulas for Inverse Functions
tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1-xy)) if xy < 1
tan⁻¹x - tan⁻¹y = tan⁻¹((x-y)/(1+xy))
2tan⁻¹x = tan⁻¹(2x/(1-x²)) if |x| < 1
Heights and Distances
Angle of Elevation and Depression
- Angle of elevation: Angle above horizontal
- Angle of depression: Angle below horizontal
Basic Formulas
Height = Distance × tan(angle of elevation)
Distance = Height/tan(angle of elevation)
If angle changes from α to β: New height = Distance × tan β
7. Mensuration
2D Mensuration
Triangle
Area = ½ × base × height
Perimeter = a + b + c
Area = ½ab sin C (where C is angle between sides a and b)
Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Rectangle
Area = length × breadth
Perimeter = 2(length + breadth)
Diagonal = √(length² + breadth²)
Square
Area = side²
Perimeter = 4 × side
Diagonal = side√2
Parallelogram
Area = base × height
Area = ab sin θ (θ = angle between adjacent sides)
Perimeter = 2(a + b)
Rhombus
Area = base × height
Area = ½ × d₁ × d₂ (d₁, d₂ are diagonals)
Area = a² sin θ (a = side, θ = angle)
Perimeter = 4a
Trapezium
Area = ½(sum of parallel sides) × height
Area = ½(a + b) × h
Perimeter = sum of all four sides
Circle
Area = πr² = πd²/4
Circumference = 2πr = πd
Area of sector = (θ/360°) × πr² (θ in degrees)
Area of sector = ½r²θ (θ in radians)
Arc length = (θ/360°) × 2πr (θ in degrees)
Arc length = rθ (θ in radians)
Segment of Circle
Area of segment = Area of sector - Area of triangle
Area of minor segment = (θ/360°)πr² - ½r² sin θ
Area of major segment = πr² - Area of minor segment
Ellipse
Area = πab (a, b are semi-axes)
Perimeter ≈ π[3(a+b) - √((3a+b)(a+3b))] (Ramanujan's approximation)
3D Mensuration
Cube
Volume = a³
Surface Area = 6a²
Space diagonal = a√3
Face diagonal = a√2
Cuboid (Rectangular Parallelepiped)
Volume = length × breadth × height = lbh
Surface Area = 2(lb + bh + hl)
Space diagonal = √(l² + b² + h²)
Cylinder
Volume = πr²h
Curved Surface Area = 2πrh
Total Surface Area = 2πr(r + h)
If cylinder is hollow: Volume = π(R² - r²)h
Cone
Volume = ⅓πr²h
Curved Surface Area = πrl (l = slant height)
Total Surface Area = πr(r + l)
Slant height l = √(r² + h²)
Frustum of Cone
Volume = (πh/3)(R² + Rr + r²) (R, r = radii of bases)
Curved Surface Area = π(R + r)s (s = slant height of frustum)
Total Surface Area = πR² + πr² + π(R + r)s
Slant height s = √[h² + (R - r)²]
Sphere
Volume = (4/3)πr³
Surface Area = 4πr²
Diameter = 2r
Hemisphere
Volume = (2/3)πr³
Curved Surface Area = 2πr²
Total Surface Area = 3πr²
Spherical Shell
Volume = (4π/3)(R³ - r³) (R, r = outer and inner radii)
Surface Area = 4π(R² + r²)
Pyramid
Volume = ⅓ × Base Area × Height
For triangular pyramid: Volume = ⅓ × ½ × base × height of base × height of pyramid
Prism
Volume = Base Area × Height
Surface Area = 2 × Base Area + Perimeter of base × Height
Tetrahedron (Regular)
Volume = a³/(6√2) (a = edge)
Surface Area = √3 a²
Height = a√(2/3)
Octahedron (Regular)
Volume = (√2/3)a³ (a = edge)
Surface Area = 2√3 a²
Combined Solids
Cylinder with Hemisphere
Volume = πr²h + (2/3)πr³
Surface Area = 2πrh + 3πr²
Cone with Hemisphere
Volume = (1/3)πr²h + (2/3)πr³
Surface Area = πrl + 2πr²
8. Statistics & Probability
Measures of Central Tendency
Ungrouped Data
Mean (Arithmetic Mean)
Mean = (Sum of all observations)/(Number of observations)
Mean = (x₁ + x₂ + ... + xₙ)/n
If data has frequencies: Mean = Σ(f × x)/Σf
Median
For odd n: Median = ((n+1)/2)th observation
For even n: Median = [n/2 th observation + (n/2 + 1)th observation]/2
Mode
Most frequently occurring value
For grouped data with equal class intervals: Mode = L + [(f₁-f₀)/(2f₁-f₀-f₂)] × h where L = lower boundary of modal class, f₁ = frequency of modal class, f₀ = frequency of class before modal class, f₂ = frequency of class after modal class, h = class interval
Grouped Data
Mean
Direct Method: Mean = Σ(f × x)/Σf
Assumed Mean Method: Mean = A + (Σf × d)/Σf where d = x - A
Step Deviation Method: Mean = A + [(Σf × u)/Σf] × h where u = (x - A)/h
Median
Median = L + [(n/2 - CF)/f] × h where L = lower boundary of median class, CF = cumulative frequency before median class, f = frequency of median class, h = class interval
Relationship between Mean, Median, Mode
For symmetric distribution: Mean = Median = Mode
For moderately skewed distribution: Mode = 3Median - 2Mean
Mean - Mode = 3(Mean - Median)
Measures of Dispersion
Range
Range = Highest value - Lowest value
Mean Deviation
Mean Deviation from Mean = Σ|x - x̄|/n
Mean Deviation from Median = Σ|x - M|/n
Variance and Standard Deviation
For Ungrouped Data
Variance (σ²) = Σ(x - x̄)²/n
Standard Deviation (σ) = √[Σ(x - x̄)²/n]
Alternative formula: σ² = (Σx²/n) - (x̄)²
For Grouped Data
Variance = Σf(x - x̄)²/Σf
Standard Deviation = √[Σf(x - x̄)²/Σf]
Coefficient of Variation
CV = (Standard Deviation/Mean) × 100%
Probability
Basic Definitions
Probability of event E: P(E) = (Number of favorable outcomes)/(Total number of possible outcomes)
0 ≤ P(E) ≤ 1
P(E) + P(not E) = 1
P(not E) = 1 - P(E)
Addition Rule
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
For mutually exclusive events: P(A ∪ B) = P(A) + P(B)
For three events: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(A ∩ C) + P(A ∩ B ∩ C)
Multiplication Rule
P(A ∩ B) = P(A) × P(B|A) = P(B) × P(A|B)
For independent events: P(A ∩ B) = P(A) × P(B)
Conditional Probability
P(A|B) = P(A ∩ B)/P(B) provided P(B) ≠ 0
P(B|A) = P(A ∩ B)/P(A) provided P(A) ≠ 0
Bayes' Theorem
P(A|B) = [P(B|A) × P(A)]/P(B)
For multiple events: P(Aᵢ|B) = [P(B|Aᵢ) × P(Aᵢ)]/[Σ P(B|Aⱼ) × P(Aⱼ)]
Independent Events
Events A and B are independent if: P(A ∩ B) = P(A) × P(B)
Equivalently: P(A|B) = P(A) and P(B|A) = P(B)
Permutations and Combinations
Fundamental Principle of Counting
If task 1 can be done in m ways and task 2 in n ways: Total ways = m × n
Permutations
nPr = n!/(n-r)! (arrangements of r objects from n objects)
nPn = n! (arrangements of n objects)
Circular permutations: (n-1)!
Circular permutations with identical objects: (n-1)!/k! where k objects are identical
Combinations
nCr = n!/[r!(n-r)!] (selections of r objects from n objects)
nCr = nC(n-r)
nCr + nC(r-1) = (n+1)Cr
Sum of all combinations: 2ⁿ = nC0 + nC1 + nC2 + ... + nCn
Special Cases
Arrangements with repetition: If n objects where n₁ are of one type, n₂ of another type, etc.
Total arrangements = n!/(n₁! × n₂! × ... × nₖ!)
Derangements
Number of derangements of n objects: Dn = n! × [1 - 1/1! + 1/2! - 1/3! + ... + (-1)ⁿ/n!]
Binomial Distribution
P(X = r) = nCr × pʳ × q^(n-r) where q = 1-p
Mean = np
Variance = npq
Standard Deviation = √(npq)
9. Set Theory
Basic Set Notation
Set: A = {a₁, a₂, a₃, ...}
Empty Set: ∅ or {}
Universal Set: U
Cardinality: |A| or n(A) = number of elements in set A
Set Relationships
Subset: A ⊆ B (A is subset of B)
Proper Subset: A ⊂ B (A is proper subset of B)
Superset: A ⊇ B (A is superset of B)
Equal Sets: A = B ⟺ A ⊆ B and B ⊆ A
Set Operations
Union
Formula: A ∪ B = {x : x ∈ A or x ∈ B}
Properties:
- A ∪ A = A (Idempotent)
- A ∪ ∅ = A (Identity)
- A ∪ U = U (Domination)
- A ∪ B = B ∪ A (Commutative)
- (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associative)
Intersection
Formula: A ∩ B = {x : x ∈ A and x ∈ B}
Properties:
- A ∩ A = A (Idempotent)
- A ∩ ∅ = ∅ (Domination)
- A ∩ U = A (Identity)
- A ∩ B = B ∩ A (Commutative)
- (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associative)
Complement
Formula: A' = Aᶜ = {x : x ∈ U and x ∉ A}
Properties:
- (A')' = A (Double Complement)
- A ∪ A' = U
- A ∩ A' = ∅
- U' = ∅
- ∅' = U
Difference
Formula: A - B = A \ B = {x : x ∈ A and x ∉ B}
Symmetric Difference: A △ B = (A - B) ∪ (B - A) = (A ∪ B) - (A ∩ B)
De Morgan's Laws
(A ∪ B)' = A' ∩ B'
(A ∩ B)' = A' ∪ B'
General Form: (⋃ᵢAᵢ)' = ⋂ᵢAᵢ' and (⋂ᵢAᵢ)' = ⋃ᵢAᵢ'
Distributive Laws
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Inclusion-Exclusion Principle
Two Sets: |A ∪ B| = |A| + |B| - |A ∩ B|
Three Sets: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|
General Form: |⋃ᵢ₌₁ⁿ Aᵢ| = Σ|Aᵢ| - Σ|Aᵢ ∩ Aⱼ| + Σ|Aᵢ ∩ Aⱼ ∩ Aₖ| - ... + (-1)ⁿ⁺¹|A₁ ∩ A₂ ∩ ... ∩ Aₙ|
Cartesian Product
Formula: A × B = {(a,b) : a ∈ A and b ∈ B}
Cardinality: |A × B| = |A| × |B|
Properties:
- A × ∅ = ∅ × A = ∅
- A × (B ∪ C) = (A × B) ∪ (A × C)
- A × (B ∩ C) = (A × B) ∩ (A × C)
Power Set
Formula: P(A) = 2^A = {X : X ⊆ A}
Cardinality: |P(A)| = 2^|A|
10. Functions & Graphs
Function Definition
Function: f: A → B where each element in A maps to exactly one element in B
Domain: Dom(f) = A
Codomain: Cod(f) = B
Range: R(f) = {f(x) : x ∈ A} ⊆ B
Types of Functions
One-to-One (Injective)
Definition: f(x₁) = f(x₂) ⟹ x₁ = x₂
Horizontal Line Test: Each horizontal line intersects graph at most once
Onto (Surjective)
Definition: For every y ∈ B, there exists x ∈ A such that f(x) = y
Range equals Codomain: R(f) = B
Bijective
Definition: Function that is both one-to-one and onto
Has Inverse: f⁻¹ exists
Function Operations
Composition
(f ∘ g)(x) = f(g(x))
Domain: {x ∈ Dom(g) : g(x) ∈ Dom(f)}
Properties:
- (f ∘ g) ∘ h = f ∘ (g ∘ h) (Associative)
- f ∘ I = I ∘ f = f (Identity)
Arithmetic Operations
(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(f · g)(x) = f(x) · g(x)
(f/g)(x) = f(x)/g(x), where g(x) ≠ 0
Inverse Functions
Definition: f⁻¹(y) = x ⟺ f(x) = y
Condition: f must be bijective
Properties:
- f⁻¹(f(x)) = x for all x ∈ Dom(f)
- f(f⁻¹(y)) = y for all y ∈ R(f)
- (f ∘ g)⁻¹ = g⁻¹ ∘ f⁻¹
Common Function Types
Linear Functions
Form: f(x) = mx + b
Slope: m = (y₂ - y₁)/(x₂ - x₁)
y-intercept: b
Point-Slope Form: y - y₁ = m(x - x₁)
Quadratic Functions
Standard Form: f(x) = ax² + bx + c (a ≠ 0)
Vertex Form: f(x) = a(x - h)² + k
Vertex: (-b/2a, f(-b/2a)) or (h, k)
Discriminant: Δ = b² - 4ac
Roots: x = (-b ± √Δ)/(2a)
Polynomial Functions
General Form: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Degree: n (highest power)
Leading Coefficient: aₙ
Rational Functions
Form: f(x) = P(x)/Q(x) where P(x) and Q(x) are polynomials
Domain: All real numbers except where Q(x) = 0
Vertical Asymptotes: x = a where Q(a) = 0 and P(a) ≠ 0
Horizontal Asymptotes:
- If deg(P) < deg(Q): y = 0
- If deg(P) = deg(Q): y = leading coefficient of P / leading coefficient of Q
- If deg(P) > deg(Q): No horizontal asymptote
Exponential Functions
Form: f(x) = aˣ (a > 0, a ≠ 1)
Properties:
- aˣ · aʸ = aˣ⁺ʸ
- aˣ / aʸ = aˣ⁻ʸ
- (aˣ)ʸ = aˣʸ
- a⁰ = 1
- a⁻ˣ = 1/aˣ
Absolute Value Functions
Definition: |x| = x if x ≥ 0, |x| = -x if x < 0
Properties:
- |xy| = |x||y|
- |x/y| = |x|/|y| (y ≠ 0)
- |x + y| ≤ |x| + |y| (Triangle Inequality)
- ||x| - |y|| ≤ |x - y|
Graph Transformations
- Vertical Shift: f(x) + k (up if k > 0, down if k < 0)
- Horizontal Shift: f(x - h) (right if h > 0, left if h < 0)
- Vertical Stretch/Compression: af(x) (stretch if |a| > 1, compression if 0 < |a| < 1)
- Horizontal Stretch/Compression: f(bx) (compression if |b| > 1, stretch if 0 < |b| < 1)
- Reflection: -f(x) (across x-axis), f(-x) (across y-axis)
11. Logarithms
Basic Definition
Logarithm: y = log_a(x) ⟺ aʸ = x
Common Logarithm: log(x) = log₁₀(x)
Natural Logarithm: ln(x) = log_e(x) where e ≈ 2.71828
Domain and Range
Domain: x > 0
Range: All real numbers
Vertical Asymptote: x = 0
x-intercept: (1, 0)
Fundamental Properties
Basic Properties
log_a(1) = 0 (since a⁰ = 1)
log_a(a) = 1 (since a¹ = a)
log_a(aˣ) = x (since aˣ = aˣ)
a^(log_a(x)) = x (for x > 0)
Logarithm Laws
Product Rule
log_a(xy) = log_a(x) + log_a(y)
Quotient Rule
log_a(x/y) = log_a(x) - log_a(y)
Power Rule
log_a(xⁿ) = n·log_a(x)
Root Rule
log_a(ⁿ√x) = log_a(x^(1/n)) = (1/n)·log_a(x)
Change of Base Formula
log_a(x) = log_b(x) / log_b(a)
log_a(x) = ln(x) / ln(a)
log_a(x) = log(x) / log(a)
Special Relationships
log_a(b) · log_b(a) = 1
log_a(b) = 1 / log_b(a)
log_a(x) = log_b(x) · log_a(b)
Exponential-Logarithmic Relationships
If aˣ = b, then x = log_a(b)
If log_a(x) = y, then x = aʸ
a^(log_a(x)) = x (x > 0)
log_a(aˣ) = x (all real x)
Natural Logarithm Properties
ln(e) = 1
ln(1) = 0
ln(eˣ) = x
e^(ln(x)) = x (x > 0)
d/dx[ln(x)] = 1/x
∫(1/x)dx = ln|x| + C
Common Logarithm Values
log(1) = 0
log(10) = 1
log(100) = 2
log(1000) = 3
log(0.1) = -1
log(0.01) = -2
Solving Logarithmic Equations
Type 1: log_a(x) = b
Solution: x = a^b
Type 2: log_a(f(x)) = log_a(g(x))
Solution: f(x) = g(x) (with domain restrictions)
Type 3: log_a(f(x)) + log_a(g(x)) = c
Solution: f(x)·g(x) = a^c
Type 4: log_a(f(x)) - log_a(g(x)) = c
Solution: f(x)/g(x) = a^c
Solving Exponential Equations
Type 1: aˣ = b
Solution: x = log_a(b)
Type 2: a^(f(x)) = a^(g(x))
Solution: f(x) = g(x)
Type 3: a^(f(x)) = b
Solution: f(x) = log_a(b)
12. Complex Numbers
Basic Definition
Complex Number: z = a + bi
Real Part: Re(z) = a
Imaginary Part: Im(z) = b
Imaginary Unit: i² = -1, i = √(-1)
Powers of i
i⁰ = 1
i¹ = i
i² = -1
i³ = -i
i⁴ = 1
Pattern: i^n = i^(n mod 4)
Complex Number Operations
Addition
(a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction
(a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Special Case: (a + bi)(a - bi) = a² + b²
Division
(a + bi)/(c + di) = [(a + bi)(c - di)]/[(c + di)(c - di)] = [(ac + bd) + (bc - ad)i]/(c² + d²)
Complex Conjugate
Definition: If z = a + bi, then z̄ = a - bi
Properties:
- z + z̄ = 2a = 2Re(z)
- z - z̄ = 2bi = 2iIm(z)
- z·z̄ = a² + b² = |z|²
- (z₁ + z₂)‾ = z̄₁ + z̄₂
- (z₁·z₂)‾ = z̄₁·z̄₂
- (z₁/z₂)‾ = z̄₁/z̄₂
Modulus (Absolute Value)
Definition: |z| = |a + bi| = √(a² + b²)
Properties:
- |z| ≥ 0, |z| = 0 ⟺ z = 0
- |z₁·z₂| = |z₁|·|z₂|
- |z₁/z₂| = |z₁|/|z₂|
- |z₁ + z₂| ≤ |z₁| + |z₂| (Triangle Inequality)
- ||z₁| - |z₂|| ≤ |z₁ - z₂|
Polar Form
Rectangular Form: z = a + bi
Polar Form: z = r(cos θ + i sin θ) = r·cis θ
Exponential Form: z = re^(iθ)
Conversion Formulas
r = |z| = √(a² + b²)
θ = arg(z) = tan⁻¹(b/a) (with quadrant adjustment)
a = r cos θ
b = r sin θ
Argument (Angle)
Principal Argument: -π < θ ≤ π or 0 ≤ θ < 2π
General Argument: θ + 2πn where n ∈ ℤ
De Moivre's Theorem
Formula: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)
Exponential Form: (re^(iθ))ⁿ = rⁿe^(inθ)
Roots of Complex Numbers
nth Roots
Formula: If z = r(cos θ + i sin θ), then the n distinct nth roots are:
z^(1/n) = r^(1/n)[cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)]
where k = 0, 1, 2, ..., n-1
Square Roots
If z = a + bi, then √z = ±(√((|z| + a)/2) + i·sign(b)·√((|z| - a)/2))
For z = r·cis θ: √z = ±√r·cis(θ/2)
Cube Roots of Unity
1, ω, ω² where ω = cos(2π/3) + i sin(2π/3) = -1/2 + i√3/2
Properties: ω³ = 1, 1 + ω + ω² = 0
Euler's Formula
e^(iθ) = cos θ + i sin θ
e^(iπ) + 1 = 0 (Euler's Identity)
cos θ = (e^(iθ) + e^(-iθ))/2
sin θ = (e^(iθ) - e^(-iθ))/(2i)
Quadratic Equations with Complex Roots
For ax² + bx + c = 0 with Δ = b² - 4ac < 0:
Roots: x = (-b ± i√|Δ|)/(2a)
Complex roots always occur in conjugate pairs
Complex Plane
Real Axis: Horizontal axis (a-axis)
Imaginary Axis: Vertical axis (bi-axis)
Point Representation: z = a + bi ↔ (a, b)
Distance Formula: |z₁ - z₂| = distance between points z₁ and z₂
Fundamental Theorem of Algebra
Every polynomial of degree n has exactly n complex roots (counting multiplicity)
Every polynomial with real coefficients has complex roots in conjugate pairs