Exam Preparation Notes

Standard Form and Laws of Indices

• Standard Form: A way to represent very large or small numbers.

  • Written as ax \times 10^k where:

  • 1 \leq a < 10 (one non-zero digit before the decimal)

  • k \in \mathbb{Z} (k is an integer)

  • k > 0 for large numbers

  • k < 0 for small numbers

• Laws of Indices (Key rules to simplify exponents):

  1. (xy)^m = x^m \times y^m

  2. (\frac{x}{y})^m = \frac{x^m}{y^m}

  3. x^m \times x^n = x^{m+n}

  4. \frac{x^m}{x^n} = x^{m-n}

  5. (x^m)^n = x^{m \times n}

  6. x^1 = x

  7. x^0 = 1

  8. \frac{1}{x^m} = x^{-m}

  9. x^{\frac{1}{n}} = \sqrt[n]{x}

  10. x^{\frac{m}{n}} = \sqrt[n]{x^m}

  • Note: These rules are not in the formula booklet.

Partial Fractions

• Purpose: To simplify rational expressions into sums of fractions that can be integrated.

• Steps:

  1. Factor the denominator into linear factors.

  2. Split the fraction into a sum (using unknown constants A, B, C).

  3. Multiply through by the denominator to eliminate fractions.

  4. Substitute values into the identity and solve for the unknown constants.

  5. Write the expression as partial fractions.

Exponentials and Logarithmzs

• Logarithm Definition: The inverse of an exponent. If a^x = b, then \log_a b = x.

• Natural Logarithm: \ln x = \log_e x where e is approximately 2.718.

• Common Logarithm: \log x = \log_{10} x.

• Laws of Logarithms (found in the formula booklet):

  1. \loga (xy) = \loga x + \log_a y

  2. \loga (\frac{x}{y}) = \loga x - \log_a y

  3. \loga (x^m) = m \loga x

Sequences and Series

Arithmetic Sequences & Series

• Definition: A sequence with a constant difference between consecutive terms (common difference d).

• Formula for nth term: un = u1 + (n-1) d.

• Sum of n terms: Sn = \frac{n}{2} (u1 + u_n).

Geometric Sequences & Series

• Definition: A sequence with a constant ratio between consecutive terms (common ratio r).

• Formula for nth term: un = u1 r^{(n-1)}.

• Sum of n terms: Sn = u1 \frac{(1 - r^n)}{(1 - r)} (for r \neq 1).

• Sum to Infinity (if |r| < 1): S\infty = \frac{u1}{1 - r}.

Proof Techniques

• Proof by Deduction: Showing a result is true for all integers by showing it holds for one integer and the next.

• Proof by Contradiction: Assuming the negation of what you want to prove and showing this leads to a contradiction.

Function Properties

• Definition of Function: A set of operations that output a specific value for every input.

• Domain: Set of x-values that can be plugged into the function.

• Range: Set of y-values that can be produced by the function.

Types of Functions

• One-to-One Function: Each input gives a unique output; passes the horizontal line test.

• Many-to-One Function: Multiple inputs produce the same output.

• Onto Function: Every element in the co-domain has a pre-image in the domain.

• Polynomial Function: A function of the form f(x) = an x^n + … + a0 where n is a non-negative integer.

Linear Functions

• Forms:

  1. Gradient-Intercept Form: y = mx + c (where m is the slope and c is y-intercept).

  • Example: y = 2x - 1 → gradient = 2, y-intercept = -1.

  1. Standard Form: ax + by = c

  2. Point-Slope Form: y - y1 = m(x - x1): Useful for finding the equation given slope and a point.

Function Relationships

• Parallel Lines: Lines with the same slope, do not intersect.

• Perpendicular Lines: Two lines intersecting at a 90° angle.

• Intercepts: Points where the graph intersects axes (x or y).

Quadratic Functions and Graphs

• Quadratic Function: f(x) = ax^2 + bx + c.

• Roots: Points where function intersects the x-axis.

• Vertex: Max or min points of the quadratic.

• Discriminant (from the quadratic formula): Determines number of roots (
  = b^2 - 4ac),

  • If D > 0 → two distinct roots;

  • If D = 0 → one repeated root;

  • If D < 0 → no real roots.

Random Variables and Probability

• Discrete Random Variable: Has countable outcomes (e.g., dice rolls).

• Continuous Random Variable: Has infinitely many outcomes (e.g., height).

• Probability Rules:

  • Addition Rule and Multiplication Rule for independent and dependent events.

• Binomial Distribution: Models success in fixed number of trials.

• Normal Distribution: Bell-shaped curve characterized by mean and standard deviation.

Statistical Analysis

Key Terms

• Mean: Average value of a dataset.

• Median: Middle value when data points are ordered.

• Mode: Most frequently occurring value(s).

• Range: Difference between highest and lowest values.

• Standard Deviation (σ): Measure of variance from the mean.

This set of notes organizes the important concepts and formulas from the subject to aid in exam preparation. Focus on understanding and practicing these key ideas to effectively prepare for assessments.