iGCSE Further Pure Mathematics Edexcel

This subject is broken down into 54 topics in 10 modules:

  1. Core 10 topics
  2. Logarithmic Functions and Indices 4 topics
  3. The Quadratic Function 3 topics
  4. Identities and Inequalities 5 topics
  5. Graphs 2 topics
  6. Series 3 topics
  7. Scalar and Vector Quantities 7 topics
  8. Rectangular Cartesian Coordinates 5 topics
  9. Calculus 7 topics
  10. Trigonometry 8 topics
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This page was last modified on 28 September 2024.

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Further Pure Mathematics

Core

Logarithmic functions

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Logarithmic functions

Logarithmic functions

Basic Definition and Properties

  • A logarithm is the power to which a certain number, called the base, must be raised to obtain a given number.
  • The expression is written as log_b(a) = n where b is the base, a is the number and n is the power.
  • Logarithmic functions are the inverses of exponential functions.

Basic Rules of Logarithms

  • Product rule: The log of a product is the sum of the logs of its factors, i.e., log_b(a * c) = log_b(a) + log_b(c).
  • Quotient rule: The log of a quotient is the difference between the logs of the numerator and the denominator, i.e., log_b(a / c) = log_b(a) - log_b(c).
  • Power rule: The log of an exponent is the exponent times the log of the base, i.e., log_b(a^n) = n * log_b(a).

Change of Base Formula

  • Any logarithm can be computed using any other base through the change of base formula, as long as both the bases are positive and not equal to 1.
  • The change of base formula is written as log_b(a) = log_c(a) / log_c(b) where a, b, c are positive numbers and b ≠ 1.

Natural Logarithm

  • The natural logarithm or ln is a logarithm in the base e, where e is an irrational and transcendental number approximately equal to 2.71828.
  • Natural logarithms have similar properties to those mentioned above for basic logarithms.

Solving Logarithmic Equations

  • To solve equations involving logarithms, use the rules of logarithms to simplify the equations.
  • If the logs in the equation have the same base, you can set the expressions inside the logs equal to each other and solve the resulting equation.
  • In some cases, it may be helpful to rewrite the logarithmic equation as an exponential equation.

Relationship with Exponential Functions

  • The set of all exponential functions is the inverse of the set of all logarithmic functions, and vice versa.
  • The graph of a logarithm function is a reflection of the graph of the corresponding exponential function over the line y = x.

Course material for Further Pure Mathematics, module Core, topic Logarithmic functions

Further Pure Mathematics

Scalar and Vector Quantities

Addition and Subtraction of Coplanar Vectors

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Addition and Subtraction of Coplanar Vectors

Addition of Coplanar Vectors

  • Coplanar vectors are vectors that exist on the same plane.
  • Vector addition can occur geometrically or algebraically.
  • For geometric addition, place the initial point of the second vector at the terminal point of the first vector. The resultant vector is from the initial point of the first vector to the terminal point of the second vector.
  • For algebraic addition, add corresponding components of the two vectors.
  • The principle of triangle law states that the sum of two vectors is the third side of the triangle they form when arranged head-to-tail.
  • The parallelogram law of addition states that the sum of two vectors is the diagonal of the parallelogram they form when placed tail-to-tail.

Subtraction of Coplanar Vectors

  • Vector subtraction can be seen geometrically as vector addition in reverse.
  • For geometric subtraction, imagine reversing the direction of the subtracted vector and then performing vector addition.
  • Algebraic subtraction simply subtracts corresponding components of two vectors.
  • The triangle method for subtraction involves using the head-to-tail method, but for the vector being subtracted, reverse its direction.
  • Remember, the resultant vector will always start from the position of the first vector and ends at the position of the second vector.

Properties of Vector Addition and Subtraction

  • Vector addition and subtraction obey the commutative law (changing the order does not change the result) and the associative law (grouping does not change the result).
  • Addition of a vector with its negation (same vector but in opposite direction) results in the zero vector.
  • Scalar multiplication affects the magnitude and direction of the vector but does not affect operations of addition and subtraction. It distributes over the operations (distributive law).

Important Concepts and Definitions

  • A scalar has magnitude only while a vector has both magnitude and direction.
  • Vectors can be equal if they have the same magnitude and direction, regardless of their initial and terminal points.
  • The magnitude of a vector is its length or size, denoted by |A|.
  • The direction of a vector is the way from its tail to head, usually given by an angle measure.
  • Vectors are denoted typically in bold or with an arrow above a letter.

Course material for Further Pure Mathematics, module Scalar and Vector Quantities, topic Addition and Subtraction of Coplanar Vectors

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