Name:………………………………………………
AS Further Maths
Rising to the challenge
You will not be able to take AS Further Maths at Havant Sixth Form College
unless you have shown evidence that you have attempted all the problems in this
booklet. The problems are quite challenging. Try to solve them independently.
If there are any problems please e mail: john.rogers@havant.ac.uk
How
to Succeed in A level Further Maths
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All of the advice contained in the “How to succeed in AS Maths”
booklet applies equally with AS Further Maths. Not all of
Further Maths is harder than AS Maths but the feedback we
get from students is that the further pure module (FP1) poses a
particular challenge. However, if you are able to achieve a high
grade in AS maths it is high likely you could also do well in AS
Further Maths- a lot is to do with attitude and mind set. First
and for most you need to enjoy a challenge.
Contents
Section A- Problem Solving
The aim of this section is to get you interested in solving
unusual problems. Although the questions may feel quite strange
initially stick with it- only use the “hints” section at the end if
you feel after a reasonable period of reflection you still have no
idea on how to proceed.
Section B- Proof in mathematics
This section contains examples of some proofs. Proof is an
important part of mathematics but in practice we find that
students are often quite daunted with the idea. The aim of
these questions is to take away some of the mystery that
surrounds the concept of proof. There are 3 proofs for you to
try
Section A- Problem Solving
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Section B -Proof in mathematics
Proof lies at the heart of mathematics. However at A-level maths it is not taught as a
discipline in itself but it just seems to crop up everywhere – you will often be asked
to “prove that” or “show that” something is true (these two basically amount to the
same thing). You may have seen questions on proof at GCSE and of course the
principle is the same at “A” level- it is just that the level of algebra required will be
higher.
In further maths we do learn a specific strategy for proof called “proof by
induction”. Otherwise if the word proof is used then the question is likely to involve
algebra.
Example 1:
Prove that if an even number is squared the resulting number is also even,
Solution
We can see quickly by looking at examples that this is true 22 = 4, 42 = 16, 62 = 36 etc.
However no matter how many examples we write down we have not proved it is true.
We need to use algebra to prove that this is generally true=
If N is even then we can write N=2m where m is an integer
Now
this is clearly divisible by 4 since m2 is an integer.
Since
is divisible by 4 it must also be divisible by 2
Example 2:
A bag contains n ball in total 6 of which are oranges. The probability that two balls
are selected form the bag without replacement are both orange is equal to 1/3. Show
that
Hence show that there are 10 orange balls.
(This question is adapted from a question from Edexcel GCSE 2015- it caused some
controversy as it was perceived to be too hard for GCSE)
Solution
The probability that the first ball selected is orange =
Given that the first ball selected is orange the probability that the second ball
selected is also orange =
The probability that both are orange is therefore =
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This equals to 1/3
As required.
This gives n =-9 or n=10. Since we cannot have a negative number of balls this means
n=10.
Exercise
Proof 1
Prove that if an odd number is squared the resulting number is also odd,
Space for working and solution- or comment on progress.
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Proof 2
Rudolph the magician adopts a bit of maths for one of his tricks. He invites a member
of the audience to think of a number he then asks them to carry out the following
instructions:
Rudolph then predicts that their answer will be 5 more than their original number
Eg 7 – 14 – 19 – 361 – 336 – 84 –12
Rudolph’s trick is not magic. Prove that it is always true.
Space for working and solution- or comment on progress.
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Proof 3
Prove that the product of 3 consecutive numbers must necessarily be a multiple of 6.
Space for working and solution- or comment on progress.
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