FP2 Tutorials – AQA A Level Maths

Contents

Roots of Polynomial Equations
Roots of Polynomial Equations
Complex Numbers
Complex Numbers
Further Complex Numbers
De Moivres Theorem
Loci in the Complex Plane
Proof by Mathematical Induction
Sum of Series
Various Types of proofs
Series
Standard Summations
Method of Differences
Differentiation
Inverse Trigonometric Functions
Hyperbolic Functions
Hyperbolic Functions
Integration
Applications of Integration – Arcs
Applications of Integration – Surface area of revolution

Roots of Polynomial Equations

Roots of Polynomial Equations

  1. Relationships between the roots and coefficients of a quadratic equation
  2. Relationships between the roots and coefficients of a cubic equation
  3. Relationships between the roots and coefficients of a quartic equation

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Complex Numbers

Complex Numbers

  1. Complex conjugates
  2. Division of a complex number by a complex number
  3. Argand diagrams
  4. Modulus and argument of a complex number
  5. Solving quadratic equations with complex roots
  6. Solving cubic equations
  7. Solving quartic equations
  8. Exam Questions - Complex numbers
  9. Exam Questions - Finding roots
  10. Exponential Form (Euler's relation)
  11. Multiplication and division rules for mod and argument of two complex numbers
  12. Exam Questions - Further complex numbers

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Further Complex Numbers

  1. Exponential Form (Euler's relation)
  2. Multiplication and division rules for mod and argument of two complex numbers
  3. Exam Questions - Further complex numbers

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De Moivre’s Theorem

  1. De Moivre's theorem
  2. Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ
  3. Exam Questions - Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ
  4. Expressing sinnθ and cosnθ in terms of sin(kθ) and cos(kθ)
  5. nth roots of a complex number
  6. Exam Questions - nth roots of a complex number

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Loci in the Complex Plane

  1. The locus of a point moving in a circle
  2. The locus of a point moving along a perpendicular bisector
  3. The locus of a point moving along a half-line
  4. The locus of a point moving on the arc of a circle
  5. Using complex numbers to represent regions on an Argand diagram
  6. Exam Questions - Loci in the complex plane

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Proof by mathematical induction

Sum of Series

  1. Proof of the sum of the series ∑r
  2. Proof of the sum of the series ∑r²
  3. Proof of the sum of the series ∑r³
  4. Proof for other series
  5. Exam Questions - Sum of series

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Various Types of Proofs

  1. Divisibility and multiple test proofs
  2. Exam Questions - Divisibility and multiple tests
  3. Recurrence relationship proofs
  4. Exam Questions - Recurrence relations
  5. Matrix proofs

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Series

Standard Summations

  1. Sigma notation
  2. Sum of the squares of the first n natural numbers ∑r2
  3. Sum of the cubes of the first n natural numbers ∑r3
  4. Using known formulae to sum more complex series
  5. Exam Questions - Series

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Method of Differences

  1. Method of differences
  2. Exam Questions - Method of differences

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Differentiation

Inverse Trigonometric Functions

  1. Derivatives of sin-1(x), cos-1(x) and tan-1(x)

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Hyperbolic Functions

Hyperbolic Functions

  1. Definitions
  2. Graphs of sinh(x), cosh(x) and tanh(x)
  3. Graphs of sech(x), cosech(x) and coth(x)
  4. Solving equations using inverse and exponential functions
  5. Hyperbolic identities
  6. Osborn's rule
  7. Inverse hyperbolic functions and their graphs
  8. Expressing inverse hyperbolic functions as natural logarithms
  9. Solving hyperbolic equations using hyperbolic identities

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Integration

Applications of Integration – Arcs

  1. Length of an arc

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Applications of Integration – Surface area of revolution

  1. Area of surface of revolution about the x-axis

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