Hsc Advanced Maths Formula Sheet

HSC Advanced Maths Formula Sheet: Your Ultimate Guide to Success

The HSC Advanced Mathematics course is renowned for its rigor and complexity. Successfully navigating this challenging curriculum requires a deep understanding of numerous formulas and theorems. This comprehensive guide provides a detailed formula sheet covering key areas within the HSC Advanced Mathematics syllabus, along with explanations and examples to solidify your understanding. Mastering these formulas is crucial for achieving a high mark in your HSC exams. This guide serves as a valuable resource, enabling you to approach problem-solving with confidence and precision.

I. Algebra & Functions

This section covers fundamental algebraic concepts and their application to functions. A strong grasp of these basics is essential for success in more advanced topics.

1. Quadratic Equations:

• Quadratic Formula: For the equation ax² + bx + c = 0, the solutions are given by: x = [-b ± √(b² - 4ac)] / 2a
• Discriminant: The discriminant (Δ) = b² - 4ac determines the nature of the roots:
  • Δ > 0: Two distinct real roots
  • Δ = 0: One real root (repeated)
  • Δ < 0: Two complex conjugate roots
• Sum and Product of Roots: For a quadratic equation ax² + bx + c = 0 with roots α and β:
  • α + β = -b/a
  • αβ = c/a

Example: Solve the quadratic equation 2x² - 5x + 2 = 0. Using the quadratic formula, x = [5 ± √(25 - 16)] / 4 = [5 ± 3] / 4. Therefore, x = 2 or x = 1/2.

2. 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.
• Polynomial Division: Use long division or synthetic division to divide polynomials.

3. Logarithms and Exponentials:

• Logarithm Definition: If y = aˣ, then x = logₐy (where a > 0, a ≠ 1).
• Change of Base: logₐb = logₓb / logₓa
• Exponential Laws:
  • aˣ * aʸ = aˣ⁺ʸ
  • aˣ / aʸ = aˣ⁻ʸ
  • (aˣ)ʸ = aˣʸ
• Logarithm Laws:
  • logₐ(xy) = logₐx + logₐy
  • logₐ(x/y) = logₐx - logₐy
  • logₐ(xⁿ) = n logₐx

4. Functions:

• Domain: The set of all possible input values (x-values).
• Range: The set of all possible output values (y-values).
• Inverse Function: If f(x) has an inverse, denoted f⁻¹(x), then f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
• Composite Function: (f ∘ g)(x) = f(g(x))

II. Calculus

Calculus forms a significant portion of the HSC Advanced Mathematics syllabus. A solid understanding of differentiation and integration is paramount.

1. Differentiation:

• Derivative Definition: f'(x) = lim (h→0) [(f(x + h) - f(x)) / h]
• 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)
• Trigonometric Derivatives:
  • d/dx (sin x) = cos x
  • d/dx (cos x) = -sin x
  • d/dx (tan x) = sec² x
• Exponential and Logarithmic Derivatives:
  • d/dx (eˣ) = eˣ
  • d/dx (ln x) = 1/x

2. Integration:

• Power Rule: ∫xⁿ dx = (xⁿ⁺¹) / (n+1) + C (where n ≠ -1)
• Integration by Substitution: A technique used to simplify integrals.
• Integration by Parts: ∫u dv = uv - ∫v du
• Definite Integral: ∫ᵇₐ f(x) dx represents the area under the curve y = f(x) from x = a to x = b.
• Fundamental Theorem of Calculus: Connects differentiation and integration.

3. Applications of Calculus:

• Finding Stationary Points: Setting the derivative equal to zero to find maximum and minimum points.
• Determining Concavity: Using the second derivative to determine whether a stationary point is a maximum, minimum, or point of inflection.
• Related Rates: Problems involving rates of change of related variables.
• Optimization Problems: Finding maximum or minimum values subject to constraints.

III. Coordinate Geometry

Coordinate geometry involves the application of algebraic techniques to geometric problems.

1. Straight Lines:

• Equation of a Line: y - y₁ = m(x - x₁) (point-slope form), where m is the slope and (x₁, y₁) is a point on the line.
• Slope: m = (y₂ - y₁) / (x₂ - x₁)
• Parallel Lines: Parallel lines have the same slope.
• Perpendicular Lines: Perpendicular lines have slopes that are negative reciprocals of each other (m₁m₂ = -1).
• Distance between two points: √[(x₂ - x₁)² + (y₂ - y₁)²]
• Midpoint of a line segment: [(x₁ + x₂) / 2, (y₁ + y₂) / 2]

2. Circles:

• Equation of a Circle: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

3. Conics:

• Parabola: A curve defined by a quadratic equation. The general form is y = ax² + bx + c or x = ay² + by + c.
• Ellipse: A curve defined by the equation (x²/a²) + (y²/b²) = 1.
• Hyperbola: A curve defined by the equation (x²/a²) - (y²/b²) = 1 or (y²/a²) - (x²/b²) = 1.

IV. Vectors

Vectors are quantities that have both magnitude and direction.

1. Vector Operations:

• Vector Addition: Add corresponding components.
• Vector Subtraction: Subtract corresponding components.
• Scalar Multiplication: Multiply each component by the scalar.
• Dot Product: a · b = |a||b|cosθ, where θ is the angle between the vectors.
• Cross Product: a x b = |a||b|sinθn, where n is a unit vector perpendicular to both a and b.

2. Vector Equations of Lines and Planes:

• Vector Equation of a Line: r = a + λb, where r is the position vector of a point on the line, a is the position vector of a point on the line, b is the direction vector of the line, and λ is a scalar parameter.
• Vector Equation of a Plane: r · n = a · n, where r is the position vector of a point on the plane, n is the normal vector to the plane, and a is the position vector of a known point on the plane.

V. Complex Numbers

Complex numbers extend the real number system to include imaginary numbers.

1. Complex Number Representation:

• z = a + bi, where a and b are real numbers and i = √(-1).
• a is the real part (Re(z)) and b is the imaginary part (Im(z)).

2. Complex Number Operations:

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
• Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
• Division: (a + bi) / (c + di) = [(a + bi)(c - di)] / (c² + d²)

3. Polar Form:

• z = r(cos θ + i sin θ) = rcisθ, where r = |z| is the modulus and θ is the argument.

4. De Moivre's Theorem:

• (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

VI. Further Calculus

This section delves into more advanced calculus concepts.

1. Implicit Differentiation:

• Used to differentiate equations that are not explicitly solved for y.

2. Parametric Equations:

• Equations that express variables in terms of a parameter.

3. Differential Equations:

• Equations involving derivatives.

VII. Probability and Statistics

A foundational understanding of probability and statistics is vital.

1. Probability:

• Probability of an Event: 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. Discrete Random Variables:

• Expected Value: E(X) = Σ [xP(X = x)]
• Variance: Var(X) = E(X²) - [E(X)]²

3. Continuous Random Variables:

• Probability Density Function: A function that describes the probability of a continuous random variable taking on a given value.
• Expected Value: E(X) = ∫xf(x)dx
• Variance: Var(X) = E(X²) - [E(X)]²

VIII. Further Topics

This area often includes topics like matrices and vectors in three dimensions, as well as more advanced probability distributions.

This comprehensive formula sheet provides a solid foundation for your HSC Advanced Mathematics studies. Remember that consistent practice and understanding the underlying concepts are key to success. Work through numerous examples and past papers to solidify your knowledge and build confidence. Good luck with your studies!