A Level Mathematics CCEA

This subject is broken down into 143 topics in 26 modules:

  1. Algebra and Functions (Pure Mathematics) 18 topics
  2. Partial Fractions (Pure Mathematics) 4 topics
  3. Exponentials and Logarithms (Pure Mathematics) 7 topics
  4. Differentiation (Pure Mathematics) 9 topics
  5. Polynomials (Pure Mathematics) 3 topics
  6. Integration (Pure Mathematics) 7 topics
  7. Parametric Equations (Pure Mathematics) 2 topics
  8. Trigonometry (Pure Mathematics) 13 topics
  9. Circle Geometry (Pure Mathematics) 4 topics
  10. Coordinate Geometry (Pure Mathematics) 4 topics
  11. Sequences and Series (Pure Mathematics) 5 topics
  12. Vectors (Pure Mathematics) 7 topics
  13. Quantities and Units in Mechanics (Applied Mathematics) 1 topics
  14. Kinematics (Applied Mathematics) 3 topics
  15. Forces and Newton's Laws (Applied Mathematics) 9 topics
  16. Impulse and Momentum (Applied Mathematics) 6 topics
  17. Moments (Applied Mathematics) 7 topics
  18. Probability (Applied Mathematics) 8 topics
  19. Sampling (Applied Mathematics) 3 topics
  20. Histograms (Applied Mathematics) 1 topics
  21. Statistical Measures (Applied Mathematics) 2 topics
  22. Correlation (Applied Mathematics) 3 topics
  23. Data Presentation and Interpretation (Applied Mathematics) 5 topics
  24. Binomial Distribution (Applied Mathematics) 2 topics
  25. Hypothesis Testing (Applied Mathematics) 7 topics
  26. Normal Distribution (Applied Mathematics) 3 topics
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  • 26
    modules
  • 143
    topics
  • 51,963
    words of revision content
  • 6+
    hours of audio lessons

This page was last modified on 28 September 2024.

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Mathematics

Algebra and Functions (Pure Mathematics)

Indices

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Indices

Basic Rules of Indices

  • Any number raised to the power of 1 is the number itself, e.g: a^1 = a
  • Any number raised to the power of 0 is 1, e.g: a^0 = 1
  • If two terms with the same base are multiplied together, the powers are added, e.g: a^m * a^n = a^(m+n)
  • If a term with a base of "a" raised to a power "m" is itself raised to a power "n", multiply the powers, e.g: (a^m)^n = a^(mn)
  • If two terms with the same base are divided, subtract the exponent of the denominator from the exponent of the numerator, e.g: a^m / a^n = a^(m-n)
  • If a term in the denominator has a negative exponent, it can be moved to the numerator and made positive, e.g: 1 / a^-n = a^n
  • The n-th root of a number "a" can be denoted by a ^ (1/n)

Laws of Indices Involving Fractions

  • a^-n = 1 / a^n, which means a reciprocal of a number can be written as a negative exponent.
  • The n-th root of a number can be expressed as a power with a fractional exponent, e.g: n√a = a^(1/n)
  • If the power of a term is a fraction, the denominator of the fraction is the root, and the numerator is the power. For example, a^(m/n) = ( n√a ) ^m

Simplification Using Indices

  • In order to simplify expressions with indices, use the laws of indices to combine terms.
  • Simplify the expression step by step, until no further simplification is possible.
  • When dealing with algebraic expressions, it's important to remember that these rules apply only to terms with the same base.

Indices and Surds

  • A Surd is an expression that includes a root (√). Surds are dealt with using the laws of indices.
  • The method of 'rationalising the denominator' is used to remove surds from the denominator of a fraction.
  • To simplify a surd, look for the largest square number which divides into the number under the surd, and use the rule √ab = √a * √b.

Exponential Equations

  • Exponential equations are those where the variable is in the exponent.
  • They can be solved using the principle that if a^m = a^n, then m = n.
  • In cases where this cannot be applied directly, logarithms may be used to solve the equation.

Course material for Mathematics, module Algebra and Functions (Pure Mathematics), topic Indices

Mathematics

Sequences and Series (Pure Mathematics)

Convergence, divergence and oscillation

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Convergence, divergence and oscillation

Understanding Convergence, Divergence, and Oscillation

Convergence

  • A sequence is said to converge if it tends towards a specific value as the terms increase.
  • This specific value is known as the limit of the sequence.
  • For a sequence (aₙ) to converge to a limit L, for any positive number ɛ, there exists a positive integer N such that for all natural numbers n > N, the absolute difference between aₙ and L is less than ɛ: |aₙ - L| < ɛ.
  • If a sequence has a limit, it is called a convergent sequence.

Divergence

  • If a sequence does not tend towards any value as the terms increase, it is said to diverge.
  • Contrary to convergent sequences, a divergent sequence does not have a specific limit.
  • Divergence can also refer to a sequence where the terms increase or decrease without bound.
  • A sequence that oscillates between two or more values is also considered to diverge.

Oscillation

  • Oscillation in a sequence refers to a situation where the sequence alternates between specified values.
  • A well-known oscillating sequence is (-1)ᵛ, which oscillates between -1 and 1.
  • A sequence that oscillates does not converge because it does not approach a specific value. However, its subsequence(s) may converge.

Tests for Convergence and Divergence

The Monotone Convergence Theorem

  • If a sequence is increasing and bounded above, or decreasing and bounded below, then it is convergent.
  • This is a key theorem to determine whether a sequence converges or diverges.

The Divergence Test

  • If the limit of a sequence (aₙ) as n approaches infinity is not equal to zero, then the series sum from n=1 to infinity of (aₙ) is divergent.

The Ratio Test

  • If a sequence has term aᵢ and |aᵢ₊₁/aᵢ| approaches L as i tends to infinity, and if L < 1, the series is absolutely convergent; if L > 1, the series is divergent; if L = 1, the test is inconclusive.

Recognizing Convergence or Divergence

  • It's possible for a sequence to look like it's converging or diverging but do the opposite.
  • Mathematical proof is required to definitively determine the convergence or divergence of a sequence.
  • In some cases, graphical representation of a sequence’s behaviour can assist in understanding whether it is converging, diverging, or oscillating.

Applying Convergence, Divergence, and Oscillation

  • Recognizing convergence, divergence, and oscillation in a sequence or series is an essential part of many mathematical and scientific fields, such as computer science, physics, and engineering.
  • The concepts are particularly useful in calculus and analysis, where they form the basis for defining and working with integrals and differential equations.

Course material for Mathematics, module Sequences and Series (Pure Mathematics), topic Convergence, divergence and oscillation

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