Mathematics Lecture Notes

Cube Number

• Definition: A number raised to the power of 3.
• Example: 2^3 = 8.

Natural Number

• Definition: Any positive whole number.
• Example: 243.

Square Number

• Definition: A number raised to the power of 2.
• Example: 2^2 = 4.

Prime Number

• Definition: A number divisible only by 1 and itself.
• Example: 7.

Common Multiple

• Example: First common multiple of 4 and 17 is 68.
• Calculation: 4 * 4 * 2 * 5 = 160.

Upper Bound Calculation

• Concept: To calculate the upper bound, you divide the nearest measurement by two.
• Example: Nearest mm is 0.1 / 2 = 0.05.
• Application: Upper bound = 6.5 + 0.05 = 6.55.

Original Price Calculation

• Steps:
  • Reduced Price: 105.
  • Reduction Percent: 16%.
• Formula: 105 / (100 - 16) * 100 = 125.

Simple Interest & Compound Interest

• Formulas:
  • Simple Interest: I = PRT.
  • Compound Interest: A = P(1 + r)^t.
• Variables:
  • I = Interest earned.
  • P = Principal amount (borrowed/invested).
  • R = Rate of interest.
  • T = Time.
  • A = Accumulated amount.

Exchange Rates

• Example: $1 = 24.3 Japanese Yen. For $80, 80 * 24.3.

Speed, Distance, and Time

• Formula: Speed = Distance/Time.
• Example:
  • Speed = 18.
  • Time = 55 minutes = 55/60 hours.
  • Distance = Speed * Time = 16.5 km.

Algebraic Expressions

• Example Equations:
  • 5 * 3 + 5N - 3N - N^2 = N^2 + 2N + 15.
  • Solve for X: 11X - 3X = -7 - 5; X = -2.75.
  • Substitution Method: 2X - (X \ / 2) = 1; X = 2/3; Y = 1.
• Subject of a Formula: Square and rearrange to find the variable.

Variations and Proportions

• Example: Y is inversely proportional to X^2, Y = K/X^2, K = 32, substitute to find Y.

Laws of Indices

• Rules:
  • a^0 = 1.
  • a^-1 = 1/a.
  • a^m * a^n = a^(m + n).
  • a^m/ a^n = a^(m - n).
  • (a^m)^n = a^(mn).
  • a^(m/n) = n√(a^m).

Geometry & Angles

• Formulas for Polygons:
  • Sum of Interior Angles: (N - 2) * 180°.
  • Each Interior Angle of a Regular Polygon: ((N - 2) * 180°) / N.
  • Sum of Exterior Angles: 360°.

Circle Theorems

• Key Concepts: Review different theorems related to circles.

Pythagoras Theorem

• Formula: a^2 + b^2 = c^2.
• Example: Calculate missing side.

Symmetry

• Example: Rhombus has 2 lines of symmetry.

Similar Triangles

• Example: 15/5 = 16.5/EF; EF = 5.5 after cross-multiplying.

Trigonometry

• SOHCAHTOA: Sine, Cosine, Tangent ratios.
• Example: sin(52) = 8.6/ BC; BC = 8.6/sin(52).

Equation of a Line

• Formula: Y = MX + C.
• Concepts:
  • Parallel lines: Same gradient.
  • Perpendicular lines: Product of gradients = -1.

Sets

• Types:
  • Proper Subset: Elements are within another set.
  • Disjoint Sets: No common elements.
  • Intersection & Union.
  • Complement: Elements not in the given set.

Vectors

• Example: Vector subtraction and components.

Transformations

• Types:
  • Reflection: Mirror image.
  • Rotation: Degree-specific rotation.
  • Translation: Horizontal and vertical shifts.
  • Enlargement: Scale factor changes.

Probability

• Example: Total = 20; Red = 6/20; Blue = 9/20; White = 5/20.

Functions

• Method: Substitute and solve.

Angles of a Sector

• Calculation: Finding arc length, area, and perimeter.

Differentiation

• Example: Differentiate 6x^2 - 3.