Algebra

Basic Algebraic Formulas

Square Formulas

(a + b)²
= a² + 2ab + b²
(a - b)²
= a² - 2ab + b²
(a + b)(a - b)
= a² - b²

Cube Formulas

(a + b)³
= a³ + 3a²b + 3ab² + b³
(a - b)³
= a³ - 3a²b + 3ab² - b³

Quadratic Equations

For equation ax² + bx + c = 0

Roots Formula
x = [-b ± √(b² - 4ac)] / 2a
Sum of Roots
x₁ + x₂ = -b/a
Product of Roots
x₁ × x₂ = c/a

Geometry

Area Formulas

2D Shapes

Rectangle
Area = length × width
Square
Area = side²
Triangle
Area = ½ × base × height
Circle
Area = πr²

3D Shapes

Cube
Volume = side³
Surface Area = 6 × side²
Sphere
Volume = (4/3)πr³
Surface Area = 4πr²
Cylinder
Volume = πr²h
Surface Area = 2πr(r + h)

Trigonometry

Basic Ratios

Sine
sin θ = Opposite/Hypotenuse
Cosine
cos θ = Adjacent/Hypotenuse
Tangent
tan θ = Opposite/Adjacent

Important Identities

Pythagorean Identity
sin²θ + cos²θ = 1
Tangent Identity
tan θ = sin θ/cos θ

Calculus

Differentiation Rules

Power Rule
d/dx(xⁿ) = n×xⁿ⁻¹
Product Rule
d/dx(u×v) = u×dv/dx + v×du/dx
Chain Rule
d/dx(f(g(x))) = f'(g(x))×g'(x)

Integration Rules

Power Rule
∫xⁿdx = xⁿ⁺¹/(n+1) + C
Basic Integrals
∫sin(x)dx = -cos(x) + C
∫cos(x)dx = sin(x) + C

Statistics and Probability

Central Tendency

Mean (Arithmetic)
x̄ = (x₁ + x₂ + ... + xₙ)/n
Median (Odd n)
(n+1)/2th term
Median (Even n)
Average of n/2th and (n/2+1)th terms
Mode
Most frequent value

Dispersion

Range
Highest value - Lowest value
Variance
σ² = Σ(x - μ)²/n
Standard Deviation
σ = √(Σ(x - μ)²/n)

Probability

Basic Probability
P(E) = Number of favorable outcomes / Total number of possible outcomes
Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
Multiplication Rule
P(A and B) = P(A) × P(B|A)

Matrices and Determinants

Matrix Operations

Addition
[aᵢⱼ] + [bᵢⱼ] = [aᵢⱼ + bᵢⱼ]
Scalar Multiplication
k[aᵢⱼ] = [kaᵢⱼ]
Matrix Multiplication
[AB]ᵢⱼ = Σ(aᵢₖ × bₖⱼ)

Determinants

2×2 Matrix
|A| = ad - bc, where A = [a b; c d]
3×3 Matrix
|A| = a(ei-fh) - b(di-fg) + c(dh-eg)

Coordinate Geometry

Distance and Point Formulas

Distance Formula
d = √[(x₂-x₁)² + (y₂-y₁)²]
Section Formula
x = (mx₂ + nx₁)/(m+n), y = (my₂ + ny₁)/(m+n)
Midpoint Formula
x = (x₁+x₂)/2, y = (y₁+y₂)/2

Straight Lines

Slope
m = (y₂-y₁)/(x₂-x₁)
Point-Slope Form
y - y₁ = m(x - x₁)
Slope-Intercept Form
y = mx + c

Circles

Standard Form
(x-h)² + (y-k)² = r²
General Form
x² + y² + 2gx + 2fy + c = 0

Number Systems

Number Properties

Divisibility Rules
2: Last digit divisible by 2
3: Sum of digits divisible by 3
4: Last two digits divisible by 4
5: Last digit is 0 or 5
9: Sum of digits divisible by 9

Number Theory

LCM Formula
LCM(a,b) × GCD(a,b) = a × b
Arithmetic Sequence
aₙ = a₁ + (n-1)d
Sₙ = n/2[2a₁ + (n-1)d]
Geometric Sequence
aₙ = a₁rⁿ⁻¹
Sₙ = a₁(1-rⁿ)/(1-r)

Commercial Mathematics

Profit and Loss

Cost Price (CP)
CP = SP - Profit
CP = SP + Loss
Selling Price (SP)
SP = CP + Profit
SP = CP - Loss
Profit Percentage
Profit% = (Profit/CP) × 100
Loss Percentage
Loss% = (Loss/CP) × 100
Marked Price (MP)
MP = SP + Discount
Discount Percentage
Discount% = (Discount/MP) × 100

Percentage

Basic Formula
Percentage = (Value/Total Value) × 100
Percentage Change
% Change = [(New Value - Original Value)/Original Value] × 100
Successive Percentages
Net% = [(100 ± x%)(100 ± y%)/100] - 100

Partnership

Profit Sharing
Profit Share = (Individual Investment × Time × Total Profit)/(Total Investment × Time)
Simple Partnership
Profit Ratio = Investment Ratio
Compound Partnership
Profit Ratio = Investment × Time

Simple Interest

Interest Formula
SI = (P × R × T)/100
Where P = Principal, R = Rate%, T = Time (years)
Amount
A = P + SI
Principal
P = (100 × SI)/(R × T)
Rate
R = (100 × SI)/(P × T)
Time
T = (100 × SI)/(P × R)

Compound Interest

Annual Compounding
A = P(1 + R/100)ᵗ
CI = A - P
Semi-Annual Compounding
A = P(1 + R/200)²ᵗ
Quarterly Compounding
A = P(1 + R/400)⁴ᵗ
Principal
P = A/(1 + R/100)ᵗ
Time (using logs)
T = log(A/P)/log(1 + R/100)

Time Value Comparisons

CI vs SI Difference
Difference = CI - SI
CI for 2 years
CI = P[(1 + R/100)² - 1]
CI for 3 years
CI = P[(1 + R/100)³ - 1]

Mathematical Tricks and Fast Calculations

Multiplication Tricks

Multiply by 5

Method
Divide by 2 and multiply by 10
Example: 76 × 5 = (76 ÷ 2) × 10 = 38 × 10 = 380
Multiply by 25
Divide by 4 and multiply by 100
Example: 48 × 25 = (48 ÷ 4) × 100 = 12 × 100 = 1200
Multiply by 11
For 2-digit numbers: Sum the digits and place between them
Example: 43 × 11 = 4(4+3)3 = 473

Square Numbers

Numbers ending in 5
First digits × (First digits + 1) followed by 25
Example: 85² = 8 × 9 = 72, append 25 = 7225
Close to 100
Add/subtract difference from 100, square difference
Example: 98² = 100 - 2, square 2 = 9604
Difference of Squares
(a+b)(a-b) = a² - b²
Example: 53² - 47² = (53+47)(53-47) = 100 × 6 = 600

Division Tricks

Divide by 5
Multiply by 2 and divide by 10
Example: 85 ÷ 5 = (85 × 2) ÷ 10 = 170 ÷ 10 = 17
Divide by 25
Multiply by 4 and divide by 100
Example: 175 ÷ 25 = (175 × 4) ÷ 100 = 700 ÷ 100 = 7

Percentage Tricks

Find 10%
Move decimal point one place left
Example: 10% of 250 = 25
Find 5%
Half of 10%
Example: 5% of 250 = 25 ÷ 2 = 12.5
Find 20%
Double 10%
Example: 20% of 250 = 25 × 2 = 50

Number Properties

Divisibility by 3
Sum of digits divisible by 3
Example: 351 → 3+5+1 = 9 (divisible by 3)
Divisibility by 9
Sum of digits divisible by 9
Example: 729 → 7+2+9 = 18 → 1+8 = 9 (divisible by 9)
Divisibility by 11
Alternating sum of digits divisible by 11
Example: 143 → 1-4+3 = 0 (divisible by 11)

Mental Math Shortcuts

Multiply by 99
Multiply by 100 and subtract original number
Example: 45 × 99 = (45 × 100) - 45 = 4500 - 45 = 4455
Multiply by 101
Add two zeros and add original number
Example: 45 × 101 = 4500 + 45 = 4545
Find Square Root
For perfect squares, memorize squares up to 20
Example: √144 = 12 (since 12² = 144)