Year 12 Maths Formula Sheet

Your Ultimate Year 12 Maths Formula Sheet: A Comprehensive Guide

Navigating the complexities of Year 12 mathematics can feel overwhelming. This comprehensive guide serves as your ultimate companion, providing a detailed formula sheet covering key concepts and offering explanations to solidify your understanding. We'll delve into crucial areas, ensuring you're equipped with the tools to tackle any mathematical challenge. This resource will cover essential formulas for algebra, calculus, trigonometry, and more, making it a valuable study aid throughout your Year 12 journey.

Algebra: The Foundation of Mathematics

Algebra forms the bedrock of many mathematical concepts. Understanding these foundational formulas is crucial for success in Year 12 and beyond.

1. Quadratic Equations

• The Quadratic Formula: For a quadratic equation of the form ax² + bx + c = 0, the solutions are given by:

  x = (-b ± √(b² - 4ac)) / 2a

• Discriminant: The discriminant (b² - 4ac) determines the nature of the roots:

  • b² - 4ac > 0: Two distinct real roots
  • b² - 4ac = 0: One real root (repeated root)
  • b² - 4ac < 0: Two complex roots

• Completing the Square: Rewriting a quadratic equation in the form a(x + p)² + q = 0, where p = b/2a and q = c - b²/4a. This form is useful for finding the vertex and sketching the parabola.

• Factorization: Expressing a quadratic as a product of two linear factors. This is only possible if the discriminant is greater than or equal to zero.

2. Indices and Logarithms

• Laws of Indices:

  • aᵐ × aⁿ = aᵐ⁺ⁿ
  • aᵐ / aⁿ = aᵐ⁻ⁿ
  • (aᵐ)ⁿ = aᵐⁿ
  • a⁰ = 1 (a ≠ 0)
  • a⁻ⁿ = 1/aⁿ
  • a¹/ⁿ = ⁿ√a

• Laws of Logarithms: (Assuming base b)

  • log_b(xy) = log_bx + log_by
  • log_b(x/y) = log_bx - log_by
  • log_b(xⁿ) = n log_bx
  • log_bb = 1
  • log_b1 = 0
  • Change of Base: log_ax = log_bx / log_ba

3. Sequences and Series

• Arithmetic Sequence: aₙ = a₁ + (n-1)d, where aₙ is the nth term, a₁ is the first term, and d is the common difference.

• Arithmetic Series: Sₙ = n/2 [2a₁ + (n-1)d] or Sₙ = n/2 (a₁ + aₙ)

• Geometric Sequence: aₙ = a₁rⁿ⁻¹, where r is the common ratio.

• Geometric Series: Sₙ = a₁(1 - rⁿ) / (1 - r) (r ≠ 1) and S∞ = a₁ / (1 - r) ( |r| < 1)

4. Polynomials

• Remainder Theorem: When a polynomial P(x) is divided by (x - a), the remainder is P(a).

• Factor Theorem: (x - a) is a factor of P(x) if and only if P(a) = 0.

Calculus: The Study of Change

Calculus deals with rates of change and accumulation. Mastering these formulas is critical for success in Year 12 mathematics.

1. Differentiation

• Power Rule: d/dx (xⁿ) = nxⁿ⁻¹

• Product Rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

• Quotient Rule: d/dx [f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)] / [g(x)]²

• Chain Rule: d/dx [f(g(x))] = f'(g(x))g'(x)

• Derivatives of Trigonometric Functions:

  • d/dx (sin x) = cos x
  • d/dx (cos x) = -sin x
  • d/dx (tan x) = sec²x
  • d/dx (cosec x) = -cosec x cot x
  • d/dx (sec x) = sec x tan x
  • d/dx (cot x) = -cosec²x

• Derivatives of Exponential and Logarithmic Functions:

  • d/dx (eˣ) = eˣ
  • d/dx (ln x) = 1/x

2. Integration

• Power Rule: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (n ≠ -1)

• Integration of Trigonometric Functions:

  • ∫sin x dx = -cos x + C
  • ∫cos x dx = sin x + C
  • ∫sec²x dx = tan x + C
  • ∫cosec x cot x dx = -cosec x + C
  • ∫sec x tan x dx = sec x + C
  • ∫cosec²x dx = -cot x + C

• Integration of Exponential and Logarithmic Functions:

  • ∫eˣ dx = eˣ + C
  • ∫(1/x) dx = ln|x| + C

3. Applications of Calculus

• Finding Stationary Points: Set the first derivative equal to zero and solve for x. Use the second derivative test to determine if they are maxima, minima, or points of inflection.

• Optimization Problems: Use calculus to find maximum or minimum values in real-world scenarios.

• Kinematics: Applying differentiation and integration to solve problems involving velocity, acceleration, and displacement.

Trigonometry: The Study of Triangles

Trigonometry plays a significant role in Year 12 mathematics, particularly in calculus and geometry.

1. Basic Trigonometric Identities

• Pythagorean Identities:

  • sin²θ + cos²θ = 1
  • 1 + tan²θ = sec²θ
  • 1 + cot²θ = cosec²θ

• Reciprocal Identities:

  • sec θ = 1/cos θ
  • cosec θ = 1/sin θ
  • cot θ = 1/tan θ

• Quotient Identities:

  • tan θ = sin θ / cos θ
  • cot θ = cos θ / sin θ

2. Trigonometric Equations

Solving trigonometric equations often involves using identities to simplify the expression and then finding solutions within a given range.

3. Trigonometric Graphs

Understanding the graphs of sine, cosine, and tangent functions, including their amplitude, period, and phase shift, is crucial for solving problems involving periodic phenomena.

Probability and Statistics: Understanding Data

Probability and statistics help us to understand and interpret data.

1. Probability

• Basic Probability: P(A) = (Number of favorable outcomes) / (Total number of outcomes)

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

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

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

2. Statistics

• Mean: The average of a set of data.

• Median: The middle value in a sorted set of data.

• Mode: The most frequent value in a set of data.

• Standard Deviation: A measure of the spread or dispersion of data.

• Variance: The square of the standard deviation.

Vectors: Magnitude and Direction

Vectors possess both magnitude and direction. Understanding vector operations is important in various mathematical applications.

1. Vector Operations

• Addition: Adding vectors graphically (head-to-tail method) or component-wise.

• Subtraction: Subtracting vectors involves adding the negative of a vector.

• Scalar Multiplication: Multiplying a vector by a scalar changes its magnitude but not its direction.

• Dot Product: The dot product of two vectors results in a scalar value. It's useful for finding the angle between two vectors. a . b = |a||b|cosθ

• Cross Product: The cross product of two vectors results in a vector perpendicular to both original vectors. It's useful for finding areas and volumes.

Conclusion: Mastering Year 12 Mathematics

This comprehensive formula sheet provides a solid foundation for tackling Year 12 mathematics. Remember, understanding the underlying concepts is just as important as memorizing the formulas. Practice regularly, work through diverse problems, and don't hesitate to seek help when needed. With dedication and consistent effort, you can confidently conquer the challenges of Year 12 maths and achieve your academic goals. This guide serves as a valuable resource; use it wisely, and good luck!