Roots of Polynomials

• (still to be completed)

Further complex numbers

• Complex conjugates
• Division of a complex number by a complex number
• Argand diagram
• Modulus and argument of a complex number
• Solving quadratic equations with complex roots (complex conjugate pairs)
• Solving cubic equations
• Solving quartic equations

Exam Questions

De Moivres' Theorem

• Modulus-argument form
• Exponential Form
• Euler's relation
• Multiplication rule for the mod and argument of two complex numbers
• Division rule for the mod and argument of two complex numbers
  • Mixed examples
• (still to be completed)

Proof by mathematical induction

Series

• Proof of the sum of the series ∑r
• Proof of the sum of the series ∑r²
• Proof of the sum of the series ∑r³
• Further examples
  • Example 1
  • Example 2

Divisibility and Multiple Tests

• Proof that an expression is divisible by a certain integer (power type)
• Proof that an expression is divisible by a certain integer (non-power type)

Recurrence Relations

• Proof of the general term from a recurrence relationship (example 1)
• Proof of the general term from a recurrence relationship (example 2)

Matrices

• Proof of matrix multiplication statements (example 1)
• Proof of matrix multiplication statements (example 2)

Exam Questions

Finite series

Revision of FP1

• ∑ notation (revision)
• Sum of the first n natural numbers ∑r and the results for ∑a and ∑(ar+b)
  • Harder examples using these results
• Sum of the squares of the first n natural numbers ∑r²
• Sum of the cubes of the first n natural numbers ∑r³
• Using known formulae to sum more complex series

Method of Differences

• Method of differences
• Example using factorials

Calculus of inverse trignometric functions

• (still to be completed)

Hyperbolics functions

• (still to be completed)

Arc length and area of surface of revolution about the x-axis

• (still to be completed)