Mathematics Quick Notes - JEE/NEET Essential Concepts

📋 Introduction

These mathematics quick notes contain essential formulas, concepts, and problem-solving techniques for rapid revision before JEE Advanced and NEET exams. Focus on understanding the concepts and practicing regularly.

📐 Algebra

Quadratic Equations

🎯 Key Formulas:
Roots: x = [-b ± √(b² - 4ac)]/2a
Discriminant: Δ = b² - 4ac
Sum of roots: α + β = -b/a
Product of roots: αβ = c/a

💡 Important Points:
- Δ > 0: Two distinct real roots
- Δ = 0: Two equal real roots
- Δ < 0: Two complex conjugate roots
- Formation of equation: x² - (sum)x + (product) = 0

📌 Special Cases:
- Roots of unity: x³ = 1, roots = 1, ω, ω²
- Reciprocal equation: ax² + bx + a = 0
- Pure quadratic: bx + c = 0

⚠️ Common Mistakes:
- Wrong discriminant calculation
- Incorrect sum and product
- Wrong sign conventions

Progressions

🎯 Key Formulas:
AP: a_n = a + (n-1)d, S_n = n/2[2a + (n-1)d]
GP: a_n = ar^(n-1), S_n = a(1-r^n)/(1-r)
Sum to infinity (GP): S_∞ = a/(1-r), |r| < 1

💡 Important Points:
- Three terms in AP: a-d, a, a+d
- Three terms in GP: a/r, a, ar
- AM ≥ GM ≥ HM
- Geometric mean of positive numbers

📌 Important Relations:
- Arithmetic mean of a and b: (a+b)/2
- Geometric mean of a and b: √(ab)
- Harmonic mean of a and b: 2ab/(a+b)

⚠️ Common Mistakes:
- Wrong formula selection
- Incorrect term number
- Not checking convergence

Binomial Theorem

🎯 Key Formulas:
(a + b)ⁿ = ΣⁿCᵣa^(n-r)bʳ
General term: T_(r+1) = ⁿCᵣa^(n-r)bʳ
Middle term: depends on n being odd/even

💡 Important Points:
- ⁿCᵣ = ⁿC_(n-r)
- Sum of coefficients: 2ⁿ
- Sum of even coefficients: 2^(n-1)
- Sum of odd coefficients: 2^(n-1)

📌 Special Cases:
- (1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + ...
- Valid for |x| < 1 when n is not positive integer
- Pascal's triangle for coefficients

⚠️ Common Mistakes:
- Wrong coefficient calculation
- Incorrect general term
- Not checking convergence

Permutations and Combinations

🎯 Key Formulas:
nPr = n!/(n-r)!
nCr = n!/(r!(n-r)!)

💡 Important Points:
- Permutations: arrangements matter
- Combinations: selections only
- nC₀ = nCₙ = 1
- nC₁ = nC_(n-1) = n

📌 Special Cases:
- Circular permutations: (n-1)!
- With repetitions: n^r
- Division into groups: n!/(n₁!n₂!...nₖ!)

⚠️ Common Mistakes:
- Confusing P and C
- Wrong formula selection
- Not considering identical items

Matrices and Determinants

🎯 Key Formulas:
det([[a b],[c d]]) = ad - bc
A⁻¹ = Adj(A)/det(A)
det(AB) = det(A) × det(B)

💡 Important Points:
- Matrix multiplication: AB ≠ BA generally
- Identity matrix: I
- Zero matrix: all elements zero
- Symmetric matrix: A = Aᵀ

📌 Properties:
- det(Aᵀ) = det(A)
- det(kA) = kⁿdet(A) for n×n matrix
- det(A⁻¹) = 1/det(A)

⚠️ Common Mistakes:
- Wrong determinant calculation
- Incorrect matrix multiplication
- Not checking invertibility

📈 Calculus

Limits and Continuity

🎯 Key Formulas:
limₓ→0 (sin x)/x = 1
limₓ→a (xⁿ - aⁿ)/(x - a) = n·a^(n-1)
limₓ→∞ (1 + 1/x)ˣ = e

💡 Important Points:
- 0/0 and ∞/∞: indeterminate forms
- L'Hospital's rule for indeterminate forms
- Continuity: limit exists and equals function value
- Removable vs non-removable discontinuities

📌 Standard Limits:
- limₓ→0 (tan x)/x = 1
- limₓ→0 (1 - cos x)/x = 0
- limₓ→0 (eˣ - 1)/x = 1
- limₓ→0 (aˣ - 1)/x = ln a

⚠️ Common Mistakes:
- Direct substitution for indeterminate forms
- Wrong L'Hospital application
- Not checking continuity conditions

Differentiation

🎯 Key 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)

💡 Important Points:
- Derivative of sin x = cos x
- Derivative of cos x = -sin x
- Derivative of eˣ = eˣ
- Derivative of ln x = 1/x

📌 Important Derivatives:
- tan x: sec²x
- sec x: sec x tan x
- aˣ: aˣ ln a
- sin⁻¹x: 1/√(1-x²)

⚠️ Common Mistakes:
- Wrong rule application
- Chain rule omission
- Incorrect derivative of inverse functions

Applications of Derivatives

🎯 Key Concepts:
- Increasing function: f'(x) > 0
- Decreasing function: f'(x) < 0
- Maximum: f'(x) = 0, f''(x) < 0
- Minimum: f'(x) = 0, f''(x) > 0

💡 Important Points:
- First derivative test for extrema
- Second derivative test for concavity
- Tangent: y - y₁ = f'(x₁)(x - x₁)
- Normal: y - y₁ = -1/f'(x₁)(x - x₁)

📌 Applications:
- Rate of change problems
- Optimization problems
- Tangent and normal problems
- Approximation using differentials

⚠️ Common Mistakes:
- Wrong derivative calculation
- Incorrect maxima/minima identification
- Not checking second derivative

Integration

🎯 Key Formulas:
∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ -1)
∫1/x dx = ln|x| + C
∫eˣ dx = eˣ + C
∫sin x dx = -cos x + C

💡 Important Points:
- Integration by parts: ∫u·dv = uv - ∫v·du
- Substitution method: u = g(x)
- Definite integration: ∫ₐᵇ f(x)dx = F(b) - F(a)
- Properties of definite integrals

📌 Integration Techniques:
- Partial fractions for rational functions
- Trigonometric substitutions
- Special integrals
- Reduction formulas

⚠️ Common Mistakes:
- Wrong integral formula
- Incorrect substitution
- Missing constant of integration

Differential Equations

🎯 Key Types:
- Variable separable: dy/dx = f(x)g(y)
- Linear: dy/dx + P(x)y = Q(x)
- Homogeneous: dy/dx = f(y/x)
- Exact: Mdx + Ndy = 0

💡 Important Points:
- Integrating factor for linear equations
- Substitution for homogeneous equations
- Initial value problems
- Order and degree

📌 Solution Methods:
- Separate variables and integrate
- Find integrating factor: e^(∫Pdx)
- Use substitution: y = vx
- Check exactness condition

⚠️ Common Mistakes:
- Wrong equation identification
- Incorrect integrating factor
- Missing initial conditions

📊 Coordinate Geometry

Straight Lines

🎯 Key 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

💡 Important Points:
- Slope: m = (y₂ - y₁)/(x₂ - x₁)
- Parallel lines: same slope
- Perpendicular lines: m₁m₂ = -1
- Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]

📌 Special Cases:
- Equation of line parallel to ax + by + c = 0
- Equation of line perpendicular to ax + by + c = 0
- Distance from point to line

⚠️ Common Mistakes:
- Wrong slope calculation
- Incorrect line equation form
- Sign errors in distance formula

Conic Sections

🎯 Circle:
- Standard: (x - h)² + (y - k)² = r²
- Center: (h, k), Radius: r
- General: x² + y² + 2gx + 2fy + c = 0

Parabola:
- Standard: y² = 4ax
- Focus: (a, 0), Directrix: x = -a
- Latus rectum: 4a

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

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

💡 Important Points:
- Identify conic from equation
- Find center, focus, directrix
- Calculate eccentricity
- Sketch using key points

⚠️ Common Mistakes:
- Wrong conic identification
- Incorrect parameter calculation
- Not checking special cases

3D Geometry

🎯 Key Formulas:
- Distance between points: √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
- Direction cosines: l = a/√(a²+b²+c²)
- Equation of line: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
- Equation of plane: ax + by + cz + d = 0

💡 Important Points:
- Direction ratios and cosines
- Angle between lines
- Angle between line and plane
- Distance from point to plane

📌 Important Concepts:
- Skew lines
- Shortest distance between lines
- Family of planes
- Intersection of line and plane

⚠️ Common Mistakes:
- Wrong distance formula
- Incorrect direction cosines
- Not considering 3D geometry

📐 Trigonometry

Trigonometric Identities

🎯 Basic Identities:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = cosec²θ
- tan θ = sin θ/cos θ

💡 Important Points:
- Reciprocal identities
- Co-function identities
- Even-odd identities
- Periodicity

📌 Compound Angles:
- 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)

⚠️ Common Mistakes:
- Wrong identity application
- Sign errors
- Domain restrictions

Multiple Angles

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

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

💡 Important Points:
- Convert to single angle
- Use identities appropriately
- Check domain restrictions

⚠️ Common Mistakes:
- Wrong formula selection
- Incorrect simplification
- Not checking validity

Inverse Trigonometry

🎯 Key Functions:
- 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)

💡 Important Points:
- Principal values only
- Domain restrictions
- Relationship: sin⁻¹x + cos⁻¹x = π/2
- Properties of inverse functions

📌 Important Identities:
- sin⁻¹(-x) = -sin⁻¹x
- cos⁻¹(-x) = π - cos⁻¹x
- tan⁻¹(-x) = -tan⁻¹x

⚠️ Common Mistakes:
- Wrong domain/range
- Incorrect principal values
- Not using relationships

Properties of Triangles

🎯 Key Formulas:
- Sine rule: a/sin A = b/sin B = c/sin C = 2R
- Cosine rule: a² = b² + c² - 2bc cos A
- Area: Δ = (1/2)ab sin C = √[s(s-a)(s-b)(s-c)]
- s = (a + b + c)/2 (semi-perimeter)

💡 Important Points:
- Apply to any triangle
- Use appropriate formula based on given data
- Check triangle inequality
- Calculate angles correctly

📌 Special Cases:
- Right triangle: Pythagoras theorem
- Equilateral triangle: all angles 60°
- Isosceles triangle: base angles equal

⚠️ Common Mistakes:
- Wrong rule selection
- Incorrect angle calculation
- Not checking triangle validity

📊 Statistics and Probability

Statistics

🎯 Key Formulas:
- Mean: x̄ = (Σx)/n
- Median: Middle value (or average of middle two)
- Mode: Most frequent value
- Variance: σ² = Σ(x - x̄)²/n
- Standard deviation: σ = √σ²

💡 Important Points:
- Measures of central tendency
- Measures of dispersion
- Skewness and kurtosis
- Correlation coefficient

📌 Important Concepts:
- Range: Maximum - Minimum
- Quartiles: Q₁, Q₂, Q₃
- Percentiles
- Coefficient of variation

⚠️ Common Mistakes:
- Wrong formula selection
- Incorrect data ordering
- Not using correct denominator

Probability

🎯 Key Concepts:
- P(A) = n(A)/n(S)
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- P(A|B) = P(A ∩ B)/P(B)
- Independent events: P(A ∩ B) = P(A) × P(B)

💡 Important Points:
- Sample space
- Mutually exclusive events
- Exhaustive events
- Bayes' theorem

📌 Important Distributions:
- Binomial: P(X = k) = nCk p^k (1-p)^(n-k)
- Normal distribution
- Poisson distribution
- Uniform distribution

⚠️ Common Mistakes:
- Wrong probability calculation
- Not identifying independent events
- Incorrect conditional probability

🧮 Mathematical Reasoning

Logical Statements

🎯 Key Concepts:
- Statement: Declarative sentence with truth value
- Compound statement: Using logical connectives
- Truth table: Shows truth values for all combinations
- Tautology: Always true, Contradiction: Always false

💡 Important Points:
- Logical connectives: AND (∧), OR (∨), NOT (¬), IMPLIES (→)
- Equivalent statements: Same truth table
- Contrapositive: If not Q then not P
- Converse: If Q then P

📌 Important Laws:
- De Morgan: ¬(P ∧ Q) ≡ ¬P ∨ ¬Q
- Distributive: P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
- Associative: (P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R)

⚠️ Common Mistakes:
- Wrong truth table
- Incorrect logical equivalence
- Not using proper notation

🎯 Problem-Solving Strategies

General Approach

1. Understand the problem
2. Identify given information
3. Choose appropriate method
4. Solve step-by-step
5. Check the answer

Subject-Specific Tips

Algebra:
- Check domain restrictions
- Verify solutions
- Consider special cases

Calculus:
- Check continuity/differentiability
- Verify using graphs
- Consider limits

Geometry:
- Draw diagrams
- Label all points
- Use coordinate geometry

Trigonometry:
- Check angle ranges
- Use appropriate identities
- Consider periodicity

Common Mistakes to Avoid

- Unit errors
- Sign mistakes
- Formula misapplication
- Calculation errors
- Not checking answers

📊 Final Exam Checklist

Before Exam:

• Review all formulas
• Practice key problem types
• Check calculator functions
• Prepare formula sheet
• Get adequate rest

During Exam:

• Read all questions carefully
• Manage time effectively
• Show all work
• Check answers
• Don’t panic on difficult questions

Use these quick notes for rapid revision and last-minute preparation! Regular practice with these concepts will significantly improve your problem-solving speed and accuracy. 🎯