IAS EDEXCEL MATH P2 NOTES

IAS EDEXCEL P2 NOTES

by Gabriel Bengo

1 Proof

• A proof is a logical argument that shows that a mathematical statement is always true.

The Structure of a Proof:

1. List already proven assumptions that will be used in your answer

2. Write down each step of the proof in a logical, chronological manner

3. Make sure all possibilities are covered

4. End the proof with a brief statement

Proof by Exhaustion

• A better method to use when a small output of results is required

• It involves trying all possible results in a given data range

→ You must use multiple examples over the given range to definitively prove a statement.

Disproof by Counter

• This method is used to disprove another statement made through the use of a singular example.

• This shows that the statement made is not true for every possible case and is, therefore, incorrect

→ In this case, only one example is sufficient to disprove a statement

​QUESTION WALKTHROUG https://www.youtube.com/watch?v=AeGc2JeXhpg&list=PLquyU_6YLv6clQ0MER2cH5ywcqmGwXcES&index=6

Proof by Deduction

What is proof by deduction?

Proof by deduction is when a mathematical and logical argument is used to show whether or not a result is true

Sets of numbers

• ℕ - the set of natural numbers

• ℤ - the set of integers

• ℚ - the set of quotients/rational numbers

• ℝ - the set of real numbers

2 Algebra & Functions

Polynomial Division

​Question Walkthrough https://www.youtube.com/watch?v=DxMKqbTjr5c&list=PLquyU_6YLv6dRpHCg13r7p62S_gIJEFY5&index=1

Factor Theorem

What do I need to know about the factor theorem?

• For a polynomial f(x) the factor theorem states that:

  • If f(p) = 0, then (x - p) is a factor of f(x)

  AND

• If (x - p) is a factor of f(x), then f(p) = 0

​ https://www.youtube.com/watch?v=bIVSqfx7aQ4&list=PLquyU_6YLv6dRpHCg13r7p62S_gIJEFY5&index=13

Remainder Theorem

• The factor theorem is actually a special case of the more general remainder theorem

• The remainder theorem states that when the polynomial f(x) is divided by (x - a) the remainder is f(a)

  • You may see this written formally as f(x) = (x - a)Q(x) + f(a)

  • In polynomial division

    • Q(x) would be the result (at the top) of the division (the quotient)

    • f(a) would be the remainder (at the bottom)

    • (x - a) is called the divisor

  • In the case when f(a) = 0, f(x) = (x - a)Q(x) and hence (x - a) is a factor of  f(x)– the factor theorem!

Remainder Theorem

How do I solve problems involving the remainder theorem?

• If it is the remainder that is of particular interest, the remainder theorem saves the need to carry out polynomial division in full

  • e.g. The remainder from  (x^2 - 2X) /(x-3) is 3^2 - 2(3) = 3

  • This is because if f(x) = x²- 2x and a = 3

• If the remainder from a polynomial division is known, the remainder theorem can be used to find unknown coefficients in polynomials

  • g. The remainder from is 8 so the value of p can be found by solving , leading to 

  • In harder problems there may be more than one unknown in which case simultaneous equations would need setting up and solving

• The more general version of remainder theorem is if f(x) is divided by (ax - b) then the remainder is 

  • The shortcut is still to evaluate the polynomial at the value of x that makes the divisor (ax - b) zero but it is not necessarily an integer 

Polynomial Factorisation

What is polynomial factorisation?

• Factorising a polynomial combines the factor theorem with the method of polynomial division

• The goal is to break down a polynomial as far as possible into a product of linear factors

​ https://www.youtube.com/watch?v=CEpaYxqPsdE&list=PLquyU_6YLv6dRpHCg13r7p62S_gIJEFY5&index=10

3 Coordinate Geometry

Equation of a Circle

Finding the Centre & Radius

How can I find the centre and radius from any form of the equation of a circle?

• A circle equation in a different form can always be rearranged into (x- a)² + (y - b)² = r² form in order to find the centre and radius

• This will often involve completing the square

Bisection of Chords

How can I find the equation of a perpendicular bisector?

• The perpendicular bisector of a line segment:

  • is perpendicular to the line segment

  • goes through the midpoint of the line segment

Angle in a Semicircle

The angle in a semicircle property says that If a triangle is right-angled, then its hypotenuse is a diameter of its circumcircle

• It also says that any angle at the circumference in a semicircle is a right

  angle

How can I use the angle in a semicircle property to find the equation of a circle?

• Because the hypotenuse of a right-angled triangle is a diameter of the triangle's circumcircle you also know that:

  1. the radius of the circumcircle is half the length of the hypotenuse

  2. the centre of the circumcircle is the midpoint of the hypotenuse

Radius & Tangent

What is the relationship between tangents and radii?

• A tangent is a line that touches a circle at a single point but doesn't cut across the circle

• A tangent to a circle is perpendicular to the radius of the circle at the point of intersection

​WMA12/01 IAL (Edexcel) P2 January 2022 Q6(a), (b) Equations of Circles (youtube.com)



WMA12/01 IAL (Edexcel) P2 January 2022 Q6(c) Circles, Equations of Tangents (youtube.com)

4 Sequences & Series

Binomial Expansion

​WMA12/01 IAL (Edexcel) P2 January 2022 Q3 Binomial Expansion (youtube.com)

Arithmetic Sequences

Arithmetic Series

How do I find the sum of an arithmetic series?

An arithmetic series is the sum of the terms of an arithmetic sequence

The following formulae will let you find the sum of the first n terms of an arithmetic series:

• a is the first term

• d is the common difference

• l is the last term

   

Arithmetic Series

How do I derive the arithmetic series formula?

• Learn this proof of the arithmetic series formula – you can be asked to give it on the exam:

  • Write the terms out once in order

  • Write the terms out again in reverse order

  • Add the two sums together

    • The terms will pair up to give the same sum 

    • There will be n of these terms

  • Divide by two as two of the sums have been added together

Geometric Sequences

What do I need to know about geometric sequences?

• In a geometric sequence, there is a common ratio between consecutive terms in the sequence

• You need to know the n^th term formula for a geometric sequence

Geometric Series

How do I find the sum of a geometric series?

• A geometric series is the sum of the terms of a geometric sequence

• The following formulae will let you find the sum of the first n terms of a geometric series:

Geometric Series

How do I prove the geometric series formula?

• Learn this proof of the geometric series formula – you can be asked to give it in the exam:

  • Write out the sum once

  • Write out the sum again but multiply each term by r

  • Subtract the second sum from the first

    • All the terms except two should cancel out

  • Factorise and rearrange to make S the subject

Sum to Infinity

• If (and only if!) |r| < 1, then the geometric series converges to a finite value given by the formula

• S_∞ is known as the sum to infinity

• If |r| ≥ 1 the geometric series is divergent and the sum to infinity does not exist