What is this?
Hello and welcome to a new series of blog posts! This new series focuses on going through some of the most challenging A-Level Maths questions that I could find. As I explain each question, I also recap and teach several maths concepts that are vital in answering the question. Note, however, that these questions - although very difficult - are not beyond the scope of the maths specification. In fact, these are past exam questions or questions that have been provided/endorsed by exam boards, such as AQA, Edexcel or OCR to name a few. I hope that this series of posts will enable you to become more confident when encountering such difficult questions in a mock, or even the actual, exam.
But, you may be thinking, "I am not aiming for an A*; why should I bother with really hard questions?". Well, here at OnlyPhysics, we don't believe in settling for a grade, or anything for that matter; but, let's say, you are solely aiming for a grade A.
In order to achieve an A grade, you don't have to answer all of the questions. So, knowing this, you may say; "Well if an A grade is like 60%, then surely I should only be revising easier questions and leave the harder questions out in the exam?". In fact, this is what a lot of students say and think. But the answer is a big, fat NO.
Let me explain my point in more detail while using the Edexcel grade boundaries (it doesn't matter what exam board you are - this is just the exam board that I studied). Have a look below at the 2019 Grade Boundaries for Edexcel A-Level Maths:
An A* is 72%, an A is 55%, and a B is about 45%. Notice how the gap between a B and an A is smaller than the gap between an A and an A*. It is easier to drop a grade than it is to go up a grade from an A. 72% for an A* is fairly low relative to other subjects. Many students tend to see such 'low' grade boundaries and their eyes light up with joy. However, I hate to break it to you but lower grade boundaries means that the exam questions are much more difficult. Also, you must consider that grade boundaries tend to go up, especially since the current A-Level specification is new and it has taken schools a few years to get used to it.
Each maths paper is 100 marks. In order to get an A grade, you have to obtain 55 marks on average. That means you can lose 45 marks and just about scrape an A. So why should you bother learning how to answer the most difficult of questions if you only need 55%? Well, it is because that 55% will quickly turn to a 79% and higher if you don't. Let me explain.
The majority of the so-called challenging questions that we will be looking at are actually very involved and so they carry a lot of marks. For example, one question we will be dealing with is worth 11 marks. If you have no clue how to answer it as you have never bothered doing so before, you will be forfeiting those 11 potential marks. Chances are there is going to be more than one difficult question - let's say 3 (if the examiners are feeling nice) in a single paper. In total they may be worth 25-30 marks. If you can't make any progress on these type of questions as you have never practiced on very challenging questions before, you will likely be leaving out 30 marks worth of questions every paper. Your paper is now essentially out of 70 marks, but you still have to get a minimum of 55 marks - your percentage has now risen to 79% for an A and say goodbye to that dream of an A* - eek.
So, believe me, practising on the hardest questions is very beneficial. They force you to apply your knowledge, switch between many different mathematical principles, and, most importantly, even if you can't solve the question, you will get a load of working out marks that will help you on your journey to a B, an A and beyond!
Enough rambling, we will now start off this series by looking at our very first hard exam question.
Question #1 - Binomial Expansion
Before we begin solving the question, I just want to add that this question tests your knowledge of binomial expansion, algebraic division, and partial fractions. If you haven't learnt one or more of the topics, you are going to find it very challenging to answer the question. I will do my best to briefly explain each topic as I explain my solution, but please make sure to have a look at each respective topic in more detail.
Part a:
Okay so part a seems to be like a standard question that you would find when doing exam questions on Binomial Expansion - a show that question. Therefore, we already know the answer, but we just need to show our working out and how we mathematically reached that answer.
First things first, we need to somehow simplify the expression. We can't perform any binomial expansion yet as there are two algebraic factors on the denominator. We need to simplify the expression in such a way as to get each fraction with only one algebraic factor on its denominator. You may be thinking "aha, I'll just use partial fractions", but you are only half right. You see, we can't split this up into two partial fractions like we normally do as the highest exponent in the numerator (x^2) is the same as the highest exponent in the denominator, if you were two expand the bottom two brackets.
Thus, there is a number that is 'outside' of the partial fractions that we need to calculate in order for us to be able to express the fraction into two partial fractions. I have included a mathematical explanation in an image below:
You see the A denotes that numerical 'remainder' that I was talking about. Now, we have to work out the value of A first before we can do anything else. We can't multiply both sides by (1-x)(2+x) and try to sub in values to find the values A, B and C as the highest power in both the numerator and denominator is the same. This will make more sense in a minute. You must, instead, use algebraic division in order to find out the value of A and then you will be able to express the rest as a partial fraction. Chances are you wouldn't have thought to do that in an exam as this skill is hardly ever tested in class but has come up in many exam questions as a sort of "curveball".
Let's now perform algebraic division in order to find out the value of A. If you have learnt about algebraic division before, then it should be pretty straightforward to follow. I have included an image of how to do this below.
The number 9 is the quotient and is therefore the value of A. It is the result of the algebraic division. The remainder is 2-x over the initial divisor (1+x)(2+x). So, what does this mean?
Now that we know the value of A and the remainder, we know that the remainder is equal to our previous two partial fractions. I have include an image below to explain this better:
Now, we can multiply both sides by (1+x)(2+x) and work out the values of B and C. Beforehand, if we didn't do any algebraic division, and multiplied both sides by (1+x)(2+x), we would have gotten completely different - and wrong - answers for the variables A, B, and C. You must do algebraic division when the highest power in the numerator is the same as the highest power in the denominator. Let us now work out the value of B and C. I have included my workings below - it should be easy to follow:
We have now successfully expressed the algebraic fraction into its respective partial fractions. All that work and we haven't even begun doing the actual binomial expansion. But, keep in mind, its 7 marks. Those 7 marks could prove critical. I know of people who missed out on an A or A* grade by 1 mark. Missing out a 7 marker is a big no no.
The binomial expansion should be pretty simple now that we have set up our partial fraction. Using the binomial expansion formula (which I have included below) we can expand the two partial fractions, up to the x^3 term.
If you learnt how to use the above formula, you should be able to perform the relevant binomial expansion. I have included an image below for you to check your working out:
Note above that in order for you to use the binomial expansion formula, the number inside the bracket must be 1. Note, therefore, that I have factored out a 2 from one of the relevant brackets. Since, however, the bracket is raised to the power of -1, the number 2 becomes 2^-1 or 1/2 when we take it out of the bracket.
Well, we have now worked out the answer to Part a. It took a long time, but, with practice, you can learn how to answer similar questions much more quickly and comfortably.
Part b:
This is the 'easy' part of the question. You need to, however, be able to answer Part a in order to achieve full marks for Part b. Firstly, we must sub in x=0.1 into the original fraction and write down the answer to as many decimal places as possible (or just save it on your calculator). Subsequently, sub in x=0.1 into the binomial expansion we just worked out. We can then work out our percentage error and give it to two significant figures as required in the question. Note that when we calculate the percentage error we always do (Original Value - New Value) / (Original Value) and multiply this decimal by 100%.
That is it! We have answered arguably the hardest A-Level maths question involving binomial expansion. However, remember that it is likely that a similar question will come up in any mock exam that you do, or even your real A-Level Exam. So make sure to learn how to answer questions like this one!
Thank you very much for reading this post! I hope that it has been helpful to you and has shown you why you shouldn't have the mentality of leaving out the 'hardest' questions. Oftentimes, the most difficult questions aren't really that difficult - they just require you to approach the question in a new way. In this case, it was doing algebraic division before you express the fraction into partial fractions and do the relevant binomial expansion. If you have any questions, post them down in the comments below. Stay tuned for Question #2!
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