Mathematics Comprehensive Formula Sheet - JEE/NEET Essential Formulas

📋 Introduction

This comprehensive mathematics formula sheet contains all essential formulas, equations, and mathematical relationships needed for JEE Advanced and NEET preparation. Formulas are organized by topic and include conditions, applications, and problem-solving strategies.

📐 Algebra

Basic Algebraic Identities

Expansion Formulas:
(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
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

Factorization Formulas:
a³ + b³ = (a + b)(a² - ab + b²)
a³ - b³ = (a - b)(a² + ab + b²)
a² - b² = (a + b)(a - b)
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Important Relations:
(a + b)² + (a - b)² = 2(a² + b²)
(a + b)² - (a - b)² = 4ab
(a + b + c)³ = a³ + b³ + c³ + 3(a + b)(b + c)(c + a)

Quadratic Equations

Standard Form: ax² + bx + c = 0, a ≠ 0
Solutions (Roots): x = [-b ± √(b² - 4ac)]/2a

Discriminant: Δ = b² - 4ac
Nature of Roots:
Δ > 0: Two distinct real roots
Δ = 0: Two equal real roots
Δ < 0: Two complex conjugate roots

Sum of Roots: α + β = -b/a
Product of Roots: αβ = c/a

Formation of Quadratic Equation:
x² - (sum of roots)x + (product of roots) = 0

Roots with Special Conditions:
- Both roots positive: c/a > 0, -b/a > 0
- Both roots negative: c/a > 0, -b/a < 0
- Roots of opposite signs: c/a < 0

Maximum/Minimum Values:
Vertex: x = -b/(2a), y = -Δ/(4a)
Maximum value (a < 0): y_max = -Δ/(4a)
Minimum value (a > 0): y_min = -Δ/(4a)

Progressions

Arithmetic Progression (AP):
First term: a
Common difference: d
n-th term: a_n = a + (n-1)d
Sum of n terms: S_n = n/2[2a + (n-1)d] = n/2(a + a_n)

Important AP Properties:
a_m + a_n = a_p + a_q when m + n = p + q
Three terms in AP: a - d, a, a + d
Four terms in AP: a - 3d, a - d, a + d, a + 3d

Geometric Progression (GP):
First term: a
Common ratio: r
n-th term: a_n = ar^(n-1)
Sum of n terms: S_n = a(1 - r^n)/(1 - r), r ≠ 1
Sum to infinity: S_∞ = a/(1 - r), |r| < 1

Important GP Properties:
Three terms in GP: a/r, a, ar
Four terms in GP: a/r³, a/r, ar, ar³

Harmonic Progression (HP):
Terms are reciprocals of AP
If a, b, c are in HP: 2/b = 1/a + 1/c

Arithmetic Mean (AM): (a + b)/2
Geometric Mean (GM): √(ab)
Harmonic Mean (HM): 2ab/(a + b)
Relation: AM ≥ GM ≥ HM

Binomial Theorem

Positive Integer Index:
(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁a^(n-1)b + ⁿC₂a^(n-2)b² + ... + ⁿCₙbⁿ
General term: T_(r+1) = ⁿCᵣa^(n-r)bʳ

Binomial Coefficients:
ⁿCᵣ = n!/(r!(n-r)!)
ⁿC₀ = ⁿCₙ = 1
ⁿC₁ = ⁿC_(n-1) = n
ⁿCᵣ = ⁿC_(n-r)

Properties of Binomial Coefficients:
Sum of coefficients: ⁿC₀ + ⁿC₁ + ... + ⁿCₙ = 2ⁿ
Sum of even coefficients: ⁿC₀ + ⁿC₂ + ⁿC₄ + ... = 2^(n-1)
Sum of odd coefficients: ⁿC₁ + ⁿC₃ + ⁿC₅ + ... = 2^(n-1)

Middle Term:
If n is even: (n/2 + 1)th term
If n is odd: Two middle terms - (n+1)/2 and (n+3)/2 terms

Binomial Expansion for Any Index:
(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ...
Valid when |x| < 1

Permutations and Combinations

Fundamental Principle:
If m ways to do first thing and n ways to do second thing:
Total ways = m × n

Permutations (Arrangements):
nPr = n!/(n-r)!
nPr = number of ways to arrange r objects from n distinct objects

Special Cases:
nPn = n!
nP1 = n
nP0 = 1

Circular Permutations:
n distinct objects: (n-1)!
When clockwise and anti-clockwise are same: (n-1)!/2

Combinations (Selections):
nCr = n!/(r!(n-r)!)
nCr = number of ways to select r objects from n distinct objects

Properties:
nCr = nC_(n-r)
nCr + nC_(r+1) = (n+1)C_(r+1)
ΣnCr = 2ⁿ

Division into Groups:
n distinct objects into two groups of r and (n-r): nCr
n distinct objects into groups of r₁, r₂, ..., r_k: n!/(r₁!r₂!...r_k!)

Matrices and Determinants

Matrix Operations:
Addition: A + B = [a_ij + b_ij]
Scalar multiplication: kA = [ka_ij]
Matrix multiplication: AB = [Σa_ik × b_kj]

Types of Matrices:
Square matrix: Same number of rows and columns
Diagonal matrix: Only diagonal elements non-zero
Scalar matrix: Diagonal elements equal
Identity matrix: Diagonal elements = 1, others = 0
Zero matrix: All elements = 0

Determinant Properties:
det(AB) = det(A) × det(B)
det(kA) = kⁿdet(A) for n × n matrix
det(A^T) = det(A)

Determinant of 2×2 Matrix:
|a b|
|c d| = ad - bc

Determinant of 3×3 Matrix:
|a b c|
|d e f| = a(ei - fh) - b(di - fg) + c(dh - eg)
|g h i|

Inverse of Matrix:
A⁻¹ = Adj(A)/det(A)
AA⁻¹ = A⁻¹A = I

Solving Linear Equations:
For system AX = B
If det(A) ≠ 0: Unique solution X = A⁻¹B
If det(A) = 0: Either no solution or infinite solutions

📈 Calculus

Limits and Continuity

Standard Limits:
limₓ→a (xⁿ - aⁿ)/(x - a) = n·a^(n-1)
limₓ→0 (sin x)/x = 1
limₓ→0 (tan x)/x = 1
limₓ→0 (1 - cos x)/x = 0
limₓ→0 (aˣ - 1)/x = ln a
limₓ→0 (eˣ - 1)/x = 1
limₓ→∞ (1 + 1/x)ˣ = e

L'Hospital's Rule:
For 0/0 or ∞/∞ forms
limₓ→a f(x)/g(x) = limₓ→a f'(x)/g'(x)

Continuity:
f(x) is continuous at x = a if:
1. f(a) exists
2. limₓ→a f(x) exists
3. limₓ→a f(x) = f(a)

Discontinuities:
Removable: Limit exists but ≠ function value
Jump: Left and right limits exist but are different
Infinite: Function approaches ±∞

Differentiation

Basic Rules:
Power Rule: d/dx(xⁿ) = nx^(n-1)
Product Rule: d/dx(uv) = u'v + uv'
Quotient Rule: d/dx(u/v) = (u'v - uv')/v²
Chain Rule: d/dx[f(g(x))] = f'(g(x)) × g'(x)

Derivatives of Functions:
d/dx(sin x) = cos x
d/dx(cos x) = -sin x
d/dx(tan x) = sec² x
d/dx(cot x) = -cosec² x
d/dx(sec x) = sec x tan x
d/dx(cosec x) = -cosec x cot x

d/dx(eˣ) = eˣ
d/dx(aˣ) = aˣ ln a
d/dx(ln x) = 1/x
d/dx(logₐx) = 1/(x ln a)

Inverse Trigonometric Functions:
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))

Higher Order Derivatives:
d²y/dx² = d/dx(dy/dx)
d³y/dx³ = d/dx(d²y/dx²)

Applications of Derivatives

Increasing/Decreasing Functions:
f'(x) > 0: Increasing
f'(x) < 0: Decreasing
f'(x) = 0: Critical point

Maxima and Minima:
First Derivative Test:
- f'(x) changes from + to -: Maximum
- f'(x) changes from - to +: Minimum

Second Derivative Test:
f''(x) < 0: Maximum
f''(x) > 0: Minimum
f''(x) = 0: Test inconclusive

Tangent and Normal:
Equation of tangent: y - y₁ = f'(x₁)(x - x₁)
Equation of normal: y - y₁ = -1/f'(x₁)(x - x₁)
Slope of tangent: m = f'(x₁)
Slope of normal: m = -1/f'(x₁)

Rate of Change:
dy/dx: Rate of change of y with respect to x
Related Rates: Chain rule application

Approximation:
dy = f'(x)dx
Δy ≈ dy for small changes

Integration

Basic Integration Formulas:
∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ -1)
∫1/x dx = ln|x| + C
∫eˣ dx = eˣ + C
∫aˣ dx = aˣ/ln a + C

Trigonometric Integrals:
∫sin x dx = -cos x + C
∫cos x dx = sin x + C
∫sec² x dx = tan x + C
∫cosec² x dx = -cot x + C
∫sec x tan x dx = sec x + C
∫cosec x cot x dx = -cosec x + C

Integration by Parts:
∫u·dv = uv - ∫v·du
LIATE rule for choosing u:
L - Logarithmic, I - Inverse trigonometric, A - Algebraic, T - Trigonometric, E - Exponential

Integration by Substitution:
∫f(g(x))g'(x)dx = ∫f(u)du where u = g(x)

Definite Integration:
∫ₐᵇ f(x)dx = F(b) - F(a)
Properties:
∫ₐᵇ f(x)dx = -∫ᵇₐ f(x)dx
∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx
∫₀ᵃ f(x)dx = ∫₀ᵃ f(a-x)dx
∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx (if f is even)
∫₋ₐᵃ f(x)dx = 0 (if f is odd)

Differential Equations

First Order Differential Equations:
Variable Separable: dy/dx = f(x)g(y)
Solution: ∫dy/g(y) = ∫f(x)dx

Linear: dy/dx + P(x)y = Q(x)
Integrating factor: I.F. = e^(∫Pdx)
Solution: y·I.F. = ∫Q·I.F.·dx + C

Homogeneous: dy/dx = f(y/x)
Substitution: y = vx
dy/dx = v + x(dv/dx)

Exact Equation: Mdx + Ndy = 0
Condition: ∂M/∂y = ∂N/∂x
Solution: ∫Mdx + ∫(terms of N not in M)dy = C

Second Order Linear Differential Equations:
Homogeneous: ay'' + by' + cy = 0
Characteristic equation: ar² + br + c = 0
Solution depends on roots r₁, r₂

📊 Coordinate Geometry

Straight Lines

Slope: m = (y₂-y₁)/(x₂-x₁)
Equation Forms:
Slope-intercept: y = mx + c
Point-slope: y - y₁ = m(x - x₁)
Two-point: (y - y₁)/(x - x₁) = (y₂ - y₁)/(x₂ - x₁)
General: ax + by + c = 0
Intercept form: x/a + y/b = 1

Angle Between Lines:
tan θ = |(m₂ - m₁)/(1 + m₁m₂)|
Lines are parallel if m₁ = m₂
Lines are perpendicular if m₁m₂ = -1

Distance Formula:
Distance between (x₁,y₁) and (x₂,y₂): d = √[(x₂-x₁)² + (y₂-y₁)²]
Distance from (x₀,y₀) to line ax + by + c = 0: d = |ax₀ + by₀ + c|/√(a² + b²)

Area of Triangle:
A = (1/2)|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Collinear points if area = 0

Conic Sections

Circle:
Standard equation: (x - h)² + (y - k)² = r²
Center: (h, k), Radius: r
General equation: x² + y² + 2gx + 2fy + c = 0
Center: (-g, -f), Radius: √(g² + f² - c)

Parabola:
Standard forms:
y² = 4ax (right opening)
y² = -4ax (left opening)
x² = 4ay (upward opening)
x² = -4ay (downward opening)
Focus: (a, 0) or (0, a)
Directrix: x = -a or y = -a

Ellipse:
Standard equation: x²/a² + y²/b² = 1 (a > b)
Center: (0, 0)
Major axis: 2a, Minor axis: 2b
Foci: (±c, 0) where c² = a² - b²
Eccentricity: e = c/a < 1

Hyperbola:
Standard equation: x²/a² - y²/b² = 1
Center: (0, 0)
Transverse axis: 2a, Conjugate axis: 2b
Foci: (±c, 0) where c² = a² + b²
Eccentricity: e = c/a > 1
Asymptotes: y = ±(b/a)x

3D Geometry

Distance Between Points:
d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]

Direction Cosines:
If direction ratios are a, b, c:
Direction cosines: l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²)
l² + m² + n² = 1

Equation of Line:
Vector form: r = a + λb
Cartesian form: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c

Equation of Plane:
General form: ax + by + cz + d = 0
Normal vector: (a, b, c)
Distance from origin: |d|/√(a²+b²+c²)

Angle Between Planes:
cos θ = |a₁a₂ + b₁b₂ + c₁c₂|/[√(a₁²+b₁²+c₁²) × √(a₂²+b₂²+c₂²)]

📐 Trigonometry

Trigonometric Identities

Basic Identities:
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ

Reciprocal Identities:
cosec θ = 1/sin θ
sec θ = 1/cos θ
cot θ = 1/tan θ

Co-function Identities:
sin(90° - θ) = cos θ
cos(90° - θ) = sin θ
tan(90° - θ) = cot θ

Even-Odd Identities:
sin(-θ) = -sin θ
cos(-θ) = cos θ
tan(-θ) = -tan θ

Compound Angle Formulas

Sum Formulas:
sin(A + B) = sin A cos B + cos 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)

Difference Formulas:
sin(A - B) = sin A cos B - cos 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)

Multiple Angle Formulas

Double Angle Formulas:
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
tan 2θ = 2 tan θ/(1 - tan²θ)

Triple Angle Formulas:
sin 3θ = 3 sin θ - 4 sin³θ
cos 3θ = 4 cos³θ - 3 cos θ
tan 3θ = (3 tan θ - tan³θ)/(1 - 3 tan²θ)

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]

Product to Sum:
sin A cos B = [sin(A+B) + sin(A-B)]/2
cos A cos B = [cos(A+B) + cos(A-B)]/2
sin A sin B = [cos(A-B) - cos(A+B)]/2

Inverse Trigonometric Functions

Domain and Range:
sin⁻¹x: Domain [-1,1], Range [-π/2, π/2]
cos⁻¹x: Domain [-1,1], Range [0, π]
tan⁻¹x: Domain R, Range (-π/2, π/2)

Properties:
sin⁻¹(-x) = -sin⁻¹x
cos⁻¹(-x) = π - cos⁻¹x
tan⁻¹(-x) = -tan⁻¹x

Principal Values:
sin⁻¹x + cos⁻¹x = π/2
tan⁻¹x + cot⁻¹x = π/2
sec⁻¹x + cosec⁻¹x = π/2

Properties of Triangles

Sine Rule:
a/sin A = b/sin B = c/sin C = 2R

Cosine Rule:
a² = b² + c² - 2bc cos A
b² = a² + c² - 2ac cos B
c² = a² + b² - 2ab cos C

Area of Triangle:
Δ = (1/2)ab sin C = (1/2)bc sin A = (1/2)ca sin B
Δ = √[s(s-a)(s-b)(s-c)] (Heron's formula)
Δ = abc/(4R)

Tangent Rule:
(a - b)/(a + b) = tan[(A - B)/2]/tan[(A + B)/2]

📊 Statistics and Probability

Measures of Central Tendency

Mean (Arithmetic Mean):
x̄ = (Σx)/n (for ungrouped data)
x̄ = (Σfx)/(Σf) (for grouped data)

Median:
Middle value when arranged in order
For even number of terms: Average of middle two values
For grouped data: Median class calculation

Mode:
Most frequently occurring value
For grouped data: Modal class calculation

Relationship:
Mean - Mode = 3(Mean - Median)

Measures of Dispersion

Range: Maximum - Minimum
Mean Deviation: MD = Σ|x - x̄|/n
Variance: σ² = Σ(x - x̄)²/n
Standard Deviation: σ = √[Σ(x - x̄)²/n]

For grouped data:
σ² = [Σf(x - x̄)²]/Σf

Coefficient of Variation:
CV = (σ/x̄) × 100

Probability

Basic Probability:
P(A) = n(A)/n(S) = (Number of favorable outcomes)/(Total outcomes)

Complementary Events:
P(A') = 1 - P(A)

Mutually Exclusive Events:
P(A ∪ B) = P(A) + P(B)

Independent Events:
P(A ∩ B) = P(A) × P(B)

Conditional Probability:
P(A|B) = P(A ∩ B)/P(B)

Bayes' Theorem:
P(A|B) = [P(B|A) × P(A)]/[P(B|A) × P(A) + P(B|A') × P(A')]

🧮 Mathematical Reasoning

Logical Statements

Compound Statements:
Conjunction (AND): p ∧ q (both true)
Disjunction (OR): p ∨ q (at least one true)
Conditional: p → q (if p then q)
Biconditional: p ↔ q (p if and only if q)

Truth Tables:
Basic operations and their truth values
Tautology: Always true
Contradiction: Always false
Contingency: Sometimes true, sometimes false

Logical Equivalence:
Two statements with same truth values
De Morgan's Laws:
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q

🎯 Usage Tips

Formula Selection Strategy:

1. Identify the problem type and relevant concepts
2. Check conditions required for formula validity
3. Ensure consistent units throughout calculations
4. Verify answer using alternative methods when possible

Memory Techniques:

1. Group related formulas by topic or concept
2. Create acronyms for formula sequences
3. Practice derivation of important formulas
4. Use visual associations for complex relationships
5. Regular practice for better retention

Use this comprehensive mathematics formula sheet as your quick reference guide for JEE/NEET preparation! Regular practice with these formulas will significantly enhance your problem-solving speed and accuracy. 🎯