9th Class Math Notes (New Book)

9th Class Math Notes are available for students with step-by-step instructions. These notes are made for proper understanding of Math concepts.

9th Class Math Notes (New Book)

Class 9 New Book Full Concepts

Chapter 1: Real Numbers

• Key Topics:
  • Rational Numbers: Expressing decimals as fractions (e.g., 0.4=4/9​).
  • Properties of Real Numbers: Commutative, Associative, Distributive, Additive Identity, Multiplicative Identity, etc.
  • Rationalizing Denominators: Simplifying expressions like 13/(4+√3).
  • Exponents and Roots: Simplifying expressions like (81/16)^3/4.
  • Word Problems: Solving problems involving consecutive integers, ages, and profit/loss.
• Important Formulas:
  • a(b+c)=ab+ac (Distributive Property).
  • x+1/x and x²+1/x²​ for x=3+√8​.

Chapter 2: Logarithms

• Key Topics:
  • Scientific Notation: Expressing numbers like 2000000 as 2×10⁶.
  • Logarithmic and Exponential Forms: Converting between log⁡_b(x)=y and by=x.
  • Laws of Logarithms: log⁡(ab)=log⁡a+log⁡b, log⁡(a/b)=log⁡a−log⁡b.
  • Solving Logarithmic Equations: Finding x in equations like log⁡_x64=3.
  • Applications: Using logarithms to solve real-world problems like population growth.
• Important Formulas:
  • log⁡_b(xy)=log⁡_b(x)+log⁡_b(y).
  • log⁡_b(x/y)=log⁡_b(x)−log⁡_b(y).

Chapter 3: Sets and Functions

• Key Topics:
  • Set Notation: Writing sets in roster and set-builder forms (e.g., {x∣x=2n,n∈N}).
  • Set Operations: Union (A∪B), Intersection (A∩B), Complement (A′).
  • Venn Diagrams: Visualizing set relationships.
  • Functions: Understanding domain, range, and function notation (e.g., f(x)=ax+b).
  • Applications: Solving problems involving sets (e.g., number of students playing sports).
• Important Formulas:
  • n(A∪B)=n(A)+n(B)−n(A∩B).
  • f(x)=ax+b for linear functions.

Chapter 4: Factorization and Algebraic Manipulation

• Key Topics:
  • Factorization: Factoring expressions like x²+7x+10=(x+5)(x+2).
  • HCF and LCM: Finding HCF and LCM of polynomials (e.g., HCF of 21x²y and 35xy² is 7xy).
  • Square Roots: Finding square roots of polynomials (e.g., x²−8x+16=±(x−4)).
  • Applications: Using factorization to solve real-world problems like profit maximization.
• Important Formulas:
  • a²−b²=(a−b)(a+b).
  • a³+b³=(a+b)(a²−ab+b²).

Chapter 5: Linear Equations and Inequalities

Key Topics:

1. Solving Linear Equations
   • Techniques for solving equations of the form ax+b=c.
   • Examples: 12x+30=−6, x³+6=−12.
2. Solving Inequalities
   • Representing solutions on a real line.
   • Examples: x−6≤−2, −9>−16+x.
3. Graphing Linear Inequalities
   • Shading solution regions for inequalities in the xyplane.
   • Examples: 2x+y≤6, 3x+7y≥21.
4. Linear Programming
   • Maximizing or minimizing functions subject to constraints.
   • Examples: Maximize f(x,y)=2x+5y subject to 2y−x≤8, x−y≤4.

Important Formulas:

1. Linear Equation Solution:ax+b=c  ⟹  x=(c−b)/a​
2. Inequality Representation:x≤a  ⟹ Shaded region to the left of a on the real line.
3. Graphing Linear Inequalities:
   • For ax+by≤c, shade the region below the line.
   • For ax+by≥c, shade the region above the line.
4. Linear Programming:
   • Objective function: f(x,y)=ax+by.
   • Constraints: g(x,y)≤c, h(x,y)≥d.

Chapter 6: Trigonometry

Key Topics:

1. Angles and Quadrants
   • Identifying the quadrant in which an angle lies.
   • Examples: 65^∘ (1st quadrant), 135^∘ (2nd quadrant).
2. Conversion of Angles
   • Converting decimal degrees to degrees, minutes, and seconds.
   • Examples: 123.456^∘ to 123^∘27′22′′.
3. Trigonometric Ratios
   • Definitions of sine, cosine, tangent, cosecant, secant, and cotangent.
   • Examples: sin⁡θ=perpendicular/hypotenuse​.
4. Trigonometric Identities
   • Fundamental identities and their proofs.
   • Examples: sin⁡²θ+cos⁡²θ=1, tan⁡θ=sin⁡θ/cos⁡θ​.
5. Right-Angled Triangles
   • Solving for unknown sides and angles using trigonometric ratios.
   • Examples: Finding x,y,z in right-angled triangles.
6. Applications of Trigonometry
   • Real-world problems involving angles and distances.
   • Examples: Calculating the height of a wall using a ladder.

Important Formulas:

1. Trigonometric Ratios: sin⁡θ=perpendicular/hypotenuse, cos⁡θ=base/hypotenuse, tan⁡θ=perpendicular/base
2. Pythagorean Identity: sin⁡²θ+cos⁡²θ=1
3. Angle Conversion: 1^∘=60′and 1′=60′′
4. Area of a Sector: Area=1/2r²θ (θ in radians)
5. Arc Length: Arc Length=rθ (θ in radians)
6. Pythagorean Theorem: a²+b²=c²(for right-angled triangles)
7. Trigonometric Values for Common Angles: sin⁡30^∘=12, cos⁡45^∘=1/√2, tan⁡60^∘=√3,