A Level Maths : Common Mistakes/misconceptions Watch

Simrankpy
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Give a suitable domain on which y=x^2(2x-3) does have an inverse function?. Can someone help with this please it's only 1 mark but I'm unable to understand it. I know that 1 to many function does not have an inverse function, and that one to one & many to 1 are functions. So I'm thinking that the above equation is a 1 to many but it also has a set of x values for which it is 1 to 1 function? But what does that actually mean???
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Sir Cumference
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A common mistake in mechanics is to include forces in your calculations that aren't relevant. E.g. you have this situation where a rod AB is freely hinged to the wall at A and supported by the strut CD:



If you are resolving forces or taking moments about the rod then you should not include forces not acting directly on the rod e.g. the reaction of the wall on the strut at C. These forces do indirectly affect the rod (as do trillions of other forces in the universe) but they are all accounted for in the force that acts on the rod by the strut at D.

It is common for students to label every force they can see on their diagram and use all of them in their calculations but you need to be more careful than that. If you're unsure then it can be best to draw separate force diagrams for every object. So in this case you would draw a force diagram just for the rod and then one for the strut and another for the wall if necessary.
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Sir Cumference
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(Original post by Simrankpy)
Give a suitable domain on which y=x^2(2x-3) does have an inverse function?. Can someone help with this please it's only 1 mark but I'm unable to understand it. I know that 1 to many function does not have an inverse function, and that one to one & many to 1 are functions. So I'm thinking that the above equation is a 1 to many but it also has a set of x values for which it is 1 to 1 function? But what does that actually mean???
Please start a separate thread.
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Sir Cumference
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This one may be more opinion based than factual but I think that students should know all the trig identities off-by-heart without having to look in the formula book and this should automatically happen from doing lots of trig questions. The key reason being that knowing how to use the identities is important but recognising where to use them is also important. E.g. if a student was faced with this integral:

\displaystyle \int \frac{1}{\sin x \cos x} \ dx

The students who would be the most successful and also the quickest would probably be those who recognise the \sin x \cos x pattern from a trig identity and are then able to turn this into

\displaystyle \int 2 \ \mathrm{cosec} \ 2x \ dx

From there the formula formula book can be used (I know there are other methods). If you've had lots of practice using these identities then you're more likely to recognise these types of things.
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Pangol
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(Original post by Sir Cumference)
This one may be more opinion based than factual but I think that students should know all the trig identities off-by-heart without having to look in the formula book and this should automatically happen from doing lots of trig questions.
This is again great advice. On a related note, there are trig identities that are not in the formula book, that can be derived from sin^2 x + cos^2 x = 1, by either dividing by sin^2 x (to get 1 + cot^2 x = cosec^2 x) or by cos^2 x (to get tan^2 x + 1 = sec^2 x). My advice here would be the opposite of the above - don't remember them, derive them every time. It only takes 30 seconds or so, and I find that I can never reliably remember which one has cot, sec, tan or cosec in, and which side the +1 is on. Better to start from scratch than risk remembering them and getting them wrong.

The same applies to the various forms of cos 2x. It is worth remembering that cos 2x = cos^ x - sin^2 x, but then I would derive the other forms by replacing the sin^2 x by 1 - cos^ 2 (to get 2 cos^2 x -1) or by replacing the cos^2 x by 1 - sin^2 x (to get 1 - 2 sin^2 x), rather than half-remembering which one starts 1 - ... and which one ends ... - 1.
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Sir Cumference
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It's common for students to forget (or not be told) that the geometric proofs of the trig addition formulae (sin(A ± B) etc.) are part of the spec so they should be learnt and understood fully.
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Sir Cumference
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Another commonly missed part of the spec is, "Understand and use integration as the limit of a sum". For Edexcel you need to recognise and understand this:

\displaystyle \int_a^b f(x) \ dx =  \lim_{\delta x \rightarrow 0} \sum_{x=a}^b f(x) \delta x

For other specs the formula may be a bit different but the concept will be the same.
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Pangol
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(Original post by Sir Cumference)
It's common for students to forget (or not be told) that the geometric proofs of the trig addition formulae (sin(A ± B) etc.) are part of the spec so they should be learnt and understood fully.
Another great one! Which reminds me:

Although students are allowed to quote and use the formula for the sum of the first n terms of an arithmetic or geometric series from the formula book, they may be asked to prove either of them from scratch.

(I don't think either of these has come up yet, and I suspect that if they did there would be some hint to get started, but you never know, there might be a requirement to prove them with no help at all.)
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3pointonefour
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(Original post by Sir Cumference)
Another commonly missed part of the spec is, "Understand and use integration as the limit of a sum". For Edexcel you need to recognise and understand this:

\displaystyle \int_a^b f(x) \ dx =  \lim_{\delta x \rightarrow 0} \sum_{x=a}^b f(x) \delta x

For other specs the formula may be a bit different but the concept will be the same.
A question on this actually came up in the 2019 exams and I was genuinely left wondering whether that was in the spec or not (since I only answered it cos of further reading). Like everyone else in the cohort flopped it because it's either not in the Edexcel book or it's not at all taught. It's a shame really since that question was an easy 4 marks.
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the bear
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(Original post by Sir Cumference)
This one may be more opinion based than factual but I think that students should know all the trig identities off-by-heart without having to look in the formula book and this should automatically happen from doing lots of trig questions. The key reason being that knowing how to use the identities is important but recognising where to use them is also important. E.g. if a student was faced with this integral:

\displaystyle \int \frac{1}{\sin x \cos x} \ dx

The students who would be the most successful and also the quickest would probably be those who recognise the \sin x \cos x pattern from a trig identity and are then able to turn this into

\displaystyle \int 2 \ \mathrm{cosec} \ 2x \ dx

From there the formula formula book can be used (I know there are other methods). If you've had lots of practice using these identities then you're more likely to recognise these types of things.
it is interesting to use the substitution u = sinx instead
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Sir Cumference
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(Original post by 3pointonefour)
A question on this actually came up in the 2019 exams and I was genuinely left wondering whether that was in the spec or not (since I only answered it cos of further reading). Like everyone else in the cohort flopped it because it's either not in the Edexcel book or it's not at all taught. It's a shame really since that question was an easy 4 marks.
It's not explained properly in textbooks and for Edexcel I really don't like the limit equation they use. If they are going to use notation that contradicts what students are used to then they should properly explain it in their textbooks.
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Sidd1
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(Original post by Sir Cumference)
It's common for students to forget (or not be told) that the geometric proofs of the trig addition formulae (sin(A ± B) etc.) are part of the spec so they should be learnt and understood fully.
So does this mean we should be able to derive them and may possibly be asked to in the exam? I guess it really is hard knowing what they can test you on.... my teacher also told us to learn the proof for product and chain rule because these new exams are just so unpredictable!
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Sir Cumference
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(Original post by Sidd1)
So does this mean we should be able to derive them and may possibly be asked to in the exam? I guess it really is hard knowing what they can test you on.... my teacher also told us to learn the proof for product and chain rule because these new exams are just so unpredictable!
The spec says that you need to “understand” the proofs. I think it’s unlikely that you’ll have to prove the formulae from scratch but if you want to be fully prepared for the exam then I recommend learning/understanding them fully. They could e.g. ask you a question that uses similar techniques to those used in the proofs and someone who knows the proofs will have an advantage.
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Sidd1
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(Original post by Sir Cumference)
It's not explained properly in textbooks and for Edexcel I really don't like the limit equation they use. If they are going to use notation that contradicts what students are used to then they should properly explain it in their textbooks.
I feel like a lot of things are not explained properly/ unclear in the textbook that's why I rely more on the spec. Also, some teachers fail to even mention these things too!
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Gent2324
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\frac{a}{b} \not\equiv  \frac{a^2}{b^2}
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Sir Cumference
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Missing invalid solutions can sometimes happen e.g. I remember this question came up in an exam a while back:


Solve the equation

\displaystyle \frac{\sin \theta}{\cos \theta + 1} + \frac{\cos \theta}{\sin \theta + 1} = 1

for 0^o \leq \theta \leq 360^o


The majority of students ended up with these solutions : 0, 180, 360, 90, 270. What they failed to realise was that 180 and 270 were not solutions to the equation since they produce denominators equal to 0.

This has been mentioned before but it's really important to remember that if A implies B then this does not necessarily mean that B implies A. Students are so used to starting with an equation, performing some algebra steps and ending up with solutions - this is how you've all done algebra since year 7 and it's always worked. Most of the time this will be fine but there will be certain occasions like the above where some of the steps are invalid when reversed.
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nonnymanny
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Thank you to Sir Cumference for starting this thread

could u do one for AS?
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ThiagoBrigido
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(Original post by Sidd1)
I feel like a lot of things are not explained properly/ unclear in the textbook that's why I rely more on the spec. Also, some teachers fail to even mention these things too!
Completely agree with you! However the scale of wrong answers in the textbook used by many students as a last resource is a fact to be challenge. Somewhat it feels like a intent to provoke some confusion.
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Sir Cumference
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Always read the front of the exam carefully. If it says something like this:

"Answers should be given to three significant figures unless otherwise stated."

then you must follow this for all questions or risk losing marks. There have been a number of exam questions over the years which have caused students to lose silly marks by forgetting to round. As an example, if you look at the answer to Q4b here, 2043 would lose 1 mark because it hasn't been rounded to 3sf. These are model answers given by an experienced teacher so it's not just students who make these mistakes!
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Pangol
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(Original post by Sir Cumference)
Always read the front of the exam carefully. If it says something like this:

"Answers should be given to three significant figures unless otherwise stated."

then you must follow this for all questions or risk losing marks. There have been a number of exam questions over the years which have caused students to lose silly marks by forgetting to round. As an example, if you look at the answer to Q4b here, 2043 would lose 1 mark because it hasn't been rounded to 3sf. These are model answers given by an experienced teacher so it's not just students who make these mistakes!
So many of your tips remind me of related ones....

As well as this 3 s.f. advice, remember to do all intermediate working to at least 4 s.f. to avoid rounding errors. Having said that, there is no need for mush more than this (5 s.f. should be the very most you will need). I once had a student who insisted on writing all of the digits on her calculator, and I never managed to convince her that it was unnecessary and inappropriate.
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